Spectral technique for solving variable‐order fractional Volterra integro‐differential equations

Abstract

This article, presented a shifted Legendre Gauss‐Lobatto collocation (SL‐GL‐C) method which is introduced for solving variable‐order fractional Volterra integro‐differential equation (VO‐FVIDEs) subject to initial or nonlocal conditions. Based on shifted Legendre Gauss‐Lobatto (SL‐GL) quadrature, we treat with integral term in the aforementioned problems. Via the current approach, we convert such problem into a system of algebraic equations. After that we obtain the spectral solution directly for the proposed problem. The high accuracy of the method was proved by several illustrative examples.

A new approach for solving variable order differential equations based on Bernstein polynomials with Prabhakar function

“…For example, Laplace transform method, 28 finite difference methods, 29 Adomians decomposition method, 30 variational iteration method, 31 Bernstein polynomials (BPs) method, 32,33 a numerical method for solving the coupled systems by using BPs, 34 a numerical method base on the inverse Laplace transform method, 35 discrete mollification method, 36 second-order Runge-Kutta method, 37 Bernstein collocation method, 38 Bernoulli wavelet method, 39 Lagrange multiplier optimization, 40 explicit finite-difference method, 41 shifted Legendre polynomials method, 42 predictor-corrector method, 43 the numerical method based on the shifted Chebyshev cardinal functions, 44 the proposed method base on the Chebyshev cardinal functions for solving the variable-order fractional pantograph equation, 45 explicit and implicit Euler method, 46 and Chebyshev wavelets. 47 Also for more details we refer the interested reader to 18,[48][49][50][51][52][53][54][55][56][57][58][59] and references therein. There are several different definitions of fractional derivative and integral such as Riemann-Liouville, Grünwald-Letnikov and Caputo 2 and actually used in applied models.…”

A numerical scheme to solve variable order diffusion-wave equations

“…Hassani and Naraghirad [23] wavelets. Jiang and Guo [25] applied the reproducing kernel method to solve 2-D VO anomalous sub-diffusion equation and see [26,27].…”

Analysis of two Legendre spectral approximations for the variable-coefficient fractional diffusion-wave equation

“…In most cases, analytical methods do not work well on most of FDEs, and even if they can be solved, the expressions of solution often contain infinite series or special functions, which are complicated or difficult to calculate [12,13], so it is natural to resort to numerical approaches. Up to now, there are several numerical techniques to solve FDEs, such as the finite difference method (FDM) [14][15][16][17], the finite element method (FEM) [18][19][20], the discontinuous Galerkin method [21,22], the spectral method [23][24][25][26][27][28], and so on.…”