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

Pal, Kamal
; 
Liu, Fang
; 
Yan, Yubin
; 
Roberts, Graham
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Affiliation
University of Chester
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2015-06-17
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Mathematics
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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(Δx^(3−α )),1<α<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(Δt+Δx^( min(3−α,β)) ),1<α<2,β>0 , where Δt,Δx denote the time and space step sizes, respectively. Numerical examples are presented and compared with the exact analytical solution for its order of convergence.
Citation
Pal, K., Liu, F., Yan, Y. & Roberts, G. (2015). Finite difference method for two-sided space-fractional partial differential equations. In I. Dimov, I. Farago & L. Vulkov (Eds.), Finite difference methods, theory and applications. 6th International Conference, FDM 2014 (pp. 307-314). Springer.
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Springer
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http://hdl.handle.net/10034/582252
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Type
Book chapter
Language
en
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ISBN
9783319202396
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http://www.springer.com/gp/book/9783319202389