High-Order Regularity and a Fully Discrete Fourier Spectral Method for a Partially Dissipative Viscoelastic Timoshenko System with Memory

This paper investigates a class of partially dissipative viscoelastic Timoshenko systems with memory, where dissipation is induced by a Volterra-type memory term acting only on the shear variable. The well-posedness of weak and strong solutions is established on finite time intervals, including existence, uniqueness, stability, and higher-order regularity under compatibility conditions consistent with mixed boundary conditions. For the numerical approximation, a Fourier spectral fully discrete scheme is constructed: sine and cosine basis expansions are used in space for unknowns satisfying Dirichlet and Neumann boundary conditions, respectively; in time, a central difference scheme is applied to the second-order derivatives, and the composite trapezoidal rule is used to approximate the memory convolution term. Based on a discrete energy method, the positivity of the constructed discrete energy is proved, and the error estimate for the fully discrete scheme with second-order convergence in time and \(q\)-th order in space is established for any q \in \mathbb{N}. Numerical experiments are given to verify the theoretical convergence rates and to compare the dynamic responses of the local and nonlocal models, demonstrating that the memory term effectively captures energy dissipation and vibration attenuation behavior in viscoelastic materials.