A new coupling algorithm for fractional diffusion equations based on multiscale functions

We discuss a new discrete-continuous coupling algorithm for fractional diffusion equations based on piecewise interpolation polynomials and multiscale functions. This method considers to discretize the time fractional derivative using the L1-2 scheme while maintaining the continuity of the spatial variable. Then, the multiscale orthogonal polynomial algorithm is used to solve the spatial dimension equations, ultimately yielding the approximate solutions for the factional diffusion equations. Additionally, we provide the theoretical foundations of the algorithm and prove its convergence and stability. Through the test results of four examples, we validate the effectiveness and accuracy of the proposed approach.