Galerkin–Legendre spectral method for the distributed-order time fractional fourth-order partial differential equation

ABSTRACT

In this paper, we consider the Galerkin–Legendre spectral method for solving the two-dimensional distributed-order time fractional fourth-order partial differential equation. By utilizing the composite Simpson formula to discretize the distributed-order integral, we transform the considered equation into a multi-term time fractional sub-diffusion equation. Then the $L2$-$1_{\sigma }$ formula is used to approximate the multi-term Caputo fractional derivatives and the Legendre spectral method is employed for the spatial discretization. The scheme is proved to be unconditionally stable and convergent in both $L^2$- and $L^{\mathrm{\infty }}$-norms with fourth-order accuracy in distributed order, second-order accuracy in time and spectral accuracy in space. Finally, some numerical tests are performed to verify the theoretical results.

KEYWORDS:

• Fourth-order differential equation
• distributed-order derivative
• Legendre spectral method
• $L2$-$1_{\sigma }$ formula
• stability and convergence
• spectral accuracy

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35R11
• 65M70

Acknowledgements

The authors would like to thank the editor and the referees for their valuable comments and suggestions which helped us improve the quality of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by NSF of China (National Natural Science Foundation of China) [No. 11771163].