Computationally Efficient Laplace Transform with Modified Variational Iteration Method for Solving Fourth-Order Fractional Integro-Differential Equations

Abstract

In this paper, linear and nonlinear fourth-order Fractional Integro Differential Equations (FIDEs) with boundary value problems are  solved by Laplace Transform with Modified Variational IterationMethod (LT-MVIM). A new technique based on the VIM is introduced to remove the random choice of initial guess by setting a specific rule depends on unknown parameters. These parameters contributed to the increase in the number of terms of the polynomial approximation and its degree, which, in turn, accelerates the convergence and increases the accuracy from one iteration compared to the standard method, where the initial approximation is still randomly chosen. Moreover, the standard method requires an infinite number of iterations, which need massive calculations in each iteration.
Some examples are given in order to show the accuracy of the solutions obtained by the proposed method. Furthermore, comparisons are made between the solutions obtained by the proposed method and Laplace Transform Variational Iteration Method (LT-VIM) based on the exact solutions, revealing that the LT-MVIM contributes to accelerating the convergence of approximate solution to the  exact solution by reducing the computational work to obtain the approximate solution using one iteration. Whereas, LT-VIM needs more iterations to obtain a suitable approximate solution, which results in an increase in the computational workload.