Laplace discrete decomposition method for solving nonlinear Volterra-Fredholm integro-differential equations

Abstract

In this article, a new modification of the Adomian Decomposition Method (ADM) that is called the Laplace Discrete Adomian Decomposition Method (LDADM) is applied to non-homogeneous nonlinear Volterra-Fredholm integro-differential equations. This method is based upon the Laplace Adomian decomposition method coupled with some quadrature rules of numerical integration. The performance of the proposed method is verified through absolute error measures between the approximated solutions and exact solutions. The series of experimental numerical results show that our proposed method performs in high accuracy and efficiency. The study highlights that the proposed method could be used to overcome the analytical approaches in solving nonlinear Volterra-Fredholm integro-differential equations.

Factorization of abstract operators into two second degree operators and its applications to integro-differential equations

“…There are several types of decomposition methods for solving BVPs with IDEs. The most popular is Adomyan decomposition method and its modifications, where the Adomyan polynomials are used, and approximate solutions of given BVPs are obtained [11][12][13][14][15][16][17][18][19]. Other types of decomposition method were considered in [4], [5], [20].…”

An efficient numerical method for a singularly perturbed Volterra-Fredholm integro-differential equation

“…The first and the second order difference schemes were proposed in [4,34]. In recent years, many authors have applied different methods such as homotopy analysis method, modified variational iteration method, Adomian decomposition method that is named Laplace discrete Adomian decomposition method, modified homotopy perturbation method to obtain approximate analytical solutions for Volterra, Fredholm, Volterra-Fredholm, fuzzy Volterra-Fredholm integro-differential equations in [8,9,15,[17][18][19].…”

Computational Scheme for the First-Order Linear Integro-Differential Equations Based on the Shifted Legendre Spectral Collocation Method

“…Moreover, they verified that the proposed method is superior to the Full-Sweep GaussSeidel methods and Full-Sweep Successive Over-Relaxation methods. For the nonlinear problems, Dawood et al [31] proposed a Laplace Discrete Adomian Decomposition Method to solve nonlinear Volterra-Fredholm IDEs. The proposed method was simple to execute and can be effectively used to overcome the analytical approaches in solving nonlinear Fredholm IDEs.…”

“…Combining with the method of cutting functions, squaring method, the Cauchy-Bunyakovsky inequality, and the integral inequality method, Iskandarov [32] studied a weakly first order nonlinear implicit Volterra IDE on the semiaxis. All in all, the methods proposed in reference [29][30][31][32] are synthetical and possess good computational accuracy for the solution of IDEs.…”