Numerical solutions of fourth‐order fractional sub‐diffusion problems via parametric quintic spline

Abstract

In this paper, we develop a numerical scheme for a fourth-order fractional subdiffusion problem using parametric quintic spline and a non-uniform approximation for Caputo fractional derivatives. The solvability, convergence and stability of the scheme are established in maximum norm, and it is shown that the convergence order is higher than some earlier work done. Four numerical experiments are further carried out to demonstrate the efficiency of the proposed scheme as well as to compare with other methods. K E Y W O R D Sfractional differential equation, numerical solution, parametric spline, quintic spline, sub-diffusion equation

A new approximation for the generalized fractional derivative and its application to generalized fractional diffusion equation

“…The next example further investigates the performance of the scheme (2.50) when the regularity of the exact solution falls short of the theoretical assumptions. Indeed, as pointed out in [32], the solution may have some singularities at t = 0, which is a possible scenario in engineering problems.Example Consider the generalized fractional diffusion problem where 0 < α < 1. The exact solution is not known for general z ( t ), so we shall compute the ‘exact’ solution by using our numerical scheme with sufficiently small spatial step size and sufficiently large N = 10,000.…”

“…The next example further investigates the performance of the scheme (2.50) when the regularity of the exact solution falls short of the theoretical assumptions. Indeed, as pointed out in [32], the solution may have some singularities at t = 0, which is a possible scenario in engineering problems. where 0 < < 1.…”

“…Clearly, the classical fractional diffusion equation is a special case of the generalized fractional diffusion equation (1.5) with z(t) = t. Therefore, our numerical scheme is also applicable to the classical equation. Many numerical methods have been developed for the classical fractional diffusion equations [17,19,21,22,[25][26][27][28][29][30][31][32][33][34][35]48], among them Ji and Sun [19] give the best result of order 3 in the temporal convergence. Since our numerical scheme can achieve order 4 in the temporal convergence, hence we also improve the numerical methods for classical fractional diffusion equations in the literature.…”

A higher order numerical scheme for generalized fractional diffusion equations

“…It is a fact that the analytical solution of a classical fractional differential equation is difficult to obtain, thus many numerical methods have been developed such as finite difference method, 13 finite element method, 14 spectral method, 15 nonpolynomial spline method, 16‐21 and other methods involving fractional‐order Lagrange polynomials, 22 Legendre‐Laguerre polynomials, 23 Genocchi hybrid functions, 24 Bernoulli wavelets, 25 to name a few. As one would expect, it is even more difficult to obtain analytical solutions of equations involving generalized fractional derivatives, and numerical treatment is more appropriate for such problems.…”

Exponential spline for the numerical solutions of linear Fredholm integro-differential equations

“…Xuhao Li and Patricia J.Y. Wong in [26][27][28][29] have successfully applied non-polynomial spline to fractional diffusion problems. Besides, non-polynomial splines have also been applied to solve a system of second-order boundary value problems in the mid-knots of the mesh [14].…”