Abstract The fractional reaction-diffusion equation has an important physical and theoretical meaning, but its analytical solution poses considerable problems. This paper develops an efficient numerical process, the local radial basis function generated by the finite difference (named LRBF-FD) method, for finding the approximation solution of the time-fractional fourth-order reaction-diffusion equation in the sense of the Riemann-Liouville derivative. The time fractional derivative is approximated using the second-order accurate formulation, while the spatial terms are discretized by means of the LRB-FFD method. The advantage of the local collocation method is to approximate the differential operators by a weighted sum of the values of the function on a local set of nodes (local support) by deriving the RBF expansion. Furthermore, the ill-conditioned matrix resulting from using the global collocation method is avoided. The unconditional stability property and convergence analysis of the time-discrete approach are thoroughly proven and verified numerically. Three numerical examples are presented confirming the theoretical formulation and effectiveness of the new approach.

中文翻译：

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复合环境下时间分数四阶反应扩散模型的数值解

摘要 分数阶反应扩散方程具有重要的物理和理论意义，但其解析解存在相当大的问题。本文开发了一种有效的数值过程，即由有限差分（称为 LRBF-FD）方法生成的局部径向基函数，用于在黎曼 -刘维尔导数。时间分数阶导数使用二阶精确公式近似，而空间项则通过 LRB-FFD 方法离散化。局部搭配方法的优点是通过导出 RBF 展开，通过局部节点集（局部支持）上的函数值的加权和来近似差分算子。此外，避免了使用全局搭配方法导致的病态矩阵。时间离散方法的无条件稳定性和收敛性分析得到了彻底的证明和数值验证。三个数值例子证实了新方法的理论公式和有效性。