Analytical Approximate Solutions of Caputo Fractional KdV-Burgers Equations Using Laplace Residual Power Series Technique

Abstract

The KdV-Burgers equation is one of the most important partial differential equations, established by Korteweg and de Vries to describe the behavior of nonlinear waves and many physical phenomena. In this paper, we reformulate this problem in the sense of Caputo fractional derivative, whose physical meanings, in this case, are very evident by describing the whole time domain of physical processing. The main aim of this work is to present the analytical approximate series for the nonlinear Caputo fractional KdV-Burgers equation by applying the Laplace residual power series method. The main tools of this method are the Laplace transform, Laurent series, and residual function. Moreover, four attractive and satisfying applications are given and solved to elucidate the mechanism of our proposed method. The analytical approximate series solution via this sweet technique shows excellent agreement with the solution obtained from other methods in simple and understandable steps. Finally, graphical and numerical comparison results at different values of α are provided with residual and relative errors to illustrate the behaviors of the approximate results and the effectiveness of the proposed method.