The Nonlinear Fisher's Reaction-Diffusion Equation: A Novel Fractional Derivation and Associated Theorems Numerical Simulations

Abstract

The fractional-order derivative introduced by Caputo-Fabrizio has important tools and applications, which this research examines. Used the new derivative on the nonlinear Fisher's reaction-diffusion problem to solve the updated equation iteratively. Our methodology is robust, as shown by fixed-point theory. Various fractional-order values were simulated numerically, along with essential theorems and proofs. Fractional diffusion equations, especially when particle clouds disperse faster than classical theory predicts, substitute the second spatial derivative with a fractional derivative of order less than two. Asymmetric solutions spread faster than conventional solutions. Complex diffusion and reaction are combined in fractional reaction-diffusion equations. We provide a practical numerical method for solving these equations via operator splitting. Assess numerical solution attributes and give numerical simulations to support the method. Additionally, we investigate biological applications where the response term reflects species expansion and the diffusion term indicates movement.