Numerical solution of system of Fredholm-Volterra integro-differential equations using Legendre polynomials

Abstract

In this paper, two collocation methods based on the shifted Legendre polynomials are proposed for solving system of nonlinear Fredholm-Volterra integro-differential equations. The equation considered in this paper involves the derivative of unknown functions in the integral term, which makes its numerical solution more complicated. We first introduce a single-step Legendre collocation method on the interval [0, 1]. Next, a multi-step version of the proposed method is derived on the arbitrary interval [0, T] that is based on the domain decomposition strategy and specially suited for large domain calculations. The first scheme converts the problem to a system of algebraic equations whereas the later solves the problem step by step in subintervals and produces a sequence of systems of algebraic equations. Some error estimates for the proposed methods are investigated. Numerical examples are given and comparisons with other methods available in the literature are done to demonstrate the high accuracy and efficiency of the proposed methods.

Haar wavelets method for solving class of coupled systems of linear fractional Fredholm integro-differential equations

“…The works by Abbaszadeh et al [27] , Alnobani and Al Yaqin [28] , Shi et al [29] , and Shirani et al [30] focus on specific wavelet-based methods such as Legendre wavelets and Chebyshev wavelets. These studies propose efficient algorithms and numerical techniques for solving different types of integro-differential equations, including Fredholm-Volterra equations and coupled differential-integral equations.…”

Robust QLM-SCFTK matrix approach applied to a biological population model of fractional order considering the carrying capacity

“…In addition to Chebyshev polynomials [13,18], different (orthogonal) polynomial functions have been utilized inside the spectral collocation approaches in the literature. Among others, we mention Dickson [25], Bessel [14,36,48], Legendre [37], Benoulli [2], Vieta-Fibonacci [1], and Jacobi [47], to name a few.…”