A semi-analytical collocation Trefftz scheme for solving multi-term time fractional diffusion-wave equations

This study presents a semi-analytical boundary-only collocation technique for solving multi-term time-fractional diffusion-wave equations. In the present collocation technique, the Laplace transformation is first implemented to convert time-fractional diffusion-wave equation to a series of time-independent nonhomogeneous equations in Laplace domain. Then the composite multiple reciprocity method (CMRM) is applied to construct a high-order homogeneous equation, which has the same solution with one of time-independent nonhomogeneous equations in Laplace domain. The collocation Trefftz scheme with high-order T-complete functions is used to obtain the semi-analytical solution of high-order homogeneous equation with boundary-only collocation in Laplace domain. Finally the numerical Laplace inversion scheme is introduced to invert the Laplace domain solutions back into the time-dependent solutions of time-fractional diffusion-wave equations. The proposed method is easy-to-implement and flexible for irregular domain problems. It evades costly convolution integral calculation in time fractional derivation approximation, and avoids the effect of time step on numerical accuracy and stability. Error analysis and numerical investigations show that the proposed collocation method is highly accurate, computationally efficient, and numerically stable for multi-term time fractional diffusion-wave equations.