Solution of initial and boundary value problems by the variational iteration method

Date

2014-03-15

Author

Altintan, D.
Uğur, Ömür

Metadata

Show full item record

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Item Usage Stats

246
views

0
downloads

The Variational Iteration Method (VIM) is an iterative method that obtains the approximate solution of differential equations. In this paper, it is proven that whenever the initial approximation satisfies the initial conditions, vim obtains the solution of Initial Value Problems (IVPs) with a single iteration. By using this fact, we propose a new algorithm for Boundary Value Problems (BVPs): linear and nonlinear ones. Main advantage of the present method is that it does not use Green's function, however, it has the same effect that it produces the exact solution to linear problems within a single, but simpler, integral. In order to show the effectiveness of the method we give some examples including linear and nonlinear BVPs.

Subject Keywords

Variational iteration method, Initial value problems, Boundary value problems, Green's function

URI

https://hdl.handle.net/11511/32157

Journal

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

DOI

https://doi.org/10.1016/j.cam.2013.07.012

Collections

Graduate School of Applied Mathematics, Article

Suggestions

OpenMETU
Core

A Rayleigh–Ritz Method for Numerical Solutions of Linear Fredholm Integral Equations of the Second Kind Kaya, Ruşen; Taşeli, Hasan (2022-01-01) A Rayleigh–Ritz Method is suggested for solving linear Fredholm integral equations of the second kind numerically in a desired accuracy. To test the performance of the present approach, the classical one-dimensional Schrödinger equation -y″(x)+v(x)y(x)=λy(x),x∈(-∞,∞) has been converted into an integral equation. For a regular problem, the unbounded interval is truncated to x∈ [ - ℓ, ℓ] , where ℓ is regarded as a boundary parameter. Then, the resulting integral equation has been solved and the results are co...
Variational iteration method for Sturm-Liouville differential equations ALTINTAN, DERYA; Uğur, Ömür (2009-07-01) In this article, He's variational iteration method is applied to linear Sturm-Liouville eigenvalue and boundary value problems, including the harmonic oscillator. In this method, solutions of the problems are approximated by a set of functions that may include possible constants to be determined from the boundary conditions. By computing variations, the Lagrange multipliers are derived and the generalised expressions of variational iterations are constructed. Numerical results show that the method is simple...
Application of a Robust Multigrid Technique for the Parallel Solution of Initial-Boundary Value Problems Martynenko, S.I.; Gökalp, İskender; Bakhtin, V.A.; Karaca, Mehmet; Toktaliev, P.D.; Semenev, P.A. (2022-12-01) This article is devoted to the construction of a parallel multigrid algorithm for the numerical solution of (non)linear initial-boundary value problems (implicit schemes) based on a robust multigrid technique (RMT). A distinctive feature of the proposed algorithm is the possibility of the parallel solution of initial-boundary value problems and initial-boundary value problems in a unified manner involving 3m independent computers (threads, if the OpenMP parallelization technology is used), m = 1, 2, 3, …. C...
Generalisation of the Lagrange multipliers for variational iterations applied to systems of differential equations ALTINTAN, DERYA; Uğur, Ömür (2011-11-01) In this paper, a new approach to the variational iteration method is introduced to solve systems of first-order differential equations. Since higher-order differential equations can almost always be converted into a first-order system of equations, the proposed method is still applicable to a wide range of differential equations. This generalised approach, unlike the classical method, uses restricted variations only for nonlinear terms by generalising the Lagrange multipliers. Consequently, this allows us t...
Boundary value problems for higher order linear impulsive differential equations Uğur, Ömür; Akhmet, Marat (2006-07-01) In this paper higher order linear impulsive differential equations with fixed moments of impulses subject to linear boundary conditions are studied. Green's formula is defined for piecewise differentiable functions. Properties of Green's functions for higher order impulsive boundary value problems are introduced. An appropriate example of the Green's function for a boundary value problem is provided. Furthermore, eigenvalue problems and basic properties of eigensolutions are considered. (c) 2006 Elsevier In...

Citation Formats

D. Altintan and Ö. Uğur, “Solution of initial and boundary value problems by the variational iteration method,” JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, pp. 790–797, 2014, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/32157.