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

Altintan, D.
Uğur, Ömür
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.


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...
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...
An Asymptotic-Numerical Hybrid Method for Solving Singularly Perturbed Linear Delay Differential Equations
Cengizci, Süleyman (Hindawi Limited, 2017)
In thiswork, approximations to the solutions of singularly perturbed second-order linear delay differential equations are studied. We firstly use two-term Taylor series expansion for the delayed convection term and obtain a singularly perturbed ordinary differential equation (ODE). Later, an efficient and simple asymptotic method so called Successive Complementary Expansion Method (SCEM) is employed to obtain a uniformly valid approximation to this corresponding singularly perturbed ODE. As the final step, ...
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: