Application of the dual reciprocity boundary integral equation approach to solve fourth-order time-fractional partial differential equations

ABSTRACT

This paper proposes a numerical approach to approximate the unknown solution of some high-order fractional partial differential equations. The main idea of this approach is to transform the original problem into an equivalent integral equation that depends only on the boundary values. The linear radial basis functions are used as the main tool for approximating the non-homogeneous terms and time derivative. Also the Caputo's sense is applied to approximate time derivatives. Numerical results demonstrate the order of time steps is $O\left({\tau }^{2-\alpha }\right)$ and $O\left({\tau }^{3-\alpha }\right)$ when $0<\alpha <1$ and $1<\alpha <2$, respectively. Finally to overcome the nonlinear terms, predictor–corrector scheme is employed. The efficiency and usefulness of proposed method are demonstrated by some numerical examples.

KEYWORDS:

• Fourth-order fractional partial differential equations
• dual reciprocity boundary elements method
• boundary integral equation method
• predictor–corrector scheme
• radial basis functions

2000 AMS SUBJECT CLASSIFICATIONS:

• 65N38
• 35R11

Acknowledgments

We would like to thank the two anonymous referees for providing us with constructive comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.