PERTURBED QUENCHING PHENOMENON FOR A CONFORMABLE FRACTIONAL ORDER REACTION DIFFUSION EQUATION

Abstract

This paper is concerned with the study of the quenching behavior of the solution for the following initial-boundary value problem:
$$
\begin{cases}C_\alpha(u(x, t))(t)=\frac{\partial^2 u}{\partial x^2}+\gamma(1-u(x, t))^{-p}, & (x, t) \in(-l, l) \times(0, T), \\ u(-l, t)=0, u(l, t)=0, & t \in(0, T), \\ u(x, 0)=u_0(x)>0, & x \in(-l, l),\end{cases}
$$
where $p>1, l>0, \gamma \geq 0,0<\alpha \leq 0, u_0$ is bounded and symmetric function. $C_\alpha(u)(t)$ is the "conformable fractional derivative" of $u$ in relation to the time variable of order $\alpha$ defined as follows:
$$
C_\alpha(u(x, t))(t)=\lim _{\varepsilon \rightarrow 0} \frac{u\left(x, t+\varepsilon t^{1-\alpha}\right)-u(x, t)}{\varepsilon}
$$
for all $t>0, \alpha \in(0,1]$.

We find the numerical quenching solution for a fractional order equation with homogeneous Dirichlet boundary conditions and show that the quenching occurs on the boundary. Also, we find some additional conditions on the initial data, and the dimension of the domain under which the numerical solution of our problem quenches, estimate the numerical quenching time and establish a convergence result for this quenching time. Finally, we give some numerical results to illustrate our analysis.

Received: April 2, 2023
Accepted: May 25, 2023