Evolution system which contains fractional derivatives can give rise to useful mathematical model for describing some important real-life or physical scenarios. Here, we suggest some numerical techniques for solving fractional-in-time reaction-diffusion models in the sense of Caputo operator. The suggested schemes are formulated with difference scheme and Fourier-spectral algorithm. In the simulation framework, it was observed that the spectral method retains the advantage of spectral accuracy over its finite difference counterpart. Both techniques are easy to adapt and extend to high-dimensions in space and time. The existence of solution, uniqueness of solution, linear stability analysis as well as the calculation of the Lyapunov exponent of the main system, are well established. Suitability of the suggested numerical techniques are tested on non-diffusive model, one-dimensional diffusive example and two-dimensional diffusive experiments.

包含分数导数的演化系统可以产生有用的数学模型，用于描述一些重要的现实生活或物理场景。在这里，我们建议一些数值技术来解决Caputo算子意义上的时间分数反应扩散模型。提出的方案由差分方案和傅里叶谱算法组成。在仿真框架中，观察到光谱方法保留了光谱精度优于其有限差分方法的优势。两种技术都很容易适应，并且可以扩展到时空的高维度。很好地确定了解的存在性，解的唯一性，线性稳定性分析以及主系统的Lyapunov指数的计算。