Paper Info
Title
WSGD-OSC Scheme for Two-Dimensional Distributed Order Fractional Reaction-Diffusion Equation.
Abstract
In this paper, a new numerical approximation is discussed for the two-dimensional distributed-order time fractional reaction–diffusion equation. Combining with the idea of weighted and shifted Grünwald difference (WSGD) approximation (Tian et al. in Math Comput 84:1703–1727, 2015; Wang and Vong in J Comput Phys 277:1–15, 2014) in time, we establish orthogonal spline collocation (OSC) method in space. A detailed analysis shows that the proposed scheme is unconditionally stable and convergent with the convergence order \(\mathscr {O}(\tau ^2+\Delta \alpha ^2+h^{r+1})\), where \(\tau , \Delta \alpha , h\) and r are, respectively the time step size, step size in distributed-order variable, space step size, and polynomial degree of space. Interestingly, we prove that the proposed WSGD-OSC scheme converges with the second-order in time, where OSC schemes proposed previously (Fairweather et al. in J Sci Comput 65:1217–1239, 2015; Yang et al. in J Comput Phys 256:824–837, 2014) can at most achieve temporal accuracy of order which depends on the order of fractional derivatives in the equations and is usually less than two. Some numerical results are also given to confirm our theoretical prediction.
Year
DOI
Venue
2018
10.1007/s10915-018-0672-3
J. Sci. Comput.
Keywords
Field
DocType
Distributed order fractional equation, WSGD operator, Orthogonal spline collocation scheme, Stability, Error estimate, 65M70, 65M12, 65M15, 35R11
Convergence (routing),Mathematical analysis,Spline collocation,Degree of a polynomial,Fractional calculus,Numerical approximation,Reaction–diffusion system,Mathematics
Journal
Volume
Issue
ISSN
76
3
0885-7474
Citations 
PageRank 
References 
1
0.34
13
Authors
3
Name
Order
Citations
PageRank
Xuehua Yang1455.38
Haixiang Zhang26412.19
Da. Xu37411.27