Dogan Kaya

A numerical comparison of partial solutions in the decomposition method for linear and nonlinear partial differential equations

In this study, the decomposition method for solving the linear heat equation and nonlinear Burgers equation is implemented with appropriate initial conditions. The application of the method demonstrated that the partial solution in the x-direction requires more computational work when compared with the partial solution developed in the t-direction but the numerical solution in the x-direction are performed extremely well in terms of accuracy and efficiency.

On the solution of a korteweg-de vries like equation by the decomposition method

A new approach to a linear Korteweg-de Vries like equation is implemented by the Adomian decomposition method. The approach is based on the choice of a suitable differential operator which may be ordinary or partial, linear or nonlinear, deterministic or stochastic [1–4]. It allows to obtain a decomposition series analytic solution of the equation which is calculated in the form of a convergent power series with easily computable components. The inhomogeneous problem is quickly solved by observing the self-canceling “noise” terms where the sum of components vanishes in the limit. Many test modeling problems from mathematical physics, linear and nonlinear, are discussed to illustrate the effectiveness and the performance of the decomposition method. This paper is particularly concerned with the accuracy for the modeling of various linear Korteweg-de Vries like equations by the Adomian decomposition method. Its remarkable accuracy is finally demonstrated in the study of several test problems.

An application for a generalized KdV equation by the decomposition method

Abstract The explicit solutions to a generalized Korteweg–de Vries equation (KdV for short) with initial condition are calculated by using the Adomian decomposition method. Using this approach we obtained for the numerical solutions of initial-value KdV equation. Numerical illustrations on well-known KdV equation with the rational and solitary wave solutions indicate that the decomposition method is efficient and accurate. In addition, an illustration of the self canceling phenomena is also given.

Solitary wave solutions for a generalized Hirota-Satsuma coupled KdV equation

The solitary wave solutions of a generalized Hirota-Satsuma coupled KdV equation are obtained exactly by using the decomposition method. The solutions were calculated in the form of a convergent power series with easily computable components. A variety of initial value system is considered, the convergence of the method as applied to the coupled KdV equation is illustrated numerically. The present algorithm performs extremely well in terms of accuracy and simplicity.

Comparing numerical methods for the solutions of systems of ordinary differential equations

In this article, we implement a relatively new numerical technique, the Adomian decomposition method, for solving linear and nonlinear systems of ordinary differential equations. The method in applied mathematics can be an effective procedure to obtain analytic and approximate solutions for different types of operator equations. In this scheme, the solution takes the form of a convergent power series with easily computable components. This paper will present a numerical comparison between the Adomian decomposition and a conventional method such as the fourth-order Runge-Kutta method for solving systems of ordinary differential equations. The numerical results demonstrate that the new method is quite accurate and readily implemented.

A numerical solution of the sine-Gordon equation using the modified decomposition method

The decomposition method for solving the sine-Gordon equation has been implemented. By using a number of initial values, the explicit and numerical solutions of the equation are calculated in the form of convergent power series with easily computable components. The present method performs extremely well in terms of accuracy, efficiency, simplicity, stability and reliability.

An application of the ADM to seven-order Sawada-Kotara equations

We implemented the Adomian decomposition method (for short, ADM) for approximating the solution of the seventh-order Sawada-Kotera (for short, sSK) and a Laxs seventh-order KdV (for short, LsKdV) equations. By using this scheme, explicit exact solution is calculated. We obtain the exact solitary-wave solutions and numerical solutions of the LsKdV and sSK equations for the initial conditions. The numerical solutions are compared with the known analytical solutions. Their remarkable accuracy are finally demonstrated for the both seven-order equations.

A convergence analysis of the ADM and an application

In this paper by considering the decomposition scheme, we solve the modified Korteweg-de Vries equation (MKdV for short) for the initial condition. We prove the convergence of Adomian decomposition method (ADM) applied to the MKdV equation.

The Decomposition Method Applied to Solve High-order Linear Volterra-Fredholm Integro-differential Equations

In this paper, Adomians decomposition method is applied to solving high-order linear Volterra-Fredholm integro-differential equations. Comparison with the results obtained in terms of Taylors polynomials reveals that the Adomians decomposition method is more accurate and is more efficient than the Taylors expansion technique. The decomposition method also minimizes the computational difficulties of Taylors expansion approach in that the components of the decomposition series are determined elegantly by using simple integrals.

Finite difference method for solving fourth-order obstacle problems

In this article, we introduce and develop a new finite difference method for solving a system of fourth-order boundary value problems associated with obstacle, unilateral and contact problems. The convergence analysis of the new method has been discussed and it was shown that the order is four and it gives approximations, which are better than those produced by other collocation, finite difference and spline methods. Numerical examples are presented to illustrate the applications of this method. E-mail: dkaya@firat.edu.tr E-mail: eisasaid@ksu.edu.sa E-mail: kamel.alkhaled@uaeu.ac.ae