A local radial basis function collocation method to solve the variable-order time fractional diffusion equation in a two-dimensional irregular domain

The local radial basis function (RBF) method is a promising solver for variable-order time fractional diffusion equation (TFDE), as it overcomes the computational burden of the traditional global method. Application of the local RBF method is limited to Fickian diffusion, while real-world diffusion is usually non-Fickian in multiple dimensions. This article is the first to extend the application of the local RBF method to two-dimensional, variable-order, time fractional diffusion equation in complex shaped domains. One of the main advantages of the local RBF method is that only the nodes located in the subdomain, surrounding the local point, need to be considered when calculating the numerical solution at this point. This approach can perform well with large scale problems and can also mitigate otherwise ill-conditioned problems. The proposed numerical approach is checked against two examples with curved boundaries and known analytical solutions. Shape parameter and subdomain node number are investigated for their influence on the accuracy of the local RBF solution. Furthermore, quantitative analysis, based on root-mean-square error, maximum absolute error, and maximum error of the partial derivative indicates that the local RBF method is accurate and effective in approximating the variable-order TFDE in two-dimensional irregular domains.