Numerical solution of nonlinear Volterra-Fredholm integro-differential equations using Homotopy Analysis Method

Abstract

The use of homotopy analysis method to approximate the solution of nonlinear Volterra-Fredholm integro-differential equation is proposed in this paper. In this case, the existence and uniqueness of the obtained solution and convergence of the method are proved. The accuracy of the proposed numerical scheme is examined by comparing with the modified Adomian decomposition method and Taylor polynomial method in the example. Also, the cost of operations in the algorithms are obtained to demonstrate the efficiency of the presented method.

The use of homotopy methods for solving nonlinear foam drainage equation

“…(2.26), as the embedding parameter q increases from 0 to 1, ϕ (x,t; q) varies continuously from the initial approximation u 0 (x,t) to the exact solution u(x,t). Such a kind of continuous variation is called deformation in homotopy [39,40,41,42,43]. Due to Taylor's theorem, ϕ (x,t; q) can be expanded in a power series of q as follows 27) where,…”

Convergence of Iterative Methods applied to Boussinesq equation

“…(2.27), as the embedding parameter q increases from 0 to 1, ϕ (x,t; q) varies continuously from the initial approximation u 0 (x,t) to the exact solution u(x,t). Such a kind of continuous variation is called deformation in homotopy [26,27,28,29,42,43,44]. Due to Taylor's theorem, ϕ (x,t; q) can be expanded in a power series of q as follows…”

Numerical Methods for Solving the System of Volterra-Fredholm Integro-Differential Equations.

“…Recently, (Darania and Ivaz, 2008) used Taylor expansion for nonlinear Volterra-Fredholm integro-differential equations, also (Hasan and Suleiman, 2018b) used Trigonometric Functions and Laguerre Polynomials to solve mixed Volterra-Fredholm integral equation. (Fariborzi Araghi and Behzadi, 2011) used Homotopy Analysis Method for the numerical solution of nonlinear Volterra-Fredholm integro-differential equations. (Hasan and Suleiman, 2018a) shows numerical solution of Mixed Volterra-Fredholm integral equations by using Linear Programming Problem, and (Hassan T.I., Sulaiman N.A., 2017) studied Aitken method to solve Volterra-Fredholm integral equations of the second kind with Homotopy perturbation method.…”