Variational Iteration Method for Linear and Nonlinear Systems of Integro-Differential Equations (original) (raw)
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International Journal of Scientific & Engineering Research, 2020
In this paper, we introduce some basic idea of Variational iteration method for short (VIM) to solve the Volterra's integro-differential equations. The VIM is used to solve effectively, easily, and accurately a large class of non-linear problems with approximations which converge rapidly to accurate solutions. For linear problems, it's exact solution can be obtained by only one iteration step due to the fact that the Lagrange multiplier can be exactly identified. It is to be noted that the Lagrange multiplier reduces the iteration on integral operator and also minimizes the computational time. The method requires no transformation or linearization of any forms. Two numerical examples are presented to show the effectiveness and efficiency of the method. Also, we compare the result with the result from Homotopy perturbation method (HPM). Finally, we investigate the absolute difference between variational iteration method and homotopy perturbation method and draw the graph of difference function by using Mathematica.
International Journal of Mechanical and Production Engineering Research and Development (IJMPERD) , 2019
In this paper, we present a modified variational iteration method for solving Fredholm integro-differential equations. This study provides an analytical approximation to determine the behavior of the solution. Moreover, it proves the existence and uniqueness in results and convergence of the solution. Finally, some examples are included to demonstrate the validity and applicability of the proposed technique.
Variational iteration method for solving Seventh order integro-differential equations
2010
In the recent years many different method were proposed to solve boundary value problems(BVPs), such as homotopy perturbation method (HPM)[1,2], variational iteration method (VIM)[3,4] and Modified Decomposition method (MDM)[5]. Recently Sweilam [6] implemented the VIM to solve fourth order integro-differential equations. In this paper, we apply the variational iteration method proposed by Ji-Huan He [7-10] to find approximate solutions for boundary value problems of seventh order integro-differential equations . To illustrate the basic idea of VIM, we consider following general nonlinear system:
In this paper, a comparative study of Adomian Decomposition Method (ADM) and Variational Iteration Method (VIM) were considered on various types of integrodifferential equation; which are Fredholm, Volterra and Fredholm-Volterra equations. From the examples considered, it was observed that these methods were compared favorably with the exact solution. VIM has an advantage over ADM due to non-requirement of Adomian polynomial and hence converges faster to the exact solution for some nonlinear problems.
A Reliable Approach for Higher-order Integro-differential Equations
2008
In this paper, we apply the variational iteration method (VIM) for solving higher-order integro differential equations by converting the problems into system of integral equations. The proposed technique is applied to the re-formulated system of integro-differential equations. Numerical results show the accuracy and efficiency of the suggested algorithm. The fact that the VIM solves nonlinear problems without calculating Adomian's polynomials