PHS-PU-WSGI discretization of nonlinear fourth-order time-fractional wave models

This paper combines a polyharmonic spline partition-of-unity collocation method with a second-order weighted and shifted Grünwald integral formula to solve nonlinear fourth-order time-fractional wave equations. These problems are numerically challenging due to the high-order spatial operator, the nonlocal time-fractional derivative, and the nonlinear term. For spatial discretization, the computational domain is covered by overlapping subdomains within a partition-of-unity framework, where polyharmonic spline functions generate local approximations. To facilitate the treatment of both the nonlinear term and the fourth-order operator, the governing equation is reformulated into an equivalent coupled system. The nonlinear term at the new time level is approximated via a second-order time extrapolation, leading to a fully discrete linear system at each time step. The performance of the method is assessed through several numerical examples, including a test problem without an analytical solution. Numerical results indicate approximately second-order convergence in time along with high spatial accuracy. These findings demonstrate the accuracy and effectiveness of the proposed method for this class of problems.