# Finite Difference Method for Two-Sided Space-Fractional Partial Differential Equations

• Kamal Pal
• Fang Liu
• Yubin Yan
• Graham Roberts
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9045)

## Abstract

Finite difference methods for solving two-sided space-fractional partial differential equations are studied. The space-fractional derivatives are the left-handed and right-handed Riemann-Liouville fractional derivatives which are expressed by using Hadamard finite-part integrals. The Hadamard finite-part integrals are approximated by using piecewise quadratic interpolation polynomials and a numerical approximation scheme of the space-fractional derivative with convergence order $$O(\varDelta x^{3- \alpha }), \, 1<\alpha <2$$ is obtained. A shifted implicit finite difference method is introduced for solving two-sided space-fractional partial differential equation and we prove that the order of convergence of the finite difference method is $$O (\varDelta t + \varDelta x^{\min (3- \alpha , \beta )}), 1< \alpha <2, \, \beta >0$$, where $$\varDelta t, \varDelta x$$ denote the time and space step sizes, respectively. Numerical examples are presented and compared with the exact analytical solution for its order of convergence.

## Notes

### Acknowledgements

We wish to express our sincere gratitude to Professor Neville. J. Ford for his encouragement, discussions and valuable criticism during the research of this work.

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© Springer International Publishing Switzerland 2015

## Authors and Affiliations

• Kamal Pal
• 1
• Fang Liu
• 2
• Yubin Yan
• 1
Email author
• Graham Roberts
• 1
1. 1.Department of MathematicsUniversity of ChesterChesterUK
2. 2.Department of MathematicsLuliang UniversityLishiPR China