TITLE:
The Finite Volume Element Method for Time-Fractional Nonlinear Fourth-Order Diffusion Equation with Time Delay
AUTHORS:
Anran Li, Qing Yang
KEYWORDS:
Time-Fractional Nonlinear Fourth-Order Diffusion Equation with Time Delay, Finite Volume Element Method, Caputo-Fractional Derivative, Optimal Priori Error Analysis
JOURNAL NAME:
Engineering,
Vol.17 No.1,
January
21,
2025
ABSTRACT: In this article, a finite volume element algorithm is presented and discussed for the numerical solutions of a time-fractional nonlinear fourth-order diffusion equation with time delay. By choosing the second-order spatial derivative of the original unknown as an additional variable, the fourth-order problem is transformed into a second-order system. Then the fully discrete finite volume element scheme is formulated by using
L1
approximation for temporal Caputo derivative and finite volume element method in spatial direction. The unique solvability and stable result of the proposed scheme are proved. A priori estimate of
L
2
-norm with optimal order of convergence
O(
h
2
+
τ
2−α
)
where
τ
and
h
are time step length and space mesh parameter, respectively, is obtained. The efficiency of the scheme is supported by some numerical experiments.