High-Order B-spline approximation for solving time-dependent Timoshenko vibrating equations

The aim of this work is to study the time-dependent coupled Timoshenko partial differential equations (PDEs) with mixed boundary conditions. These wave equations govern the beam displacements in the fields of structural mechanics and civil engineering. The modeling algorithm is implemented in two steps: Firstly, we transform the time-dependent coupled equations into coupled transverse vibrating equations depending on the frequency parameter and space variable. A Galerkin approximation built on Normalized Uniform Polynomial Spline (NUPS) basis functions is employed to compute the pairs of solutions that are the focus of this work. Secondly, we compute the Inverse Fourier Transform (IFT) solution using some quadrature. We also exhibit comparison results for the various calculations of IFT-integral solutions. The theoretical estimates are verified by several tests of Timoshenko wave equations with known analytical solutions. Numerous numerical experiments are presented to validate the numerical convergence rates, and a few examples are also included to demonstrate the high precision, the efficiency, and the outstanding resolution capabilities for both smooth and discontinuous solutions, as well as plots of the 3D beam's field displacements.