Numerical solution of the higher-order linear Fredholm integro-differential-difference equation with variable coefficients

Abstract

a b s t r a c tThe main aim of this paper is to apply the Legendre polynomials for the solution of the linear Fredholm integro-differential-difference equation of high order. This equation is usually difficult to solve analytically. Our approach consists of reducing the problem to a set of linear equations by expanding the approximate solution in terms of shifted Legendre polynomials with unknown coefficients. The operational matrices of delay and derivative together with the tau method are then utilized to evaluate the unknown coefficients of shifted Legendre polynomials. Illustrative examples are included to demonstrate the validity and applicability of the presented technique and a comparison is made with existing results.

Improved Jacobi matrix method for the numerical solution of Fredholm integro-differential-difference equations

“…The maximum error for the corrected Jacobi polynomial solution (37) is calculated in a similar way, Present method Taylor method [23] Tau method [24] and the results are shown in Table 3 for miscellaneous values of N, M. The decrease in maximum error, as M increases, is indisputable. Finally, the third-order FIDDE has also been solved using Legendre, Gegenbauer (also Chebyshev), and Jacobi polynomials, for comparison purposes.…”

“…Also, they have been investigated using different methods by scientists [23][24][25][26][27][28]. Various numerical schemes for solving a partial integrodifferential equation are presented by Dehghan [29].…”

Legendre spectral collocation method for Fredholm integro-differential-difference equation with variable coefficients and mixed conditions

“…Problems involving these equations arise frequently in many applied areas which include engineering, mechanics, physics, chemistry, astronomy, biology, economics, potential theory, electrostatics, etc. [1][2][3][4][5][6][7].…”

“…The learned researchers Gulsu and Sezer [1] solved integro-differential-difference equations with variable coefficients by using Taylor matrix method, Laguerre collocation method (LCM) [2], and Chebyshev collocation method [3]. In [4], the higher-order linear Fredholm integro-differential-difference equations with variable coefficients have been solved by Legendre polynomials. Boubaker polynomial [5], Fibonacci collocation method (FCM) [6] and homotopy analysis method (HAM) [7] have been applied to solve Fredholm integro-differential-difference equations with variable coefficients.…”

Exact Solutions for a Local Fractional DDE Associated with a Nonlinear Transmission Line

“…where 0 < α 1, a 0 , k, and χ are arbitrary constants, ψ n is the solution of the system (11)- (13). Now, we can construct three types of exact solutions for Eq.…”