A New Method of Solving the Time-Fractional Mixed Nonlinear Diffusion and Diffusion-Wave Equation

It is well known that meshless methods are effective for solving fractional differential equations on both regular and irregular domains. However, many commonly used basis functions such as Legendre wavelets and B-splines are naturally defined on rectangular domains, which limits their applicability in certain meshless frameworks. In this paper, we are motivated to develop a new meshless method for solving the time-fractional mixed nonlinear diffusion and diffusion-wave equation on arbitrary domains, by employing a skillful extension technique. The proposed method utilizes the well-known Legendre multiwavelets to obtain the best approximate solution by seeking the minimum of approximate solutions within a reproducing kernel space. This approach avoids the need to compute complicated shape functions or other local basis functions. Moreover, the construction of the reproducing kernel Legendre multiwavelet bases presented in this paper is straightforward. Numerical examples on both rectangular domains and domains with curved boundaries confirm that the method is efficient and achieves high accuracy.