Numerical solution of fractional differential equations via a Volterra integral equation approach

Abstract

Abstract:The main focus of this paper is to present a numerical method for the solution of fractional differential equations. In this method, the properties of the Caputo derivative are used to reduce the given fractional differential equation into a Volterra integral equation. The entire domain is divided into several small domains, and by collocating the integral equation at two adjacent points a system of two algebraic equations in two unknowns is obtained. The method is applied to solve linear and nonlinear fractional differential equations. Also the error analysis is presented. Some examples are given and the numerical simulations are also provided to illustrate the effectiveness of the new method.PACS (

Application of Intelligent Paradigm through Neural Networks for Numerical Solution of Multiorder Fractional Differential Equations

“…The spectral approximation method is used by Li [ 30 ] to compute the fractional derivative and integral and also presents the pseudo-spectral approximation technique for some classes of FDEs. Esmaeili [ 31 ] developed a numerical technique in which the properties of the Caputo derivative were used to reduce the fractional differential equation into a Volterra integral equation.…”

Solution of nonlinear weakly singular Volterra integral equations using the fractional‐order Legendre functions and pseudospectral method

“…In this example, we consider the following fractional Riccati equation [45]: 2.87 10 10 4.58 10 11 D 1=5 7.93 10 6 4.08 10 9 7.72 10 11 4.12 10 12 -D 1=8 3.37 10 3 9.16 10 12 5.26 10 14 -- D 1=4 2.58 10 5 1.20 10 7 5.14 10 9 5.42 10 10 7.69 10 11 D 1=5 1.13 10 5 7.71 10 9 1.45 10 10 6.96 10 12 -D 1=8 3.37 10 3 1.73 10 11 1.25 10 13 --…”

“…The pseudospectral methods were named after their similarity to the spectral techniques. Spectral methods are a powerful tool for solving the partial and ordinary differential and integral equations and fractional problems [3][4][5][6][7][8][9][10][11][12][13]. In the pseudospectral method, the unknown function is expanded as a global polynomial interpolants based on some suitable points.…”

Nonstandard Gauss—Lobatto quadrature approximation to fractional derivatives

“…Although there is comprehensive literature on the numerical methods for solving equations involving fractional derivatives and integrals (cf. [8,9,11,3,10,35]), there seems to exist a few literature on automatic quadrature for the fractional derivatives, see e.g. [20,24,39].…”