Numerical study of the fractional fourth-order evolution problems with weak singularity arises in chemical systems

Abstract

A higher-order time-fractional evolution problems (EPs) with the Caputo time fractional derivative is considered. A weak singularity typically appears close to the initial time (\(t=0\)) in this problem’s solution, which reduces the accuracy of conventional numerical methods with uniform mesh. The technique of nonuniform mesh based on the solution’s acceptable regularity is a very efficient way to regain precision. In chemistry, these equations are often used to simulate intricate diffusion processes with memory effects, particularly whenever pattern formation, domain wall propagation in liquid crystals are involved. In the current study, we solve a time-fractional fourth-order partial differential equation with non-smooth solutions using the quintic trigonometric B-spline (QTBS) technique with temporally graded mesh. The stability and convergence of the proposed numerical scheme are discussed broadly, which illustrates clearly how the regularity of the solution and the mesh grading affect the order of convergence of the proposed scheme, allowing one to select the most effective mesh grading. The plots and tabulated results of some test problems are displayed to validate the accuracy and efficiency of the scheme using graded mesh.