Meshless analysis of two-dimensional two-sided space-fractional wave equation based on improved moving least-squares approximation

ABSTRACT

The Space-fractional wave equations (SFWE) have been found to be very adequate in describing anomalous transport and dispersion phenomena. Due to the non-local property of integro-differential operator of space-fractional derivative, it is very challenging to deal with fractional model. In this paper, a meshless analysis of two-dimensional two-sided SFWE is proposed based on the improved moving least-squares (IMLS) approximation. The trial function for the SFWE is constructed by the IMLS approximation, where the resulting algebraic equation system to obtain the shape functions is no more ill conditioned and has high computational efficiency. The Riemann–Liouville operator is discretized by the Grünwald formula. The centre difference method and the strong-forms of the SFWE are used to obtain the final fully discrete algebraic equation. And the essential boundary conditions can be directly and easily imposed on as a finite element method. Due to the adoption of IMLS approximation and strong-forms, this method will be highly accurate and efficient. Numerical results demonstrate that this method is highly accurate and computationally efficient for SFWE. Moreover, the convergence and error estimate have been analysed in our study.

KEYWORDS:

• Meshless method
• improved moving least-squares (IMLS) approximation
• space-fractional wave equation

2000 AMS SUBJECT CLASSIFICATIONS:

• 65N22
• 65M32
• 65N35

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

Project supported by the Natural Science Foundation of Ningbo City [grant numbers 2014A610028, 2015A610166, 2015A610299] and the Natural Science Foundation of Zhejiang Province of China [grant number LY15A020007] and the K. C. Wong Magna Fund in Ningbo University, China.