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

AbstractThe 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

Using hybrid of block-pulse functions and bernoulli polynomials to solve fractional fredholm-volterra integro-differential equations

Fractional integro-differential equations have been the subject of significant interest in science and engineering problems. This paper deals with the numerical solution of classes of fractional Fredholm-Volterra integro-differential equations. The fractional derivative is described in the Caputo sense. We consider a hybrid of block-pulse functions and Bernoulli polynomials to approximate functions. The fractional integral operator for these hybrid functions together with the Legendre-Gauss quadrature is used to reduce the computation of the solution of the problem to a system of algebraic equations. Several examples are given to show the validity and applicability of the proposed computational procedure

He's Variational Iteration Method for Solving a Partial Differential Equation Arising in Modelling of the Water Waves

The variational iteration method is applied to solve the Kawahara equation. This method produces the solutions in terms of convergent series and does not require linearization or small perturbation. Some examples are given. The comparison with the theoretical solution shows that the variational iteration method is an efficient method

The Legendre-tau technique for the determination of a source parameter in a semilinear parabolic equation

A numerical procedure for an inverse problem concerning diffusion equation with source control parameter is considered. The proposed method is based on shifted Legendre-tau technique. Our approach consists of reducing the problem to a set of algebraic equations by expanding the approximate solution as a shifted Legendre function with unknown coefficients. The operational matrices of integral and derivative together with the tau method are then utilized to evaluate the unknown coefficients of shifted Legendre functions. Illustrative examples are included to demonstrate the validity and applicability of the presented technique