Efficient numerical techniques for computing the Riesz fractional-order reaction-diffusion models arising in biology

On the exact and numerical solutions to a nonlinear model arising in mathematical biology

This study acquires the exact and numerical approximations of a reaction-convection-diffusion equation arising in mathematical bi- ology namely; Murry equation through its analytical solutions obtained by using a mathematical approach; the modified exp(-Ψ( η ))-expansion function method. We successfully obtained the kink-type and singular soliton solutions with the hyperbolic function structure to this equa- tion. We performed the numerical simulations (3D and 2D) of the obtained analytical solutions under suitable values of parameters. We obtained the approximate numerical and exact solutions to this equa- tion by utilizing the finite forward difference scheme by taking one of the obtained analytical solutions into consideration. We investigate the stability of the finite forward difference method with the equation through the Fourier-Von Neumann analysis. We present the L 2 and L ∞ error norms of the approximations. The numerical and exact approx- imations are compared and the comparison is supported by a graphic plot. All the computations and the graphics plots in this study are car- ried out with help of the Matlab and Wolfram Mathematica softwares. Finally, we submit a comprehensive conclusion to this study.

Analytical Study of Variable‐Order Fractional Differential Equations With Initial and Terminal Antiperiodic Boundary Conditions

This study investigates the existence, uniqueness, and stability of solutions to Riemann–Liouville fractional differential equations with fractional variable‐order and antiperiodic boundary conditions. By employing the Banach fixed point theorem, we establish conditions for the uniqueness of solutions, while Schauder’s fixed point theorem is used to prove their existence in a Banach space. We further demonstrate Ulam–Hyers–Rassias stability, ensuring solution robustness against perturbations. The variable‐order framework enables modeling of complex systems with evolving memory, offering advantages over constant‐order models in applications such as physics and epidemiology. A concrete example illustrates the practical applicability of our results. This work provides a rigorous theoretical foundation, bridging pure mathematics with potential applications in science and engineering, and sets the stage for future numerical and applied studies.

Simple mathematical models with very complicated dynamics