Abstract
In this work, the modified Laplace Adomian decomposition method (LADM) is applied to solve the Burgers’ equation. In addition, example that illustrate the pertinent features of this method is presented, and the results of the study is discussed. We prove the convergence of LADM applied to the Burgers’ equation. Also, Burgers’ equation has some discontinuous solutions because of effects of viscosity term. These discontinuities raise phenomenon of shock waves. Some explicit and implicit numerical methods have been experimented on Burgers’ equation but these schemes have not been seen proper in this case because of their conditional stability and existence of viscosity term. We consider two types of box schemes and implement on Burgers’ equation to get better results with no artificial wiggles.
Highlights
In the recent decade, the study of nonlinear partial differential equations modelling physical phenomena, has become an important tool
Agadjanov [22] solved the Duffing equation by this method. This numerical technique basically illustrates how the Laplace transform may be used to approximate the solutions of the nonlinear partial differential equations by manipulating the decomposition method
We carefully applied a reliable modification of Laplace decomposition method for this equation
Summary
The study of nonlinear partial differential equations modelling physical phenomena, has become an important tool. The ADM involves separating the equation under investigation into linear and nonlinear portions This method generates a solution in the form of a series whose terms are determined by a recursive relation using the Adomian polynomials. Agadjanov [22] solved the Duffing equation by this method This numerical technique basically illustrates how the Laplace transform may be used to approximate the solutions of the nonlinear partial differential equations by manipulating the decomposition method. Our aim in this paper is to obtain the numerical and analytical solutions by using the modified Laplace Adomian decomposition method and explicit and implicit numerical methods. The remainder of the paper is organized as follows: In Sections 2 and 3, a brief discussions for the modified Laplace Adomian decomposition method and application of this method are presented and approximate solution for one example is obtained. The solution through the modified Adomian decomposition method highly depends upon the choice of k0(x, t) and k1(x, t), where k0(x, t) and k1(x, t) represent the terms arising from the source term and prescribed initial conditions
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