Pedro Lima

1. Numerical solution of nonlinear third-kind Volterra integral equations using an iterative collocation method

K. Kherchouche, A. Bellour, P.M. Lima

Journal Paper

International Journal of Computer Mathematics, 2023

In this paper, we discuss the application of an iterative collocation method based on the use of Lagrange polynomials for the numerical solution of a class of nonlinear third-kind Volterra integral equations. The approximate solution is given by explicit formulas. The error analysis of the proposed numerical method is studied theoretically. Some numerical examples are given to confirm our theoretical results.

2. An efficient spectral method for solving third-kind Volterra integral equations with non-smooth solutions

Y. Talaei, P.M. Lima

Journal Paper

Computational and Applied Mathematics (2023)

3. Numerical Simulation of the Stochastic Neural Field Equation with Delay

Pedro Lima and Tiago Sequeira

Conference Proceeding

Proceedings of the Second International Conference , 8-10 February 2023, Universtiy of Mazandaran, Iran, pp. 488 -495

The Neural Field Equations (NFE) are used to model the synaptic interactions between neurons in a continuous neural network, called a neural field. This kind of integro-differential equations proved to be an useful tool to describe the spatiotemporal neural activity from a macroscopic point of view, allowing the study of a wide variety of neurobiological phenomena, such as the sensory stimuli processing.

The present article aims to study the effects of additive noise in one- and two-dimensional neuronal fields, while taking into account finite axonal velocity and external stimulus. A Galerkin-type method is presented, which applies Fast Fourier Transforms to optimise the computational effort required to solve these equations. The explicit Euler-Maruyama scheme is implemented to obtain the stochastic numerical solution.

A published numerical solver written in Julia is used to simulate the neural fields in study.

4. Accurate Ito-Taylor-Discretization-Based State Estimation in Stochastic Neural Field Equations with Infinte Signal Transmissiion Rate

Kulikova M.V., Lima P.M., Kulikov G.Y.

Conference Proceeding

Proceedings of ECC2022 (2022), p. 1436-1441

Abstract:

In this paper, we explore the Dynamic Neural Fields (DNFs) in the presence of model uncertainties, i.e. the stochastic DNF equations. The model describes a neural tissue affected by external stimulus where the interaction of neurons population is treated as a continuum. This yields a stochastic nonlinear integro-differential equation. The working memory mechanism modeled by the DNFs should allow the system to cope with missing sensors' information. The goal of this paper is to design an accurate reconstruction methodology of the pattern formation in a neural tissue from an incomplete data available from the sensors. We assume that the measurements are partially observed in both domains that are the time and the spatial domains. For that, we explain how the state-space representation can be adopted for the stochastic DNFs and, next, we derive the filtering method based on the Itô-Taylor expansion of order 1.5 within the Extended Kalman Filter. The numerical experiments are also provided.

5. Pattern Recognition Facilities of Extended Kallman Filtering in Stochastic Neural Fields

Kulikova M.V., Lima P.M., Kulikov G.Y.

Conference Proceeding

Proceedings of ECC2022 (2022), p.1061-1066

Abstract:

In mathematical neuroscience, a special interest is paid to a working memory mechanism in the neural tissue modeled by the Dynamic Neural Field (DNF) in the presence of model uncertainties. The working memory facility implies that the neurons' activity remains self-sustained after the external stimulus removal due to the recurrent interactions in the networks and allows the system to cope with missing sensors' information. In our previous works, we have developed two reconstruction methods of the neural membrane potential from incomplete data available from the sensors. The methods are derived within the Extended Kalman filtering approach by using the Euler-Maruyama method and the Ito-Taylor expansion of order 1.5. It was shown that the Ito-Taylor EKF -based restoration process is more accurate than the Euler-Maruyama-based alternative. It improves the membrane potential reconstruction quality in case of incomplete sensors information. The aim of this paper is to investigate their pattern recognition facilities, i.e. the quality of the pattern formation reconstruction in case of model uncertainties and incomplete information. The numerical experiments are provided for an example of the stochastic DNF with multiple active zones arisen in a neural tissue.

6. Numerical solution of the stochastic neural field equation with applications to working memory

Lima PM, Erlhagen W, Kulikova MV, Kulikov GY.

Journal Paper

Physica A: Statistical Mechanics and its Applications. 2022 Jun 15;596:127166.

Abstract
The main goal of the present work is to investigate the effect of noise in some neural fields, used to simulate working memory processes. The underlying mathematical model is a stochastic integro-differential equation. In order to approximate this equation we apply a numerical scheme which uses the Galerkin method for the space discretization. In this way we obtain a system of stochastic differential equations, which are then approximated in two different ways, using the Euler–Maruyama and the Itô–Taylor methods. We apply this numerical scheme to explain how a population of cortical neurons may encode in its firing pattern simultaneously the nature and time of sequential stimulus events. Numerical examples are presented and their results are discussed.

7. Numerical simulations of one-and two-dimensional stochastic neural field equations with delay

Sequeira TF, Lima PM

Journal Paper

Journal of Computational Neuroscience. 2022 May 27:1-3

Abstract

Neural Field Equations (NFE) are intended to model the synaptic interactions between neurons in a continuous neural network, called a neural field. This kind of integro-differential equations proved to be a useful tool to describe the spatiotemporal neuronal activity from a macroscopic point of view, allowing the study of a wide variety of neurobiological phenomena, such as the sensory stimuli processing. The present article aims to study the effects of additive noise in one- and two-dimensional neural fields, while taking into account finite axonal velocity and an external stimulus. A Galerkin-type method is presented, which applies Fast Fourier Transforms to optimise the computational effort required to solve these equations. The explicit Euler-Maruyama scheme is implemented to obtain the stochastic numerical solution. An open-source numerical solver written in Julia was developed to simulate the neural fields in study.

8. Sequential method for fast neural population activity reconstruction in the cortex from incomplete noisy measurements

Kulikova MV, Lima PM, Kulikov GY.

Journal Paper

Computers in Biology and Medicine. 2022 Feb 1;141:105103.

Abstract
During recent years there has been a growing interest in stochastic dynamic neural fields employed for modeling and predictions in biomedical and technical systems. In this paper, given some incomplete noisy data available from sensors, we propose and explore a state estimation method for fast restorations of membrane potential in the cortex based on such measurements and the Amari equation used for simulations of neural population activity in a stochastic setting. Our novel technique relies upon a Galerkin-type spectral approximation utilized within the conventional state-space approach. Translating a stochastic system into its state-space form creates a straightforward and fruitful way to the data-driven parameter estimation, filtering, prediction and smoothing. The present study is particularly focused on establishing a nonlinear stochastic Galerkin-spectral-approximation-induced system of large size, which is further estimated by the traditional extended Kalman filter (EKF). The efficiency of calculations is the main purpose of our research. That is why the fast filtering solution devised is based on processing the incoming data incrementally, that is, by processing measurements one at a time, rather than handling them as a unified high-dimensional vector. Such sequential filters suit well for dealing with large data sets as well as with real-time on-line computations. Also, their derivation and substantiation is of great interest in the context of neural network training because of large stochastic systems arisen there. In comparison to the batch filtering, our novel algorithm reduces the computational cost of membrane potential reconstructions in terms of the amount of grid nodes N accepted in the underlying spacial discretization, significantly. Apart from its computation efficiency, this sequential method is more robust to round-off errors committed within a computer-based finite precision arithmetics than the classical EKF because of the (N × N)-matrix inversion elimination from such membrane potential calculations. The superior performance of our technique is examined and confirmed in comparison to the batch one on two known scenarios in the dynamic neural field modeling.

9. Iterative continuous collocation method for solving nonlinear Volterra Integral Equations

Rouibah K., Bellour A,, Lima P.M. , RawashdehE.

Journal Paper

Kragujevac Journal of Mathematics, 46 (4) (2022), 635-648.

10. Analysis of the Euler and trapezoidal discretization methods for the numerical solution of nonlinear functional Volterra integral equations of Urysohn type

Bazm S, Lima P, Nemati S.

Journal Paper

Journal of Computational and Applied Mathematics. 2021 Dec 15;398:113628.

Abstract

In this paper, we investigate nonlinear functional Volterra–Urysohn integral equations, a class of nonlinear integral equations of Volterra type. The existence and uniqueness of the solution to the equation is proved by a technique based on the Picard iterative method. For the numerical approximation of the solution, the Euler and trapezoidal discretization methods are utilized which result in a system of nonlinear algebraic equations. Using a Gronwall inequality and its discrete version, first order of convergence to the exact solution for the Euler method and quadratic convergence for the trapezoidal method are proved. To prove the convergence of the trapezoidal method, a new Gronwall inequality is developed. Finally, numerical examples show the functionality of the methods.

11. Reconstruction of Hidden States in Stochastic Neural Field Equations with Infinite Signal Transmission Rate

Kulikova MV, Lima PM, Kulikov GY

Conference Proceeding

In 2021 25th International Conference on System Theory, Control and Computing (ICSTCC) 2021 Oct 20 (pp. 358-365). IEEE

Abstract
Due to a growing interest in using dynamic neural field models for solving practical applications, a special interest lies in developing efficient estimation methods for accurate restoration of hidden processes of mathematical models at hand. In this paper, we first propose a constructive way of setting up the nonlinear Bayesian filtering problem arisen in Galerkin-type spectral approximations applied to dynamic neural fields. Next, we propose the efficient numerical scheme for reconstructing the neural population activity over the spatial and time domains from incomplete measurements in both domains. The proposed numerical scheme is designed using the Galerkin approximation and the continuous-discrete extended Kalman filter. The accuracy of the estimation procedure is evaluated by using artificial data through exhaustive numerical tests.

12. Numerical solution of the neural field equation in the presence of random disturbance.

G.Yu. Kulikov, P.M. Lima and M.V. Kulikova

Journal Paper

Journal of Computational and Applied Mathematics , 387 (2021) , 112563.

This paper aims at presenting an efficient and accurate numerical method for treating both deterministic- and stochastic-type neural field equations (NFEs) in the presence of external stimuli input (or without it). The devised numerical integration means belongs to the class of Galerkin-type spectral approximations. The particular effort is focused on an efficient practical implementation of the solution technique because of the partial integro-differential fashion of the NFEs in use, which are to be integrated, numerically. Our method is implemented in Matlab. Its practical performance and efficiency are investigated on three variants of an NFE model with external stimuli inputs. We study both the deterministic case of the mentioned model and its stochastic counterpart to observe important differences in the solution behavior. First, we observe only stable one-bump solutions in the deterministic neural field scenario, which, in general, will be preserved in our stochastic NFE scenario if the level of random disturbance is sufficiently small. Second, if the area of the external stimuli is large enough and exceeds the size of the bump, considerably, the stochastic neural field solution’s behavior may change dramatically and expose also two- and three-bump patterns. In addition, we show that strong random disturbances, which may occur in neural fields, fully alter the behavior of the deterministic NFE solution and allow for multi-bump (and even periodic-type) solutions to appear in all variants of the stochastic NFE model studied in this paper.

13. Mathematical Modeling of Working Memory in the Presence of Random Disturbance using Neural Field Equations

Lima PM, Erlhagen W, Kulikov GY, Kulikova MV.

Conference Proceeding

InEPJ Web of Conferences 2021 (Vol. 248, p. 01021). EDP Sciences

Abstract

In this paper, we describe a neural field model which explains how a population of cortical neurons may encode in its firing pattern simultaneously the nature and time of sequential stimulus events. Moreover, we investigate how noise-induced perturbations may affect the coding process. This is obtained by means of a two-dimensional neural field equation, where one dimension represents the nature of the event (for example, the color of a light signal) and the other represents the moment when the signal has occurred. The additive noise is represented by a Q-Wiener process. Some numerical experiments reported are carried out using a computational algorithm for two-dimensional stochastic neural field equations.

14. Numerical solution of a class of third-kind Volterra integral equations using Jacobi wavelets

Nemati S, Lima PM, Torres DF

Journal Paper

Numerical Algorithms. 2021 Feb;86(2):675-91

Abstract
We propose a spectral collocation method, based on the generalized Jacobi wavelets along with the Gauss–Jacobi quadrature formula, for solving a class of third-kind Volterra integral equations. To do this, the interval of integration is first transformed into the interval [− 1, 1], by considering a suitable change of variable. Then, by introducing special Jacobi parameters, the integral part is approximated using the Gauss–Jacobi quadrature rule. An approximation of the unknown function is considered in terms of Jacobi wavelets functions with unknown coefficients, which must be determined. By substituting this approximation into the equation, and collocating the resulting equation at a set of collocation points, a system of linear algebraic equations is obtained. Then, we suggest a method to determine the number of basis functions necessary to attain a certain precision. Finally, some examples are included to illustrate the applicability, efficiency, and accuracy of the new scheme.

15. Iterative collocation method for solving a class of nonlinear weakly singular Volterra integral equations

Kherccouche K., Lima P.M., BellourA.

Journal Paper

Dolomites Research Notes on Approximation, 14 (2021), 33-41.

16. Numerical solution of variable-order fractional differential equations using Bernoulli polynomials

Nemati S., Lima P.M., Sedaghat S.

Journal Paper

Fractal and Fractional, 5 (2021), 219.

17. Numerical Solution of the Time Fractional Cable Equation

Morgado M.L., Lima P.M., Mendes M.V.

Book Chapter

Differential and Difference Equations with Applications. ICDDEA 2019. Springer Proceedings in Mathematics & Statistics, vol 333. Springer, Cham. (2020), pp.603-619.

Abstract

The time fractional diffusion equation has attracted the attention of many researchers in the last years due to its many applications in different domains. In this article we are concerned with one of these models, the time fractional cable equation, which describes the spatial and temporal dependence of transmembrane potential V(x, t) along the axial direction of a cylindrical nerve cell segment. The time derivative is of Caputo type.

We use a numerical scheme , which is based on the L1-method and on a finite-difference scheme for the time and space discretization, respectively.

18. Accuracy study in numerical simulations to stochastic neural field equations

Kulikova MV, Kulikov GY, Lima PM

Conference Proceeding

In2020 24th International Conference on System Theory, Control and Computing (ICSTCC) 2020 Oct 8 (pp. 254-261). IEEE

Abstract:

This paper elaborates accuracy issues in numerical solutions to Stochastic Neural Field Equations (SNFEs) with the infinite signal transmission speed and in the presence of external stimuli input. The numerical integration method under study belongs to the family of Galerkin-sort spectral approximations of one-dimensional SNFEs considered here. It reduces the partial integro-differential fashion of such models to a large system of ordinary Stochastic Differential Equations (SDEs). Eventually, these SDEs are solved by the Euler-Maruyama scheme of order 0.5 in MATLAB. In this paper, we devise a different-order numerical solution to the SNFE at hand and look at the difference of such stochastic simulations on average for evaluating the consistency of the solution derived. The effect of the SDE-numerical-integration-accuracy on formation of high neuron activity regions (so-called "bumps") is discussed within one SNFE with external stimuli.

19. Legendre wavelet collocation method combined with the Gauss–Jacobi quadrature for solving fractional delay-type integro-differential equations

Nemati S., Lima P.M., Sedaghat S.

Journal Paper

Applied Numerical Mathematics, 149 (2020) , 99-112

Abstract

In this work, we present a collocation method based on the Legendre wavelet combined with the Gauss–Jacobi quadrature formula for solving a class of fractional delay-type integro-differential equations. The problem is considered with either initial or boundary conditions and the fractional derivative is described in the Caputo sense. First, an approximation of the unknown solution is considered in terms of the Legendre wavelet basis functions. Then, we substitute this approximation and its derivatives into the considered equation. The Caputo derivative of the unknown function is approximated using the Gauss–Jacobi quadrature formula. By collocating the obtained residual at the well-known shifted Chebyshev points, we get a system of nonlinear algebraic equations. In order to obtain a continuous solution, some conditions are added to the resulting system. Some error bounds are given for the Legendre wavelet approximation of an arbitrary function. Finally, some examples are included to show the efficiency and accuracy of this new technique.

20. A numerical method for finite-part integrals

Diogo T., Lima P., Occorsio D.

Journal Paper

Dolomites Research Notes on Approximation, 13 (2020), p.1-11

21. Apontamentos de Matemática Computacional

Graça M.M. and Lima P. M.

Book

2020, IST, Portugal

Texbook for the course on Computational Mathematics, for students of the 2nd year of various engineering programs at IST (in portuguese).

22. Numerical Investigation of a Neural Field Model Including Dendritic Processing

Avitabile D., Coombes S. , Lima P.M.

Journal Paper

Journal of Computational Dynamics, V.7 (2020) , pp. 271-290.

Abstract: We consider a simple neural field model in which the state variable is dendritic voltage, and in which somas form a continuous one-dimensional layer. This neural field model with dendritic processing is formulated as an integro-differential equation. We introduce a computational method for approximating solutions to this nonlocal model, and use it to perform numerical simulations for neuro-biologically realistic choices of anatomical connectivity and nonlinear firing rate function. For the time discretisation we adopt an Implicit-Explicit (IMEX) scheme; the space discretisation is based on a finite-difference scheme to approximate the diffusion term and uses the trapezoidal rule to approximate integrals describing the nonlocal interactions in the model. We prove that the scheme is of first-order in time and second order in space, and can be efficiently implemented if the factorisation of a small, banded matrix is precomputed. By way of validation we compare the outputs of a numerical realisation to theoretical predictions for the onset of a Turing pattern, and to the speed and shape of a travelling front for a specific choice of Heaviside firing rate. We find that theory and numerical simulations are in excellent agreement.

23. A novel Lagrange operational matrix and tau-collocation method for solving variable-order fractional differential equations

S. Sabermahani, Y. Ordokhani, P.M.Lima

Journal Paper

Iranian Journal of Science and Technology, Transactions A: Science, 44 (2019), 127-135

The main result achieved in this paper is an operational Tau-Collocation method based on a class of Lagrange polynomials. The proposed method is applied to approximate the solution of variable-order fractional differential equations (VOFDEs). We achieve operational matrix of the Caputo’s variable-order derivative for the Lagrange polynomials. This matrix and Tau-Collocation method are utilized to transform the initial equation into a system of algebraic equations. Also, we discuss the numerical solvability of the Lagrange-Tau algebraic system in the case of a variable-order linear equation. Error estimates are presented. Some examples are provided to illustrate the accuracy and computational efficiency of the present method to solve VOFDEs. Moreover, one of the numerical examples is concerned with the shape-memory polymer model.

24. Numerical Investigation of Stochastic Neural Field Equations

P.M. Lima

Conference Proceeding

Advances in Mathematical Methods and High Performance Computing: V.K. Sing , D. Gao, A. Fischer, Eds., Springer, 2019, pp. 51-67

Abstract

We introduce a new numerical algorithm for solving the stochastic neural field equation (NFE) with delays. Using this algorithm, we have obtained some numerical results which illustrate the effect of noise in the dynamical behaviour of stationary solutions of the NFE, in the presence of spatially heterogeneous external inputs.

25. A numerical approach for solving fractional optimal control problems using modified hat functions

Nemati S., Lima P.M., Torres D.F.M.

Journal Paper

Communications in Nonlinear Science and Numerical Simulation,78 (2019) ,104849

We introduce a numerical method, based on modified hat functions, for solving a class of fractional optimal control problems. In our scheme, the control and the fractional derivative of the state function are considered as linear combinations of the modified hat functions. The fractional derivative is considered in the Caputo sense while the Riemann–Liouville integral operator is used to give approximations for the state function and some of its derivatives. To this aim, we use the fractional order integration operational matrix of the modified hat functions and some properties of the Caputo derivative and Riemann–Liouville integral operators. Using results of the considered basis functions, solving the fractional optimal control problem is reduced to the solution of a system of nonlinear algebraic equations. An error bound is proved for the approximate optimal value of the performance index obtained by the proposed method. The method is then generalized for solving a class of fractional optimal control problems with inequality constraints. The most important advantages of our method are easy implementation, simple operations, and elimination of numerical integration. Some illustrative examples are considered to demonstrate the effectiveness and accuracy of the proposed technique.

26. An improved composite collocation method for distributed-order fractional differential equations based on fractional Chelyshkov wavelets,

Rahimkani P., Ordokhani Y., Lima P.M.

Journal Paper

Applied Numerical Mathematics, 145 (2019), 1-27.

In this paper, we introduce a new family of fractional functions based on Chelyshkov wavelets for solving one- and two-variable distributed-order fractional differential equations. The concept of fractional derivative is utilized in the Caputo sense. The idea of solving these problems is based on fractional integral operator of fractional-order Chelyshkov wavelets with composite collocation method. This operator and collocation method are utilized to reduce the solution of the distributed fractional differential equations to a system of algebraic equations. The convergence of the fractional-order Chelyshkov wavelets bases is discussed. The efficiency and the applicability of the new methodology are illustrated by eight examples, in addition our findings in comparison with the existing results show the advantage of our method.

27. Effective numerical solution to two-dimensional stochastic neural field equations.

Kulikova MV, Kulikov GY, Lima PM

Conference Proceeding

In 2019 23rd International Conference on System Theory, Control and Computing (ICSTCC) 2019 Oct 9 (pp. 650-655). IEEE

Abstract:

In this paper, we explore an efficient numerical integration scheme for solving 2D Neural Field Equations(NFEs) in the presence of external stimuli input in both deterministic and stochastic scenarios. The method is based on Galerkin-type spectral approximation and our attention is paid to its efficient implementation because of the partial integro-differential fashion of the NFEs under discussion and two-dimensional spatial domain. The straightforward implementation of the proposed numerical scheme yields a very time consuming method in case of a good spatial resolution. Our efforts are focused on translating the numerical technique into a high-level computer code in MATLAB where the improved efficiency is achieved by vectorizing operations and accessing subarrays via MATLAB's colon notation. The results of numerical experiments are presented.

28. Numerical simulation of neural fields with finite transmission speed and random disturbance

Kulikov GY, Kulikova MV, Lima PM

Conference Proceeding

In 2019 23rd International Conference on System Theory, Control and Computing (ICSTCC) 2019 Oct 9 (pp. 644-649). IEEE

Abstract:

This paper aims at presenting an efficient and accurate numerical method for treating both deterministic and stochastic Neural Field Equations(NFEs) with a finite signal transmission speed and in the presence of external stimuli input. The numerical integration tool devised belongs to the class of Galerkin-type spectral approximations, and our particular effort focuses on an efficient implementation of the solution technique because of the partial integro-differential fashion of the NFEs in use. This method is intended for implementation in MATLAB. Its performance and efficiency is investigated on an NFE model with external stimuli inputs. We study both the deterministic case of the mentioned model and its stochastic counterpart to observe important differences in the solution behavior when the assumption of finite/infinite transmission speed implemented.

29. Numerical investigation of stochastic neural field equations

Lima PM

Book Chapter

Advances in Mathematical Methods and High Performance Computing. Advances in Mechanics and Mathematics, vol 41. Springer, Cham 2019:51-67

Abstract:

We introduce a new numerical algorithm for solving the stochastic neural field equation (NFE) with delays. Using this algorithm, we have obtained some numerical results which illustrate the effect of noise in the dynamical behaviour of stationary solutions of the NFE, in the presence of spatially heterogeneous external inputs.

30. Numerical simulations of two-dimensional neural fields with applications to working memory

Lima PM, Erlhagen W

Conference Proceeding

In 2018 European Control Conference (ECC) 2018 Jun 12 (pp. 2040-2045). IEEE

Abstract:

In this paper we describe a neural field model which explains how a population of cortical neurons may encode in its firing pattern simultaneously the nature and time of sequential stimulus events. From the mathematical point of view, this is obtained my means of a two-dimensional field, where one dimension represents the nature of the event (for example the color of a light signal) and the other represents the elapsed time. Some numerical experiments are reported which were carried out using a computational algorithm for two-dimensional neural field equations. These numerical experiments are described and their results are discussed.

31. Numerical solution of integro-differential equations arising from singular boundary value problems

Lima P. M., Bellour A., Bulatov M.V.

Journal Paper

Appl. Math. Comput., 336(2018) 1-15

Abstract

We consider the numerical solution of a singular boundary value problem on the half line for a second order nonlinear ordinary differential equation. Due to the fact that the nonlinear differential equation has a singularity at the origin and the boundary value problem is posed on an unbounded domain, the proposed approaches are complex and require a considerable computational effort. In the present paper, we describe an alternative approach, based on the reduction of the original problem to an integro-differential equation. Though the problem is posed on the half-line, we just need to approximate the solution on a finite interval. By analyzing the behavior of the numerical approximation on this interval, we identify the solution that satisfies the prescribed boundary condition. Although the numerical algorithm is much simpler than the ones proposed before, it provides accurate approximations. We illustrate the proposed methods with some numerical examples.

32. An effective numerical method for solving fractional pantograph differential equations using modification of hat functions

Nemati S., Lima P.M., Sedaghat S.

Journal Paper

Applied Numerical Mathematics, 131 (2018), 174-189

Abstract

In this work, a spectral method based on a modification of hat functions (MHFs) is proposed to solve the fractional pantograph differential equations. Some basic properties of fractional calculus and the operational matrices of MHFs are utilized to reduce the considered problem to a system of linear algebraic equations. The greatest advantage of using MHFs is the large number of zeros in their operational matrix of fractional integration, product operational matrix and also pantograph operational matrix. This property makes these functions computationally attractive. Some illustrative examples are included to show the high performance and applicability of the proposed method and a comparison is made with the existing results. These examples confirm that the method leads to the results of 3 . order of convergence. $O (h^{3})$

33. Numerical Solution of a Third-Kind Volterra Integral Equation Using an Operational Matrix Technique

Nemati S. and Lima P.M.

Conference Proceeding

Proceedings of 2018 European Control Conference, ECC 2018, 8550223, pp. 3215-3220

Abstract:

In this work, an operational matrix of fractional integration based on an adjustment of hat functions is used for solving a class of third-kind Volterra integral equations. We show that the application of this numerical technique reduces the problem to a linear system of equations that can be efficiently solved. Two examples are considered to demonstrate the accuracy and efficiency of the proposed method.

34. Numerical solution of nonlinear fractional integro-differential equations with weakly singular kernels via a modification of hat functions

S. Nemati and P.M.Lima

Journal Paper

Appl. Math. Comput., 327 (2018)

Abstract

In the present paper, a modification of hat functions (MHFs) has been considered for solving a class of nonlinear fractional integro-differential equations with weakly singular kernels, numerically. The fractional order operational matrix of integration is introduced. We provide an error estimation for the approximation of a function by a series of MHFs. To suggest a numerical method, the main problem is converted to an equivalent Volterra integral equation of the second kind and operational matrices of MHFs are used to reduce the problem to the solution of bivariate polynomial equations. Finally, illustrative examples are provided to confirm the accuracy and validity of the proposed method.

35. Uma abordagem matemática do Sudoku

P.M. Lima

Journal Paper

Gazeta de Matemática, 185 (2018) 16-24.

36. A Novel computational approach to singular free boundary problems in ordinary differential equations

Lima P.M, Morgado M.L., Schobinger M., Weinmueller E.B.

Journal Paper

Applied Numerical Mathematics, 114 (2017) 97-107

Abstract

We study the numerical solution of a singular free boundary problem for a second order nonlinear ordinary differential equation, where the differential operator is the degenerate m-Laplacian. A typical difficulty arising in free boundary problems is that the analytical solution may become non-smooth at one boundary or at both boundaries of the interval of integration. A numerical method proposed in [18] consists of two steps. First, a smoothing variable transformation is applied to the analytical problem in order to improve the smoothness of its solution. Then, the problem is discretized by means of a finite difference scheme.

In the present paper, we consider an alternative numerical approach. We first transform the original problem into a special parameter dependent problem sometimes referred to as an ‘eigenvalue problem’. By applying a smoothing variable transformation to the resulting equation, we obtain a new problem whose solution is smoother, and so the open domain Matlab collocation code bvpsuite [17] can be successfully applied for its numerical approximation.

37. Smoothing transformation and spline collocation for weakly singular Volterra integro-differential equations

Diogo T., Lima P.M., Pedas A., Vainikko G.

Journal Paper

Applied Numerical Mathematics, 114 (2017), 63-76

Abstract

This work is concerned with the construction and analysis of high order numerical methods for solving initial value problems for linear Volterra integro-differential equations with different types of singularities. Using an integral reformulation of the initial value problem, a smoothing transformation is applied so that the exact solution of the resulting equation does not contain any singularities in its derivatives up to a certain order. After that, the regularized equation is solved by a piecewise polynomial collocation method on a uniform or mildly graded grid. Finally, the obtained spline approximations can be used to define (typically non-polynomial) approximations for the initial value problem. The theoretical results are tested by some numerical examples.

38. Computational Methods for Two-Dimensional Neural Fields

Lima P.M. , Buckwar E.

Book Chapter

Nonlinearity: Problems, Solutions and Applications, vol. 1, L.A. Uvarova,A.B. Nadykto, A.V. Latyshev (Eds.), Nova Science Publishers, N.Y., 2017

39. Numerical Analysis and Optimization

Katkovskaya I, Kovaleva I, Krotov V., Lima PM , Volkov V., Zubko O.

Book

Bagosles, Minsk, 2017

This textbook (in Russian) was composed in the frame of the international project Applied Computation in Engineering and Science, ACES . The translation to English is in preparation.

40. Numerical investigation of noise induced changes to the solution behaviour of the discrete FitzHugh–Nagumo equation

Ford N., Lima P.M., Lumb P.

Journal Paper

Applied Mathematics and Computation , 293 (2017) 448-460.

Abstract

In this work we introduce and analyse a stochastic functional equation, which contains both delayed and advanced arguments. This equation results from adding a stochastic term to the discrete FitzHugh–Nagumo equation which arises in mathematical models of nerve conduction. A numerical method is introduced to compute approximate solutions and some numerical experiments are carried out to investigate their dynamical behaviour and compare them with the solutions of the corresponding deterministic equation.

41. On High Order Barycentric Root-Finding Methods

Graça M.M. and and Lima P.M.

Journal Paper

Tendências em Matemática Aplicada e Computacional, 17, N. 3 (2016), 321-330,

Abstract

To approximate a simple root of a real function f we construct a family of iterative maps, which we call Newton-barycentric functions, and analyse their convergence order. The performance of the resulting methods is illustrated by means of numerical examples.

42. Numerical solution of the neural field equation in the two-dimensional case

Lima P.M., Buckwar E

Journal Paper

SIAM Journal of Scientific Computing, 37 (2015) B962- B979

Abstract

We are concerned with the numerical solution of a class of integro-differential equations, known as neural field equations, which describe the large-scale dynamics of spatially structured networks of neurons. These equations have many applications in neuroscience and robotics. We describe a numerical method for the approximation of solutions in the two-dimensional case, including a space-dependent delay in the integrand function. Compared with known algorithms for this type of equation we propose a scheme with higher accuracy in the time discretization. Since computational efficiency is a key issue in this type of calculation, we use a new method for reducing the complexity of the algorithm. The convergence issues are discussed in detail and a number of numerical examples are presented, which illustrate the performance of the method.

43. Numerical Simulations in Two-Dimensional Neural Fields

Lima P.M. , Buckwar E.

Conference Proceeding

BMC Neuroscience 2015, 16(Suppl 1):P22

44. Numerical Investigation of the Two-dimensional Neural Field Equation with Delay

Lima P.M., Buckwar E.

Conference Proceeding

Second International Conference on Mathematics and Computers in Sciences and in Industry (MCSI), IEEE Conference PublicationsPages: 131 - 137

ABSTRACT

Neural Field Equations (NFEs) are integrodifferential equations which describe the electric potential field and the interaction between neurons, in certain regions of the brain. They are becoming increasingly important for the interpretation of EEG, fMRi and optical imaging data. In the present article we describe a new efficient algorithm for the numerical simulation of two-dimensional neural fields with delays. The main features of this method are discussed and its performance is illustrated by some numerical examples.

45. Root finding by high order iterative methods based on quadratures

Graça M.M. and Lima P.M.

Journal Paper

Applied Mathematics and Computation, 264 ( 2015) 466–482.

Abstract

We discuss a recursive family of iterative methods for the numerical approximation of roots of nonlinear functions in one variable. These methods are based on Newton–Cotes closed quadrature rules. We prove that when a quadrature rule with n + 1 nodes is used the resulting iterative method has convergence order at least n + 2, starting with the case n = 0 (which corresponds to the Newton’s method).

46. Mathematical Modelling and Numerical Simulations in Nerve Conduction

Ford N.J., Lima P.M. , Lumb P.M.

Conference Proceeding

Proceedings of the International Conference on Bio-inspired Systems and Signal Processing, H. Lose, A.Fred, H.Gamboa and D. Elias, eds., Scitepress 2015, pp 283-288

ABSTRACT

In this paper we are concerned with the numerical solution of the discrete FitzHugh-Nagumo equation. This equation describes the propagation of impulses across a myelinated axon. We analyse the asymptotic behaviour of the solutions of the considered equation and numerical approximations are computed by a new algorithm, based on a finite difference scheme and on the Newton method. The efficiency of the method is discussed and its performance is illustrated by a set of numerical examples.

47. An integral method for the numerical solution of nonlinear singular boundary value problems

Bulatov M.V., Lima P.M., and Than Do Tien

Journal Paper

Bulletin of the South Ural State University, Ser. Mathematical Modelling, Programming & Computer Software, 8

We discuss the numerical treatment of a nonlinear singular second order boundary value problem in ordinary dierential equations, posed on an unbounded domain, which represents the density prole equation for the description of the formation of microscopic bubbles in a non-homogeneous uid. Due to the fact that the nonlinear dierential equation has a singularity at the origin and the boundary value problem is posed on an unbounded domain, the proposed approaches are complex and require a considerable computational eort. This is the motivation for our present study, where we describe an alternative approach, based on the reduction of the original problem to an integro-dierential equation. In this way, we obtain a Volterra integro-dierential equation with a singular kernel. The numerical integration of such equations is not straightforward, due to the singularity. However, in this paper we show that this equation may be accurately solved by simple product integration methods, such as the implicit Euler method and a second order method, based on the trapezoidal rule. We illustrate the proposed methods with some numerical examples

48. Computational methods for a mathematical model of propagation of nervous signals in myelinated axons

Ford N.J., Lima P.M. and Lumb P.M.

Journal Paper

Applied Numerical Mathematics , 85 (2014), 38-53.

Abstract

This paper is concerned with the approximate solution of a nonlinear mixed type functional differential equation (MTFDE) arising from nerve conduction theory. The equation considered describes conduction in a myelinated nerve axon. We search for a monotone solution of the equation defined in the whole real axis, which tends to given values at ±∞. We introduce new numerical methods for the solution of the equation, analyse their performance, and present and discuss the results of the numerical simulations.

49. Analysis and numerical approximation of singular boundary value problems with the p-Laplacian in fluid mechanis

Kulikov G. Yu., Lima P.M., Morgado M.L.

Journal Paper

Journal of Computational and Applied Mathematics, 262 (2014) 87-104.

Abstract

This paper studies a generalization of the Cahn–Hilliard continuum model for multi-phase fluids where the classical Laplacian has been replaced by a degenerate one (i.e., the so-called $p$ -Laplacian). The solution’s asymptotic behavior is analyzed at two singular points; namely, at the origin and at infinity. An efficient technique for treating such singular boundary value problems is presented, and results of numerical integration are discussed and compared with earlier computed data.

50. Density profile equation with p-Laplacian: analysis and numerical simulation

Hastermann G., Lima P.M., Morgado M.L., and Weinmueller E.W.

Journal Paper

Applied Mathematics and Computation 225 (2013) 550–561

Abstract

Analytical properties of a nonlinear singular second order boundary value problem in ordinary differential equations posed on an unbounded domain for the density profile of the formation of microscopic bubbles in a nonhomogeneous fluid are discussed. Especially, sufficient conditions for the existence and uniqueness of solutions are derived. Two approximation methods are presented for the numerical solution of the problem, one of them utilizes the open domain Matlab code bvpsuite. The results of numerical simulations are presented and discussed.

51. Existence and uniqueness of solutions to weakly singular integral-algebraic and integrodifferential equations

Bulatov M.V., Lima P.M,. and Weinmueller E.W.

Journal Paper

Cent. Eur. J. Math. , 12 ( 2014 ) 308-321.

Abstract

We consider systems of integral-algebraic and integro-differential equations with weakly singular kernels. Although these problem classes are not in the focus of the main stream literature, they are interesting, not only in their own right, but also because they may arise from the analysis of certain classes of differential-algebraic systems of partial differential equations. In the first part of the paper, we deal with two-dimensional integral-algebraic equations. Next, we analyze Volterra integral equations of the first kind in which the determinant of the kernel matrix k(t, x) vanishes when t = x. Finally, the third part of the work is devoted to the analysis of degenerate integro-differential systems. The aim of the paper is to specify conditions which are sufficient for the existence of a unique continuous solution to the above problems. Theoretical findings are illustrated by a number of examples.

52. Analysis and numerical methods for fractional differential equations with delay,

Morgado M.L., Ford N.J., and Lima P.M.

Journal Paper

Journal of Computational and Applied Mathematics, 252 (2013)

Abstract

In this paper, we consider fractional differential equations with delay. We focus on linear equations. We summarise existence and uniqueness theory based on the method of steps and we give a theorem on the propagation of derivative discontinuities. We discuss the dependence of the solution on the parameters of the equation and conclude with a numerical treatment and examples based on the adaptation of a fractional backward difference method to the delay case.

53. Numerical solution of a class of two-dimensional nonlinear Volterra integral equations using Legendre polynomials

Nemati S., Lima P.M. , Ordokhani Y

Journal Paper

Journal of Computational and Applied Mathematics, 242 (2013) 53-69.

Abstract

In this paper, a method for finding an approximate solution of a class of two-dimensional nonlinear Volterra integral equations is discussed. The properties of two-dimensional shifted Legendre functions are presented. The operational matrices of integration and product together with the collocation points are utilized to reduce the solution of the integral equation to the solution of a system of nonlinear algebraic equations. Some results concerning the error analysis are obtained. We also consider the application of the method to the solution of certain partial differential equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

54. Analysis and Computational Approximation of a Forward-Backward Equation Arising in Nerve Conduction

P.M. Lima, M.F. Teodoro, N.J.Ford, P. M.Lumb,

Book Chapter

Differential and Difference Equations with Applications , Springer Proceedings in Mathematics & Statistics, Volume 47, 2013, pp 475-483

Abstract

This paper is concerned with the approximate solution of a nonlinear mixed-type functional differential equation (MTFDE) arising from nerve conduction theory. The equation considered describes conduction in a myelinated nerve axon. We search for a solution defined on the whole real axis, which tends to given values at ±∞.The numerical algorithms, developed previously by the authors for linear problems, were upgraded to deal with the case of nonlinear problems on unbounded domains. Numerical results are presented and discussed.

55. Método da Bissecção

Lima P.M.

Didactic Video

Ddactic video about the bissection method (in Portuguese).

56. Método da Bissecção (Exemplo)

Lima P.M.

Didactic Video

Ddactic video with an application of the bissection method (in Portuguese).

57. Método de Newton

Lima P.M.

Didactic Video

Didactic video about the Newton's method (in Portuguese).

58. Método de Newton (exemplo)

Lima P.M.

Didactic Video

Didactic video with applications of the Newton method (in Portuguese).

59. Solutions of the Discrete FitzHugh-Nagumo System, Differential and Difference Equations with Applications

Pedro A., Lima P.M.

Book Chapter

Differential and Difference Equations with Applications , Springer Proceedings in Mathematics & Statistics ,Volume 47, 2013, pp 551-560

In this work, we are concerned with a system of two functional differential equations of mixed type (with delays and advances), known as the discrete Fitzhugh-Nagumo equations, which arises in the modeling of impulse propagation in a myelinated axon.

In the case where the recovery process is neglected, this system reduces to a single equation, which is well studied in the literature. In this case it is known that for each set of the equation parameters (within certain constraints), there exists a value of τ (delay) for which the considered equation has a monotone solution v satisfying certain conditions at infinity. The main goal of the present work is to show that for sufficiently small values of the coefficients in the second equation of system (1), this system has a solution (v,w) whose first component satisfies certain boundary conditions and has similar properties to the ones of v, in the case of a single equation. With this purpose we linearize the original system as t →−∞ and t →∞ and analyze the corresponding characteristic equations. We study the existence of nonoscillatory solutions based on the number and nature of the roots of these equations.

60. Efficient computational methods for singular free boundary problems using smoothing variable substitutions

Morgado M.L. and Lima P.M.

Journal Paper

Journal of Computational and Applied Mathematics, 236 (2012) 2981–2989.

Abstract

We study a class of singular free boundary problems for the degenerate p $m$ -Laplacian. Taking into account the behavior of the solution in the neighborhood of the singular points, a variable substitution is introduced, which makes the solution smooth in all the domain. Then, a standard finite difference scheme is used to discretize the problem. The numerical results suggest that in this way the second order convergence of the finite difference scheme is recovered, in spite of the singularities.

61. Two-dimensional integral-algebraic systems: analysis and computational methods

Bulatov P.M. and Lima P.M.

Journal Paper

Journal of Computational and Applied Mathematics, 236 (2011) 132-140

Abstract

In this article we formulate sufficient conditions for the existence and uniqueness of solution to systems of two-dimensional Volterra integral equations, in which the coefficient of the main term is a singular matrix. A numerical method is introduced which can be applied to approximate the solution when the given conditions are satisfied. The convergence of this method is proved and illustrated by numerical examples.

62. Numerical solution of nonlinear two-dimensional integral equations using rationalized Haar functions

Babolian E. , Bazm S., and Lima P.M.

Journal Paper

Communications on Nonlinear Science and Numerical Simulation, 16 (2011) 1164-1175.

Abstract

Two-dimensional rationalized Haar (RH) functions are applied to the numerical solution of nonlinear second kind two-dimensional integral equations. Using bivariate collocation method and Newton–Cotes nodes, the numerical solution of these equations is reduced to solving a nonlinear system of algebraic equations. Also, some numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method.

63. Analytical and numerical investigation of mixed-type functional differential equations

Lima P.M., Teodoro M.F., Ford N.J. and Lumb P.M.

Journal Paper

Journal of Computational and Applied Mathematics 234 (2010) 2826-2837

Abstract

This paper is concerned with the approximate solution of a linear non-autonomous functional differential equation, with both advanced and delayed arguments.

64. Numerical solution of a class of singular free boundary problems involving the m-Laplace operator

P.M. Lima and M.L. Morgado

Journal Paper

Journal of Computational and Applied Mathematics, 234 (2010) 2838-2847

Abstract

For a class of singular free boundary problems with applications in electromagnetism and plasma physics, an analytical-numerical approach is proposed based on the asymptotic expansion of the solution in the neighborhood of the singular points. This approach was already used to approximate the solution of certain classes of singular boundary value problems on bounded (Lima and Morgado (2009) [14]) and unbounded domains (Konyukhova et al. (2008) [12]). Here, one-parameter families of solutions of suitable singular Cauchy problems, describing the behavior of the solution at the singularities, are derived and based on these families numerical methods for the approximation of the solution of the free boundary problems are constructed.

65. The numerical solution of forward-backward differential equations: Decomposition and related issues

Ford N.J., Lumb P.M., Lima P.M. and Teodoro M.F.

Journal Paper

Journal of Computational and Applied Mathematics, 234 (2010) 2745-2756.

This paper focuses on the decomposition, by numerical methods, of solutions to mixed-type functional differential equations (MFDEs) into sums of “forward” solutions and “backward” solutions.

$x^{'} (t) = a x (t) + b x (t - 1) + c x (t + 1)$

66. Finite element solution of a linear mixed-type functional differential equation

Lima P.M., Teodoro M.F., Ford N.J. and Lumb P.M.

Journal Paper

Numerical Algorithms, 55 (2010) 301-320.

Abstract

This paper is devoted to the approximate solution of a linear first-order functional differential equation which involves delayed and advanced arguments. We seek a solution x, defined for t ∈ (0, k − 1],(k ∈ IN ), which takes given values on the intervals [ − 1, 0] and (k − 1, k]. Continuing the work started in previous articles on this subject, we introduce and analyse a computational algorithm based on the finite element method for the solution of this problem which is applicable both in the case of constant and variable coefficients. Numerical results are presented and compared with the results obtained by other methods.

67. Finite difference solution of a singular boundary value problem for the p-Laplace operator,

Morgado M.L. and Lima P.M.

Journal Paper

Numerical Algorithms, 55 (2010) 337-348.

Abstract

We are concerned about a singular boundary value problem for a second order nonlinear ordinary differential equation. The differential operator of this equation is the radial part of the so-called N-dimensional p-Laplacian (where p > 1), which reduces to the classical Laplacian when p = 2. We introduce a finite difference method to obtain a numerical solution and, in order to improve the accuracy of this method, we use a smoothing variable substitution that takes into account the behavior of the solution in the neighborhood of the singular points.

68. Numerical modeling of oxygen diffusion in cells with Michaelis-Menten uptake kinetics

Lima P.M. and Morgado M.L.

Journal Paper

Journal of Mathematical Chemistry, 48 (2009) 145-158.

Abstract

A class of singular boundary value problems is studied, which models the oxygen diffusion in a spherical cell with Michaelis-Menten uptake kinetics. Suitable singular Cauchy problems are considered in order to determine one-parameter families of solutions in the neighborhood of the singularities. These families are then used to construct stable shooting algorithms for the solution of the considered problems and also to propose a variable substitution in order to improve the convergence order of the finite difference methods. Numerical results are presented and discussed.

69. A new approach to the numerical solution of forward-backward equations

Teodoro M.F., Lima P.M., Ford N.J., and Lumb P.M.

Journal Paper

Frontiers of Mathematics in China, V.4, N.1 (2009) 155-168.

Abstract

This paper is concerned with the approximate solution of functional differential equations having the form: x′(t) = αx(t) + βx(t - 1) + γx(t + 1). We search for a solution x, defined for t ∈ [−1, k], k ∈ ℕ, which takes given values on intervals [−1, 0] and (k-1, k]. We introduce and analyse some new computational methods for the solution of this problem. Numerical results are presented and compared with the results obtained by other methods.

70. Analytical-numerical investigation of a singular boundary value problem for a generalized Emden-Fowler equation,

P.M. Lima and M.L. Morgado

Journal Paper

Journal of Computational and Applied Mathematics, 229 (2009) 480- 487.

Abstract

In this paper we are concerned about a singular boundary value problem for a quasilinear second-order ordinary differential equation, involving the one-dimensional $p$ -laplacian. Asymptotic expansions of the one-parameter families of solutions, satisfying the prescribed boundary conditions, are obtained in the neighborhood of the singular points and this enables us to compute numerical solutions using stable shooting methods.

71. Superconvergence of collocation methods for a class of weakly singular Volterra integral equations

Diogo M.T. and Lima P.M.

Journal Paper

Journal of Computational and Applied Mathematics, 218 (2008) 307-316.

Abstract

We discuss the application of spline collocation methods to a certain class of weakly singular Volterra integral equations. It will be shown that, by a special choice of the collocation parameters, superconvergence properties can be obtained if the exact solution satisfies certain conditions. This is in contrast with the theory of collocation methods for Abel type equations. Several numerical examples are given which illustrate the theoretical results.

72. Bubbles and droplets in nonlinear physics models: analysis and numerical simulation of singular nonlinear boundary value problems

Konyukhova N.B., Lima P.M., Morgado M.L., Solovev M.B.

Journal Paper

Comp. Maths. Math. Phys. 48 (2008), n.11, 2018-2058.

Abstract

For a second-order nonlinear ordinary differential equation (ODE), a singular Boundary value problem (BVP) is investigated which arises in hydromechanics and nonlinear field theory when static centrally symmetric bubble-type (droplet-type) solutions are sought. The equation, defined on a semi-infinite interval 0 < r < ∞, possesses a regular singular point as r→ 0 and an irregular one as r→ ∞. We give the restrictions to the parameters for a correct mathematical statement of the limit boundary conditions in singular points and their accurate transfer into the neighborhoods of these points using certain results for singular Cauchy problems and stable initial manifolds. The necessary and sufficient conditions for the existence of bubble-type (droplet-type) solutions are discussed (in the form of additional restrictions to the parameters) and some estimates are obtained. A priori detailed analysis of a singular nonlinear BVP leads to efficient shooting methods for solving it approximately. Some results of the numerical experiments are displayed and their physical interpretation is discussed.

73. Analysis and numerical approximation of a free boundary problem for a singular ordinary differential equation

P.M. Lima and M.L. Morgado

Journal Paper

TEMA- Tendências da Matemática Aplicada e Computacional, Vol. 8 (2007) N.2, 259-268.

We analyse a free boundary problem for a second order nonlinear or- dinary differential equation. The asymptotic behavior of the solutions satisfying certain boundary conditions is analysed at the endpoints of the interval where the solution is sought. Based on this study, an efficient shooting method isintroduced and numerical results are obtained.

74. Collocation solutions of a weakly singular Volterra integral equation

M.T.Diogo and P.M.Lima

Journal Paper

TEMA - Tendências da Matemática Aplicada e Computacional, Vol. 8 (2007), N.2, 229-238.

	Abstract

	. The discrete superconvergence properties of spline collocation s

	olutions

	for a certain Volterra integral equation with weakly singular kernel are

	analyzed. In

	particular, the attainable convergence orders at the collocation points ar

	e examined

	for certain choices of the collocation parameters.

75. Numerical modelling of qualitative behaviour of solutions to convolution integral equations

Ford N.J., Diogo M.T., Ford J., and Lima P.M.

Journal Paper

J. Comp. Appl. Math. 205 (2007), 849-858.

We consider the qualitative behaviour of solutions to linear integral equations of convolution type,where the kernel k is assumed to be either integrable or of exponential type. After a brief review of the well-known Paley–Wiener theory we give conditions that guarantee that exact and approximate solutions of (1) are of a specific exponential type. As an example, we provide an analysis of the qualitative behaviour of both exact and approximate solutions of a singular Volterra equation with infinitely many solutions. We show that the approximations of neighbouring solutions exhibit the correct qualitative behaviour.

76. Efficient numerical solution of the density profile equation in hydrodynamics

Kitzhofer G., Koch O., Lima P.M., and Weinmueller E.B.

Journal Paper

Journal of Scientific Computing, 32 (2007), 411-424.

We discuss the numerical treatment of a nonlinear second order boundary value problem in ordinary differential equations posed on an unbounded domain which represents the density profile equation for the description of the formation of microscopical bubbles in a non-homogeneous fluid. For an efficient numerical solution the problem is transformed to a finite interval and polynomial collocation is applied to the resulting boundary value problem with essential singularity. We demonstrate that this problem is well-posed and the involved collocation methods show their classical convergence order. Moreover, we investigate what problem statement yields favorable conditioning of the associated collocation equations. Thus, collocation methods provide a sound basis for the implementation of a standard code equipped with an a posteriori error estimate and an adaptive mesh selection procedure. We present a code based on these algorithmic components that we are currently developing especially for the numerical solution of singular boundary value problems of arbitrary, mixed order, which also admits to solve problems in an implicit formulation. Finally, we compare our approach to a solution method proposed in the literature and conclude that collocation is an easy to use, reliable and highly accurate way to solve problems of the present type.

77. Solution of a singular integral equation by a split-interval method

Diogo M. T., Ford N.J., Lima P.M., and Thomas S.M.

Journal Paper

International Journal on Numerical Analysis and Modeling, 4 (2007), 63-73.

Abstract. In this paper we give details of a new numerical method for the solution of a singular integral equation of Volterra type that has an infinite class of solutions. The split-interval method we have adopted utilises a simple robust numerical method over an initial time interval (which includes the singularity) combined with extrapolation. We describe the method and give details of its order of convergence together with examples that show its effectiveness.

78. Analysis of singular boundary value problems for an Emden-Fowler equation

P.M.Lima and Morgado M.L.

Journal Paper

Communications on Pure and Applied Analysis, 5 (2006), 321-336 .

In this work we are concerned about a second order nonlinear ordinary differential equation. Our main purpose is to describe one-parameter families of solutions of this equation which satisfy certain boundary conditions. These one-parameter families of solutions are obtained in the form of asymptotic or convergent series. The series expansions are then used to approximate the solutions of two boundary value problems. We are specially interested in the cases where these problems are degenerate with respect to the unknown function and/or to the independent variable. Lower and upper solutions for each of the considered boundary value problems are obtained and, in certain particular cases, a closed formula for the exact solution is derived. Numerical results are presented and discussed.

79. Numerical solution of a nonlinear Abel type Volterra integral equation

Diogo M.T., Lima P.M. and Rebelo M.S.

Journal Paper

Communications on Pure and Applied Analysis, 5(2006), 277-288.

We are concerned with the analytical and numerical analysis of a nonlinear weakly singular Volterra integral equation. Owing to the singularity of the solution at the origin, the global convergence order of Euler's method is less than one. The smoothness properties of the solution are investigated and, by a detailed error analysis, we prove that first order of convergence can be achieved away from the origin. Some numerical results are included confirming the theoretical estimates.

80. Analytical-numerical investigation of bubble-type solutions of nonlinear singular problems

Lima P.M., Konyukhova N.B., Chemetov N.V., and Sukov A.I.

Journal Paper

Journal of Computational and Applied Mathematics, 189 (2006), 260-273

Abstract

In this work we are concerned about a singular boundary value problem for a second-order nonlinear ordinary differential equation, arising in hydrodynamics and nonlinear field theory, when centrally symmetric bubble-type solutions are sought. We are interested on solutions of this equation which are strictly increasing on the positive semi-axis and have finite limits at zero and infinity. Necessary conditions for the existence of such solutions are obtained in the form of a restriction on the equation parameters. The asymptotic behavior of certain solutions of this equation is analyzed near the two singularities (when $r \to 0 +$ and $r \to \infty$ ), where the considered boundary conditions define one-parameter families of solutions. Based on the analytic study, an efficient numerical method is proposed to compute approximately the needed solutions of the above problem. Some results of the numerical experiments are displayed and their physical interpretation is discussed.

81. Matemática Experimental

Graça M.M. and Lima P.M.

Book

2006, IST Press, Portugal

Textbook for the course Experimental Mathematics, 1st year of the degree on Applied Mathematics and Computation at IST (in Portuguese)

82. Numerical methods for a Volterra integral equations with non-smooth solutions

Diogo M.T., Ford N.J., Lima P.M. and Valtchev S.

Journal Paper

Journal of Computational and Applied Mathematics, 189 (2006), 412-423.

Abstract

We consider the numerical treatment of a singular Volterra integral equation with an infinite set of solutions, one of which is smooth and all others have infinite gradient at the origin. This equation has been the subject of previous works, where we have dealt with the approximation of the smooth solution. Here we present numerical methods which enable us to obtain approximations to any of the infinite class of solutions. Some numerical examples are given which illustrate the performance of the methods employed.

83. High order product integration methods for a Volterra integral equation with logarithmic singular kernel

Diogo T, Franco NB, Lima P

Journal Paper

Communications on Pure and Applied Analysis. 2004 Jun 1;3(2):217-36

Abstract

This work is concerned with the construction and analysis of high

order product integration methods for a class of Volterra integral equations

with logarithmic singular kernel. Sufficient conditions for the methods to be

convergent are derived and it is shown that optimal convergence orders are

attained if the exact solution is sufficiently smooth. The case of non-smooth

solutions is dealt with by making suitable transformations so that the new

equation possesses smooth solutions. Two particular methods are considered

and their convergence proved. A sample of numerical examples is included.

84. Numerical solution of a singular boundary value problem for a generalized Emden–Fowler equation

Lima PM, Oliveira AM

Journal Paper

Applied numerical mathematics;45(2003) :389-409

Abstract

In this paper we shall deal with an equation of the form

y''(x)=-g(x)xpy(x)q,

where p and q are real parameters satisfying p>−2, q<−1 and g is a positive and continuous function on [0,1].

We shall search for positive solutions which satisfy the boundary conditions:

y(0)=y(1)=0.

The initial nonlinear problem is transformed into a sequence of linear ones, each one of them is approximated by a finite difference scheme. Asymptotic expansions of the error are obtained and numerical examples are then analysed.

85. Asymptotic and numerical approximation of a nonlinear singular boundary value problem

Konyukhova NB, Lima PM, Carpentier MP

Journal Paper

Trends in Computational and Applied Mathematics. 2002 Jun 1;3(2):141-50

Abstract

In this work, we consider a singular boundary value problem for a nonlinear second-order differential equation of the form g00(u) = ug(u)q=q; (0.1) where 0 < u < 1 and q is a known parameter, q < 0. We search for a positive solution of (0.1) which satisfies the boundary conditions g0(0) = 0; (0.2) lim u!1¡ g(u) = lim u!1¡ (1 ¡ u)g0(u) = 0: (0.3) We analyse the asymptotic properties of the solution of (0.1)-(0.3) near the singularity, depending on the value of q. We show the existence of a one-parameter family of solutions of equation (0.1) which satisfy the boundary condition (0.3) and obtain convergent or asymptotic expansions of these solutions.

86. Numerical solution of a nonuniquely solvable Volterra integral equation using extrapolation methods

Lima P, Diogo T

Journal Paper

Journal of Computational and Applied Mathematics. 2002 Mar 1;140(1-2):537-57

Abstract

In this work the numerical solution of a Volterra integral equation with a certain weakly singular kernel, depending on a real parameter μ, is considered. Although for certain values of μ this equation possesses an infinite set of solutions, we have been able to prove that Euler's method converges to a particular solution. It is also shown that the error allows an asymptotic expansion in fractional powers of the stepsize, so that general extrapolation algorithms, like the E-algorithm, can be applied to improve the numerical results. This is illustrated by means of some examples.

87. Numerical methods and error estimates for a singular boundary‐value problem

Lima PM, Oliveira AM

Journal Paper

Mathematical Modelling and Analysis. 2002 Jan 1;7(2):271-84

Abstract

In this paper we analyze a class of equations of the form y? (x) = —g(x) xp (y(x)) q where p and q are real parameters satisfying p > _1 , g < _1 and g is a positive and continuous function on [0,1]. We search for positive solutions which satisfy the boundary conditions y'(0)=y(l) = 0.

Numerical approximations of the solution are obtained by means of a finite difference scheme and the asymptotic expansion of the discretization error is deduced. Some numerical examples are analyzed.

Nagrinejama viena klase antrosios eiles netiesiniu diferencialiniu lygčiu su kraštine salyga. Uždavinys yra singuliarusis viename arba abiejuose intervalo galuose. Siūlomas skaitinis metodas taikytinas atskiriems uždaviniu klases atvejams. Darbas tesia ankstesnius autoriu darbu tyrimus. Pateikti skaitinio eksperimento rezultatai, patvirtinantys teorinius iverčius.

88. Singular problems for Emden-Fowler-type second-order nonlinear ordinary differential equations

Dyshko AL, Carpentier MP, Konyukhova NB, Lima PM

Journal Paper

Computational mathematics and mathematical physics. 2001;41(4):557-80

89. Analysis of product integration methods for a class of singular Volterra integral equations

Diogo T, Lima P, Franco NB

Journal Paper

Trends in Computational and Applied Mathematics. 2000 Jun 1;1(2):373-87

Abstract

The construction and analysis of high order numerical methods for Volterra integral equations with a certain weakly singular kernel have been investigated in [6], under the assumption the the solution is sufficiently smooth.

90. Aproximaçao de problemas de valores de fronteira singulares usando subsoluçoes e supersoluçoes

Lima PM, Oliveira AM

Journal Paper

Trends in Computational and Applied Mathematics. 2000 Jun 1;1(2):401-14

91. Numerical solution of a singular boundary-value problem in non-Newtonian fluid mechanics

Lima PM, Carpentier M

Journal Paper

Computer physics communications., 126(2000) 114-20

Abstract

In the present paper we consider the second-order boundary-value problem

(1) g″=1qugq, 0⩽u<1

with q<0. We search for a positive solution of (1) which satisfies the boundary conditions

g′(0)=g(1)=0.

This problem arises in the boundary-layer theory for non-Newtonian fluids. In order to obtain numerical solutions, we use two different iterative methods and a finite-difference scheme. A variable substitution is used in order to improve the approximation and the convergence is accelerated by means of extrapolation methods. Numerical results for different values of q are given and compared with the results obtained by other authors.

92. Iterative methods for a singular boundary-value problem

Lima PM, Carpentier MP

Journal Paper

Journal of computational and applied mathematics. 1999 Nov 15;111(1-2):173-86

Abstract

We consider a second-order nonlinear ordinary differential equation of the form

y″=1qxyq, 0⩽x<1

where q<0, with the boundary conditions

y′(0)=y(1)=0.

This problem arises in boundary layer equations for the flow of a power-law fluid over an impermeable, semi-infinite flat plane. We show that classical iterative schemes, such as the Picard and Newton methods, converge to the solution of this problem, in spite of the singularity of the solution, if we choose an adequate initial approximation. Moreover, we observe that these methods are more efficient than the methods used before and may be applied to a larger range of values of q. Numerical results for different values of q are given and compared with the results obtained by other authors.

93. Asymptotic expansions and numerical approximation of nonlinear degenerate boundary-value problems

Lima PM, Carpentier MP

Journal Paper

Applied numerical mathematics. 1999 May 10;30(1):93-111

Abstract

In the present paper we are concerned with boundary-value problems (BVP) for the generalized Emden–Fowler equations. Asymptotic expansions of the solution are obtained near the endpoints. We use a finite-difference scheme to approximate the solution and the convergence is accelerated by means of extrapolation methods. A variable substitution is introduced to diminish the effect of the singularity at the origin. Numerical results, obtained by different methods, are presented for two particular cases.

94. An extrapolation method for a Volterra integral equation with weakly singular kernel

Lima P, Diogo T

Journal Paper

Applied Numerical Mathematics. 1997 Aug 1;24(2-3):131-48

Abstract

In this work we consider second kind Volterra integral equations with weakly singular kernels. By introducing some appropriate function spaces we prove the existence of an asymptotic error expansion for Euler's method. This result allows the use of certain extrapolation procedures which is illustrated by means of some numerical examples.

95. Numerical methods and asymptotic error expansions for the Emden-Fowler equations

Lima PM

Journal Paper

Journal of computational and applied mathematics. 1996 Jun 28;70(2):245-66

Abstract

In the present paper we analyse a numerical method for computing the solution of some boundary-value problems for the Emden-Fowler equations. The differential equations are discretized by a finite-difference method and we derive asymptotic expansions for the discretization error. Based on these asymptotic expansions, we use an extrapolation algorithm to accelerate the convergence of the numerical method.

96. Convergence acceleration for boundary value problems with singularities using the E-algorithm

Lima PM, Graça MM

Journal Paper

Journal of computational and applied mathematics. 1995 Jul 31;61(2):139-64

Abstract

In the present work we use the E-algorithm to accelerate the convergence of finite-difference schemes for ordinary differential equations. As a model, a boundary-value problem for a second-order differential equation is considered, where the right-hand side may have different kinds of singularities. An asymptotic error expansion is obtained, enabling the use of the E-algorithm to accelerate the convergence. The applicability and the efficiency of the E-algorithm are discussed and illustrated by numerical examples.

97. Richardson extrapolation in boundary value problems for differential equations with nonregular right-hand side

Lima PM

Journal Paper

Journal of Computational and Applied Mathematics. 1994 May 20;50(1-3):385-400

Abstract

When the finite-difference method is used to solve initial- or boundary value problems with smooth data functions, the accuracy of the numerical results may be considerably improved by acceleration techniques like Richardson extrapolation. However, the success of such a technique is doubtful in cases were the right-hand side or the coefficients of the equation are not sufficiently smooth, because the validity of an asymptotic error expansion — which is the theoretical prerequisite for the convergence analysis of the Richardson extrapolation — is not a priori obvious. In this work we show that the Richardson extrapolation may be successfully applied to the finite-difference solutions of boundary value problems for ordinary second-order linear differential equations with a nonregular right-hand side. We present some numerical results confirming our conclusions.

98. Numerical solution of boundary value problems for the Emden-Fowler equations using extrapolation methods

Lima PM

Conference Proceeding

Hellenic European Research on Mathematics and Informatics (Hermis 94), Lipitakis. E.A, ed.., Athens,1994, p. 803-812

99. Problemas de Equações Diferenciais Ordinárias

M.L.Krasnov, A.I.Kisseliov, G.I.Makarenko

Translated Book

McGraw-Hill , Lisboa

Selected problems on ordinary differential equations, translated from Russian to Portuguese by P.M.Lima.

100. A program for deriving recoupling coefficients formulae

Lima PM

Journal Paper

Computer physics communications. 1991 Jul 1;66(1):89-98.

Abstract
The program we describe in this paper derives the formulae needed to evaluate the recoupling coefficients which arise in the calculation of matrix elements by the methods of Racach algebra, using the graphical techniques introduced by Jucys et al. and El-Baz et al.. The recoupling coefficients are represented by diagrams, which are stored in the computer memory in the form of arrays. These diagrams are transformed according to given rules, so that any diagram may be decomposed into a product of simple standard diagrams. Using this method, any given recoupling coefficient may be expressed as a weighted sum of 6j-symbols products, affected by phase factors. The obtained formulae are stored in such a form that they may be directly used by another program to calculate the concrete values of the coefficients.

101. A new program for calculating matrix elements in atomic structure

Lima PM

Journal Paper

Computer physics communications. 1991 Jul 1;66(1):99-114

Abstract

The solution of many problems concerning the electronic structure of atoms requires the evaluation of the matrix elements of the Hamiltonian operator, including the electrostatic interaction. These matrix elements may be expressed as weighted sums of radial integrals. The program we describe in this paper evaluates the coefficients of the Slater integrals and, if these are given, computes all the matrix elements for a given set of configurations. This program has nearly the same purposes as Hibbert's program (Comput. Phys. Commun. 1 (1969) 359) and is also based on the Racach techniques. The main difference between this algorithm and the cited one is the method used to calculate the recoupling coefficients. While Hibbert's programs use Burke's algorithm (Comput. Phys. Commun. 1 (1970) 241) to calculate these coefficients, in our program they are computed using the graphical techniques developed by Jucys et al. (Mathematical Apparatus of the Theory of Angular Momenta, Israel Program for Scientific Translation, Jerusalem, 1962). According to this method, that we describe in another paper (Comput. Phys. Commun. 66 (1991) 89, this issue) the formulae needed to calculate the recoupling coefficients are previously derived and simplified (as a first step of the program). The use of this method may considerably reduce the running time, specially in the case of large configuration interaction matrices.

102. Fórmulas Mathemáticas

Tsypkin A. an Tsypkin G.

Translated Book

Mir Publishers, Moscow,1990

Popular handbook of mathematical formulae, translated from Russian to Portuguese by P.M.Lima.

103. Fracções Contínuas

N. Beskin

Translated Book

Mir Publishers, Moscow, 1987

This book explains the theory of continued fractions in a simple language, for students and other readers interested in Mathematics. It was translated from Russian to Portuguese by P.M. Lima and published in 1987 by Mir Publishers , Moscow ; published again in 2011 by Ulmeiro, Lisbon.

104. Princípios de Mecânica Quântica

V. Fock

Translated Book

Mir Publishers, Moscow, 1986

This famous book on Quantum Mechanics was translated from Russian to Portuguese, by the Mir Editors, and PM Lima was one of the translaters.

105. A Demonstração em Geometria

A. Fetissov

Translated Book

Mir Publishers, Moscow, 1985

This is a book for a wide audience of readers: students and other people interested in Mathematics. It is concerned with the importance of proofs in Geometry. It was translated by P.M. Lima from Rsussian to Portuguese. It was published by the Mir Publishers, in Moscow, in 1985, and later in 2001, by Ulmeiro, Lisbon.

Pedro Lima

Full Professor

Mathematics Department of Instituto Superior Tecnico

Publications

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1. Numerical solution of nonlinear third-kind Volterra integral equations using an iterative collocation method

2. An efficient spectral method for solving third-kind Volterra integral equations with non-smooth solutions

3. Numerical Simulation of the Stochastic Neural Field Equation with Delay

4. Accurate Ito-Taylor-Discretization-Based State Estimation in Stochastic Neural Field Equations with Infinte Signal Transmissiion Rate

5. Pattern Recognition Facilities of Extended Kallman Filtering in Stochastic Neural Fields

6. Numerical solution of the stochastic neural field equation with applications to working memory

7. Numerical simulations of one-and two-dimensional stochastic neural field equations with delay

8. Sequential method for fast neural population activity reconstruction in the cortex from incomplete noisy measurements

9. Iterative continuous collocation method for solving nonlinear Volterra Integral Equations

10. Analysis of the Euler and trapezoidal discretization methods for the numerical solution of nonlinear functional Volterra integral equations of Urysohn type

11. Reconstruction of Hidden States in Stochastic Neural Field Equations with Infinite Signal Transmission Rate

12. Numerical solution of the neural field equation in the presence of random disturbance.

13. Mathematical Modeling of Working Memory in the Presence of Random Disturbance using Neural Field Equations

14. Numerical solution of a class of third-kind Volterra integral equations using Jacobi wavelets

15. Iterative collocation method for solving a class of nonlinear weakly singular Volterra integral equations

16. Numerical solution of variable-order fractional differential equations using Bernoulli polynomials

17. Numerical Solution of the Time Fractional Cable Equation

18. Accuracy study in numerical simulations to stochastic neural field equations

19. Legendre wavelet collocation method combined with the Gauss–Jacobi quadrature for solving fractional delay-type integro-differential equations

Abstract

20. A numerical method for finite-part integrals

21. Apontamentos de Matemática Computacional

22. Numerical Investigation of a Neural Field Model Including Dendritic Processing

23. A novel Lagrange operational matrix and tau-collocation method for solving variable-order fractional differential equations

24. Numerical Investigation of Stochastic Neural Field Equations

25. A numerical approach for solving fractional optimal control problems using modified hat functions

26. An improved composite collocation method for distributed-order fractional differential equations based on fractional Chelyshkov wavelets,

27. Effective numerical solution to two-dimensional stochastic neural field equations.

28. Numerical simulation of neural fields with finite transmission speed and random disturbance

29. Numerical investigation of stochastic neural field equations

30. Numerical simulations of two-dimensional neural fields with applications to working memory

31. Numerical solution of integro-differential equations arising from singular boundary value problems

Abstract

32. An effective numerical method for solving fractional pantograph differential equations using modification of hat functions

Abstract

33. Numerical Solution of a Third-Kind Volterra Integral Equation Using an Operational Matrix Technique

34. Numerical solution of nonlinear fractional integro-differential equations with weakly singular kernels via a modification of hat functions

Abstract

35. Uma abordagem matemática do Sudoku

36. A Novel computational approach to singular free boundary problems in ordinary differential equations

Abstract

37. Smoothing transformation and spline collocation for weakly singular Volterra integro-differential equations

Abstract

38. Computational Methods for Two-Dimensional Neural Fields

39. Numerical Analysis and Optimization

40. Numerical investigation of noise induced changes to the solution behaviour of the discrete FitzHugh–Nagumo equation

Abstract

41. On High Order Barycentric Root-Finding Methods

Abstract

42. Numerical solution of the neural field equation in the two-dimensional case

Abstract

43. Numerical Simulations in Two-Dimensional Neural Fields

44. Numerical Investigation of the Two-dimensional Neural Field Equation with Delay

ABSTRACT

45. Root finding by high order iterative methods based on quadratures

Abstract

46. Mathematical Modelling and Numerical Simulations in Nerve Conduction

47. An integral method for the numerical solution of nonlinear singular boundary value problems

48. Computational methods for a mathematical model of propagation of nervous signals in myelinated axons

Abstract

49. Analysis and numerical approximation of singular boundary value problems with the p-Laplacian in fluid mechanis

Abstract

50. Density profile equation with p-Laplacian: analysis and numerical simulation

Abstract

51. Existence and uniqueness of solutions to weakly singular integral-algebraic and integrodifferential equations

Abstract

52. Analysis and numerical methods for fractional differential equations with delay,

Abstract

53. Numerical solution of a class of two-dimensional nonlinear Volterra integral equations using Legendre polynomials

Abstract

54. Analysis and Computational Approximation of a Forward-Backward Equation Arising in Nerve Conduction

55. Método da Bissecção

56. Método da Bissecção (Exemplo)

57. Método de Newton

58. Método de Newton (exemplo)