Numerical analysis of computational approaches to nonlinear generalized time-fractional mixed sub-diffusion-wave equations with delay

Abstract

This paper investigates computational methods for solving linear and delayed nonlinear generalized time-fractional mixed sub-diffusion-wave equations. The model admits three equivalent fractional integro-differential forms arising from its fractional structure. These forms are unified within a generalized framework based on weighted Caputo and Riemann-Liouville operators, thereby avoiding case-by-case treatment and enhancing memory modeling flexibility with time-dependent kernels. A second-order temporal finite difference scheme is constructed for the resulting generalized linear model, and unconditional stability is established. The scheme is further adapted to graded temporal meshes to improve accuracy for solutions with weak initial singularities. The approach is then extended to nonlinear problems with a time delay, where a linearized second-order scheme is developed and analyzed using the discrete fractional Grönwall inequality. To reduce the computational cost of fractional memory terms, the Sum of Exponentials technique is employed to construct fast approximations of the weighted fractional operators, leading to efficient implementations of the proposed schemes. Numerical experiments confirm the theoretical results and demonstrate the accuracy and efficiency of both the direct and fast schemes.