On the closed form exact solitary wave solutions for the time-fractional order nonlinear unsteady convection diffusion system with unique-existence analysis.

In this paper, the time-fractional two-dimensional unsteady convection diffusion system governed by nonlinear partial differential equations captured the time evaluation of disturbance and addressing modeling real-world phenomena such as turbulence, traffic flow, heat and fluid transport, and gas dynamics. Fractional derivatives are introduced to enable a more generalized representation of diffusion behaviors. To establish solution the unique existence, we apply a fixed-point strategy via Lipschitz continuity. By employing the generalized exponential rational function method, we derive exact solitary wave solutions in the forms of dark, singular, complex, combo, trigonometric, and rational waves. Furthermore, 3D plots illustrate solution dynamics under various parameter choices providing deep exploration into the system's behavior and demonstrating the applicability of fractional order models in advanced technological and scientific contexts. [ABSTRACT FROM AUTHOR]