The Use of Iterative Methods to Solve Two-Dimensional Nonlinear Volterra-Fredholm Integro-Differential Equations

Abstract

In this present paper, we solve a two-dimensional nonlinear Volterra-Fredholm integrodifferential equation by using the following powerful, efficient but simple methods: (i) Modified Adomian decomposition method (MADM), (ii) Variational iteration method (VIM), (iii) Homotopy analysis method (HAM) and (iv) Modified homotopy perturbation method (MHPM). The uniqueness of the solution and the convergence of the proposed methods are proved in detail. Numerical examples are studied to demonstrate the accuracy of the presented methods.

Numerical Study of Second Painlevé Equation

“…The VIM plays an important role in recent researches for solving various kinds of problems. see for example [23,34,1,26,27,3] and the references therein. See also [22,19].…”

Solving nonlinear Fredholm integro-differential equations via modifications of some numerical methods

“…The approximate solution for solving the nonlinear Fredholm integro-dierential equation of the second kind in the complex plane by using the properties of rationalized Haar wavelet has been obtained in [14]. A two-dimensional nonlinear Volterra-Fredholm integro-dierential equation by using some iterative methods is presented [13]. In [16], the author modied the existing homotopy perturbation method to solve the high-order integro-dierential equations through canonical polynomials basis function.…”

Modified Adomian Decomposition Method for the Solution of Integro-Differential Equations

“…Behiry [8], applied the differential transformation method to high-order nonlinear Volterra-Fredholm integro-differential equations with separable kernels. Behzadi [9], solved a two-dimensional nonlinear Volterra-Fredholm integro differential equation by using the Modified Adomian Decomposition method, Variation iteration, Homotopy analysis method and Modified homotopy perturbation method.…”