Application of Fixed Point Theory and Solitary Wave Solutions for the Time-Fractional Nonlinear Unsteady Convection-Diffusion System

In this article, the two-dimensional time fractional unsteady convection-diffusion system is under consideration. The convection-diffusion system of nonlinear partial differential equations has remained a uniform fascination for scientists owing to its energetic significance as well as its possession of a broad spectrum of practical and physical applications. In particular, these practicable implications include turbulence, heat transfer, fluid flow, traffic flow, and modeling of gas dynamics. The Caputo operator is applied for the fractional order derivatives and their inversion. The existence of results and uniqueness is proved by applying the fixed point theory with the help of some well-known results and theorems such as the contraction mapping theorem with Lipschitz condition, and Schauder’s fixed point theorem. Mainly, we find the exact solitary wave solutions of the underlying model. For this sake, the new extended direct algebraic method is applied and the solutions are gained in the form of dark, singular, complex, combo, trigonometric and rational solutions. Further, we draw 3D plots to show the behavior of these solutions by choosing the different values of parameters.