Theoretical and numerical treatment for the fractal-fractional model of pollution for a system of lakes using an efficient numerical technique

Abstract

This article proposes an efficient simulation to investigate the fractal-fractional (FF) pollution model's solution behavior for a network of three lakes connected by channels. With the aid of an efficient numerical technique for integration, the system of fractional differential equations (FDES) is numerically integrated. Special attention is given to studying the uniqueness and existence of the suggested model's solution by applying the Picard-Lindelöf's theorem and Banach's fixed point theorem. In this work, we deal with three cases (periodic, exponentially decaying, and linear) of the input models. The results are compared with that of the fourth-order Runge-Kutta technique (RK4) as well as the q-homotopy analysis transform method. The given figures for studying and simulating the proposed system through different values of FF-operators and comparisons show that the solutions for the system demonstrated the dynamic and natural behavior of the model. Our findings show that the used technique provides a straightforward and efficient tool to solve such problems. The key benefit of the suggested method is that it only requires a few easy steps, doesn't produce secular terms, and doesn't rely on a perturbation parameter.

Keywords

• Model of pollution for system of lakes
• Fractal-fractional derivative
• Picard-Lindelöf's theorem
• Banach fixed point theorem
• Numerical integration
• RK4