Galerkin finite element schemes with fractional Crank–Nicolson method for the coupled time-fractional nonlinear diffusion system

Abstract

This paper deals with two fractional Crank-Nicolson-Galerkin finite element schemes for coupled time-fractional nonlinear diffusion system. The first scheme is iterative and is based on Newton's method, while the other one is a linearized scheme. Existence-uniqueness results of the fully discrete solution for both schemes are discussed. In addition, a priori bounds and a priori error estimates are derived for proposed schemes using a new discrete fractional Grönwall-type inequality. Both the schemes yield O(Δt 2) accuracy in time and hence, superior to O(Δt 2−α) accurate L1 scheme existing in the literature. Moreover, three different numerical examples are provided to illustrate the theoretical estimates .

A novel linearized Galerkin finite element scheme with fractional Crank–Nicolson method for the nonlinear coupled delay subdiffusion system with smooth solutions

“…Unlike the second order schemes proposed by many academicians (cf. previous studies 33–36,43 and references therein), our scheme does not demand any compatibility type conditions on the exact solution. We derive the existence and uniqueness results for the fully discrete solution and provide its a priori bound, a priori error estimate in L ∞ ( L 2 ) norm.…”

“…The mixed finite element for space discretization along with second order accurate scheme based on Riemann–Liouville derivative approximation in time has been used to solve the proposed problem. Likewise, some recent research 34–36 has focused on solving complex nonlinear subdiffusion equations by utilizing second order accurate methods for approximation to the time‐fractional derivatives. We note that the second order accurate method to the Riemann–Liouville derivative (called fractional Crank–Nicolson method) used to approximate solutions of problems considered in previous works 33–36 requires regularity assumption u ∈ C 3 ([0, T ]; L 2 (Ω)) and certain compatibility conditions on the exact solution u .…”

“…Likewise, some recent research 34–36 has focused on solving complex nonlinear subdiffusion equations by utilizing second order accurate methods for approximation to the time‐fractional derivatives. We note that the second order accurate method to the Riemann–Liouville derivative (called fractional Crank–Nicolson method) used to approximate solutions of problems considered in previous works 33–36 requires regularity assumption u ∈ C 3 ([0, T ]; L 2 (Ω)) and certain compatibility conditions on the exact solution u . Unlike the fractional Crank–Nicolson method addressed in previous works, 33–36 in the present paper, we apply L 2‐1 σ (also called fractional Crank–Nicolson) method to the Caputo fractional derivative which does not require any compatibility conditions on the exact solution but only demands the solution u ∈ C 3 ([0, T ]; L 2 (Ω)).…”

“…We note that the second order accurate method to the Riemann–Liouville derivative (called fractional Crank–Nicolson method) used to approximate solutions of problems considered in previous works 33–36 requires regularity assumption u ∈ C 3 ([0, T ]; L 2 (Ω)) and certain compatibility conditions on the exact solution u . Unlike the fractional Crank–Nicolson method addressed in previous works, 33–36 in the present paper, we apply L 2‐1 σ (also called fractional Crank–Nicolson) method to the Caputo fractional derivative which does not require any compatibility conditions on the exact solution but only demands the solution u ∈ C 3 ([0, T ]; L 2 (Ω)). Moreover, due to the high accuracy of the L 2‐1 σ method over the L 1 method, recent works in the literature focus on the L 2‐1 σ method while finding the numerical solution of fractional PDEs; see other studies 37–42 and references therein.…”

Numerical Solutions for Time-Fractional Cancer Invasion System With Nonlocal Diffusion

“…Jin et al [39] solved proposed Crank-Nicolson-Galerkin finite element scheme to solve the linear time FPDEs. Kumar et al [40] proposed Crank-Nicolson-Galerkin finite element scheme to solve the time-fractional nonlinear diffusion equation using Newton's algorithm. However, according to author's knowledge there is no paper available in the literature to study the fractional order cancer invasion system using finite element method.…”