High-resolution numerical method for the time-fractional fourth-order diffusion problems via improved quintic B-spline function

Abstract

In this paper, we design and analyse a high-order numerical algorithm based on the improvised quintic B-spline collocation method for solving the fourth-order fractional diffusion equation. The time-fractional derivative is approximated by Caputo’s time derivative. The space derivative is approximated by the collocation method based on improvised quintic B-spline functions. It is shown that the proposed algorithm is unconditionally stable. Through rigorous convergence analysis, the method is shown \((2-\beta )\) order convergent in time and almost sixth-order convergent in space direction. It is also shown that the theoretical rate of convergence is the same as that acquired experimentally. To confirm the theoretical results and to test the efficiency and robustness, the method is tested on three problems. The main contribution of the developed algorithm is that the order of convergence and numerical results obtained are better than the existing methods, like the sextic B-spline collocation method (Roul and Goura in Appl Math Comput 366:124727, 2020), the quintic B-spline method (Siddiqi and Arshed in Int J Comput Math 92(7):1496–1518, 2015), and the quintic spline method (Tariq and Akram in Numer Methods Part Differ Equ 33(2):445–466, 2017). It has been proved that the order of convergence of the proposed method is six, which is two orders of magnitude higher than the other spline collocation methods.