Abstract
In this paper, we propose efficient and reliable numerical methods to solve two notable non-integer-order partial differential equations. The proposed algorithm adapts the Fourier spectral method in space, coupled with the exponential integrator scheme in time. As an advantage over existing methods, our method yields a full diagonal representation of the non-integer fractional operator, with better accuracy over a finite difference scheme. We realize in this work that evolution equations formulated in the form of fractional-in-space reaction-diffusion systems can result in some amazing examples of pattern formation. Numerical experiments are performed in two and three space dimensions to justify the theoretical results. Simulation results revealed that pattern formation in a fractional medium is practically the same as in classical reaction-diffusion scenarios.
Highlights
Systems with non-integer order are commonly referred to as fractional differential equations
3.2 Numerical techniques for fractional diffusion equation we do not intend to go into details, but we report a brief survey of some of the numerical approaches that have been used
In Figure, we present the numerical results justifying the performance of both finite difference and Fourier spectral methods at some instances of α
Summary
Systems with non-integer order are commonly referred to as fractional differential equations. In Section , we introduce the general two-components partial differential equations formulated in both classical and fractional reaction-diffusion systems. We shall examine some existing background theorems and definitions that are well established for the general two-component reaction-diffusion system We quickly summarized the conditions for a Turing (diffusion-driven) instability as (i) (s (k = )) < , (s (k = )) < , and (ii) (s (k > )) > , (s (k > )) > , with sq defined as the zeros of the quadratic equation p(s) = s + k – λa s + Dk – λa – λ a a. Riesz space-fractional derivative In a similar fashion, a space-fractional diffusion equation can be taken as
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