Error Analysis of An Explicit Finite Difference Approximation for the Space Fractional Wave Equations

In this paper, the space fractional wave equation (SFWE) is numerically studied, where the fractional derivative is defined in the sense of Caputo. An explicit finite difference approximation (EFDA) for SFWE is presented. The stability and the error analysis of the EFDA are discussed. To demonstrate the effectiveness of the approximated method, some test examples are presented

An efficient dynamical systems method for solving singularly perturbed integral equations with noise

In this paper we apply the dynamical systems method (DSM) proposed by A. G. Ramm, and the variational regularization method (VRM), to obtain numerical solution to some singularly perturbed ill-posed problems contaminated by noise. The results obtained by these methods are compared to the exact solution for the model problems. It is found that the dynamical systems method is preferable because it is easier to apply, highly stable, robust, and it always converges to the solution even for large size models

Cardinal functions for Legendre pseudo-spectral method for solving the integro-differential equations

In this article, we present a new numerical method to solve the integro-differential equations (IDEs). The proposed method uses the Legendre cardinal functions to express the approximate solution as a finite series. In our method the operational matrix of derivatives is used to reduce IDEs to a system of algebraic equations. To demonstrate the validity and applicability of the proposed method, we present some numerical examples. We compare the obtained numerical results from the proposed method with some other methods. The results show that the proposed algorithm is of high accuracy, more simple and effective

Numerical simulation of fractional Cable equation of spiny neuronal dendrites

In this article, numerical study for the fractional Cable equation which is fundamental equations for modeling neuronal dynamics is introduced by using weighted average of finite difference methods. The stability analysis of the proposed methods is given by a recently proposed procedure similar to the standard John von Neumann stability analysis. A simple and an accurate stability criterion valid for different discretization schemes of the fractional derivative and arbitrary weight factor is introduced and checked numerically. Numerical results, figures, and comparisons have been presented to confirm the theoretical results and efficiency of the proposed method

Two computational algorithms for the numerical solution for system of fractional differential equations

In this paper, two efficient numerical methods for solving system of fractional differential equations (SFDEs) are considered. The fractional derivative is described in the Caputo sense. The first method is based upon Chebyshev approximations, where the properties of Chebyshev polynomials are utilized to reduce SFDEs to system of algebraic equations. Special attention is given to study the convergence and estimate the error of the presented method. The second method is the fractional finite difference method (FDM), where we implement the Grünwald–Letnikov’s approach. We study the stability of the obtained numerical scheme. The numerical results show that the approaches are easy to implement implement for solving SFDEs. The methods introduce a promising tool for solving many systems of linear and non-linear fractional differential equations. Numerical examples are presented to illustrate the validity and the great potential of both proposed techniques

Optimal control for a fractional tuberculosis infection model including the impact of diabetes and resistant strains

The objective of this paper is to study the optimal control problem for the fractional tuberculosis (TB) infection model including the impact of diabetes and resistant strains. The governed model consists of 14 fractional-order (FO) equations. Four control variables are presented to minimize the cost of interventions. The fractional derivative is defined in the Atangana-Baleanu-Caputo (ABC) sense. New numerical schemes for simulating a FO optimal system with Mittag-Leffler kernels are presented. These schemes are based on the fundamental theorem of fractional calculus and Lagrange polynomial interpolation. We introduce a simple modification of the step size in the two-step Lagrange polynomial interpolation to obtain stability in a larger region. Moreover, necessary and sufficient conditions for the control problem are considered. Some numerical simulations are given to validate the theoretical results. Keywords: Tuberculosis model, Diabetes and resistant strains, Atangana-Baleanu fractional derivative, Lagrange polynomial interpolation, Nonstandard two-step Lagrange interpolation metho

Nonstandard finite difference method for solving the multi-strain TB model

In this paper, numerical studies for the mathematical model of tuberculosis (TB), that incorporates three strains, i.e., drug - sensitive, emerging multi - drug resistant(MDR) and extensively drug - resistant (XDR), are presented. Special class of numerical methods, known as nonstandard finite difference method (NSFDM) is introduced to solve this model. Numerical stability analysis of fixed points are studied. The obtained results by NSFDM are compared with other known numerical methods such as implicit Euler method and fourth-order Runge–Kutta method (RK4). It is concluded that NSFD scheme preserves the positivity of the solution and numerical stability in larger region than the other methods

Cluster computing for the large scale discrete fractional Cable equation

This paper presents a numerical simulation technique for the fractional Cable equation in large scale domain. Special attention is given to the parallel execution of the fractional weighted average finite difference method (FWA-FDM) on distributed system with explicit message passing, where the fractional derivative is defined in Riemann–Liouville sense. The resultant huge system of equations is studied using precondition conjugate gradient method (PCG), with the implementation of cluster computing on it. The proposed approach fulfills the suitability for the implementation on Linux PC cluster through the minimization of inter-process communication. To examine the efficiency and accuracy of the proposed method, numerical test experiments using different number of the Linux PC cluster nodes are studied. The performance metrics clearly show the benefit of using the proposed approach on the Linux PC cluster in terms of execution time reduction and speedup with respect to the sequential running in a single PC

On the numerical solution of space fractional order diffusion equation via shifted Chebyshev polynomials of the third kind

In this paper, we propose a numerical scheme to solve space fractional order diffusion equation. Our scheme uses shifted Chebyshev polynomials of the third kind. The fractional differential derivatives are expressed in terms of the Caputo sense. Moreover, Chebyshev collocation method together with the finite difference method are used to reduce these types of differential equations to a system of algebraic equations which can be solved numerically. Numerical approximations performed by the proposed method are presented and compared with the results obtained by other numerical methods. The results reveal that our method is a simple and effective numerical method