Novel Analysis of Nonlinear Seventh Order Kaup-Kupershmidt Equation via the Caputo Operator

Abstract

In this work, we use two unique methodologies namely the homotopy perturbation transform method and Yang transform decomposition method to solve the fractional nonlinear seventh order Kaup-Kupershmidt (KK) problem. The physical phenomena that arise in chemistry, physics, and engineering are mathematically explained in this equation. In particular, nonlinear optics, quantum mechanics, plasma physics, fluid dynamics and so on. The provided methods are used to solve the fractional nonlinear seventh order KK problem, along with the Yang transform and the fractional Caputo derivative. The results are significant and necessary for exploring a range of physical processes. This paper uses modern approaches and the fractional operator in this situation to develop satisfactory approximations to the offered problem. To solve the fractional KK equation, we first use the Yang transform and the fractional Caputo derivative. When the suggested approaches are used, the results are compared to the exact solution. By comparing the outcomes with the precise solution using graphs and tables, we can verify the efficacy of the offered strategies. Also, the outcomes of using the suggested methods at various fractional orders are examined, demonstrating that the findings get more accurate as the value moves from fractional order to integer order. Moreover, the offered methods are innovative, simple, and quite accurate, demonstrating that they are effective approaches for resolving any differential equations.