Lie symmetry, numerical solution with spectral method and conservation laws of Degasperis–Procesi equation by homotopy and direct methods

Abstract

ABSTRACT Partial differential equations are widely used to describe complex phenomena in various branches of science, including: physics, mechanics, etc. Therefore, obtaining high-precision target solutions and, if possible, finding exact analytical solutions of these equations play an important role in these sciences. In this paper, we first examine Lie symmetry and invariant transformations. Next, we obtain the equation solutions with the help of symmetries. We compute invariants and similarity solutions. Spectral method was used to numerically solve the equation. Then, we introduce two approaches, the Homotopy method and the direct method, which are used to construct conservation laws. Then we construct three conservation laws for the Degasperis-Procesi equation by using the direct method. Numerical solution and analysis of the Degasperis-Procesi equation by the methods proposed in this research have not been done in previous studies. Also, in Iran, this is the first time that this research has been done, which is a combination of mathematics, physics and computer science, and is considered interdisciplinary and applied.