THE PERTURBED NUMERICAL PROCESS FOR A REACTION-DIFFUSION PROBLEM WITH CONFORMABLE DERIVATIVE
Keywords:
reaction diffusion equation, quenching solution, conformable fractional derivatives (CFD), conformable Euler method (CEM), modified conformable Euler method (MCEM), perturbation method, analytical solution, homotopy perturbation method, convergence, numerical quenching time, extinction, existence, finite difference method, Maple, MATLAB-SimulinkDOI:
https://doi.org/10.17654/0975045225005Abstract
We present a numerical algorithm for time-fractional reactiondiffusion equation with boundary condition to find approximate solutions by using perturbation method. We study the quenching behavior with the conformable Euler method (CEM) for the finite difference discretization of the fractional derivative in the Caputo sense for the following initial-boundary value problem:
$$
\begin{cases}D_t^\alpha u(x, t)+\beta u_x(x, t)+\varepsilon \Delta(u(x, t))=\lambda(x) f(u(x, t)) & \text { in } \Omega \times(0, T) \\ u=0 & \text { on } \partial \Omega \times(0, T) \\ u(x, 0)=u_0(x) \geq 0, & x \in \Omega\end{cases}
$$
where $\Omega$ is a bounded domain in $\mathbb{R}^N$, with smooth boundary $\partial \Omega$, $u_0$ is a bounded and symmetric function. $D^\alpha$ is the "conformable fractional derivative" of $u$ with respect to the time variable of order $\alpha$ in the sense of Caputo. Also, we study the numerical quenching solution for a fractional diffusion equation with degenerate source term. The fractional derivative, acting on the spatial variable, contains left-sided Caputo derivative and right-sided RiemannLiouville derivative simultaneously.
Received: May 31, 2024
Revised: June 14, 2024
Accepted: November 28, 2024
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