This paper is concerned to study the well-posedness, the Mittag–Leffler stability of solutions of time-fractional nonlocal reaction–diffusion equation in bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n.$ We use the Faedo–Galerkin approximation method with initial data in $L^2\left(\mathrm{\Omega }\right)$ to show a solution in $u\in L^{\mathrm{\infty }}(0,T;L^2(\mathrm{\Omega }\left)\right)\cap L^2(0,T;H_0^1(\mathrm{\Omega }\left)\right).$ Further, we construct the suitable Lyapunov function to ensure that a solution of the proposed model is the Mittag–Leffler stable. Furthermore, we fully discretize the Galerkin finite element method for the proposed time-fractional model in two-space dimension. Here, time-fractional derivative is given in Caputo's sense and discretized using $L^1$ approximation scheme. Error analysis of the proposed numerical method is performed and error bounds are obtained for the error measured in $L^2$ norm. All the theoretical results are validated with several constructive numerical examples.

本文主要研究有界域中时间分数非局域反应-扩散方程解的适定性、Mittag-Leffler稳定性。 $\mathrm{\Omega }\subset {\mathbb{电阻}}^n.$ 我们使用 Faedo-Galerkin 近似方法，初始数据为 ${升}^2\left(\mathrm{\Omega }\right)$ 显示解决方案 $你\in {升}^{\mathrm{\infty }}(0,吨;{升}^2(\mathrm{\Omega }\left)\right)\cap {升}^2(0,吨;H_0^1(\mathrm{\Omega }\left)\right).$此外，我们构造了合适的 Lyapunov 函数以确保所提出模型的解是 Mittag-Leffler 稳定的。此外，我们在二维空间中完全离散了所提出的时间分数模型的 Galerkin 有限元方法。在这里，时间分数导数在 Caputo 的意义上给出并使用离散化${升}^1$近似方案。对所提出的数值方法进行了误差分析，并获得了测量误差的误差界限${升}^2$规范。所有的理论结果都得到了几个建设性的数值例子的验证。