dc.contributor.author Xing, Yanyuan en dc.contributor.author Yan, Yubin en dc.date.accessioned 2018-01-24T11:48:39Z dc.date.available 2018-01-24T11:48:39Z dc.date.issued 2018-01-02 dc.identifier.citation Xing, Y., & Yan, Y. (2018). A higher order numerical method for time fractional partial differential equations with nonsmooth data. Journal of Computational Physics, 357, 305-323. https://doi.org/10.1016/j.jcp.2017.12.035 en dc.identifier.issn 0021-9991 dc.identifier.doi 10.1016/j.jcp.2017.12.035 dc.identifier.uri http://hdl.handle.net/10034/620810 dc.description.abstract Gao et al. (2014) introduced a numerical scheme to approximate the Caputo fractional derivative with the convergence rate $O(k^{3-\alpha}), 0< \alpha <1$ by directly approximating the integer-order derivative with some finite difference quotients in the definition of the Caputo fractional derivative, see also Lv and Xu (2016), where $k$ is the time step size. Under the assumption that the solution of the time fractional partial differential equation is sufficiently smooth, Lv and Xu (2016) proved by using energy method that the corresponding numerical method for solving time fractional partial differential equation has the convergence rate $O(k^{3-\alpha}), 0< \alpha <1$ uniformly with respect to the time variable $t$. However, in general the solution of the time fractional partial differential equation has low regularity and in this case the numerical method fails to have the convergence rate $O(k^{3- \alpha}), 0 < \alpha <1$ uniformly with respect to the time variable $t$. In this paper, we first obtain a similar approximation scheme to the Riemann-Liouville fractional derivative with the convergence rate $O(k^{3- \alpha}), 0 < \alpha <1$ as in Gao \et \cite{gaosunzha} (2014) by approximating the Hadamard finite-part integral with the piecewise quadratic interpolation polynomials. Based on this scheme, we introduce a time discretization scheme to approximate the time fractional partial differential equation and show by using Laplace transform methods that the time discretization scheme has the convergence rate $O(k^{3- \alpha}), 0 < \alpha <1$ for any fixed $t_{n}>0$ for smooth and nonsmooth data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the theoretical results are consistent with the numerical results. dc.language.iso en en dc.publisher Elsevier en dc.relation.url https://www.sciencedirect.com/science/article/pii/S0021999117309294 en dc.subject Time fractional partial differential equation en dc.subject Error estimates en dc.subject Laplace transform en dc.subject Caputo fractional derivative en dc.title A higher order numerical method for time fractional partial differential equations with nonsmooth data en dc.type Article en dc.contributor.department Lvliang University; University of Chester en dc.identifier.journal Journal of Computational Physics en dc.date.accepted 2017-12-24 or.grant.openaccess Yes en rioxxterms.funder Unfunded en rioxxterms.identifier.project Unfunded en rioxxterms.version AM en rioxxterms.licenseref.startdate 2020-01-02 html.description.abstract Gao et al. (2014) introduced a numerical scheme to approximate the Caputo fractional derivative with the convergence rate $O(k^{3-\alpha}), 0< \alpha <1$ by directly approximating the integer-order derivative with some finite difference quotients in the definition of the Caputo fractional derivative, see also Lv and Xu (2016), where $k$ is the time step size. Under the assumption that the solution of the time fractional partial differential equation is sufficiently smooth, Lv and Xu (2016) proved by using energy method that the corresponding numerical method for solving time fractional partial differential equation has the convergence rate $O(k^{3-\alpha}), 0< \alpha <1$ uniformly with respect to the time variable $t$. However, in general the solution of the time fractional partial differential equation has low regularity and in this case the numerical method fails to have the convergence rate $O(k^{3- \alpha}), 0 < \alpha <1$ uniformly with respect to the time variable $t$. In this paper, we first obtain a similar approximation scheme to the Riemann-Liouville fractional derivative with the convergence rate $O(k^{3- \alpha}), 0 < \alpha <1$ as in Gao \et \cite{gaosunzha} (2014) by approximating the Hadamard finite-part integral with the piecewise quadratic interpolation polynomials. Based on this scheme, we introduce a time discretization scheme to approximate the time fractional partial differential equation and show by using Laplace transform methods that the time discretization scheme has the convergence rate $O(k^{3- \alpha}), 0 < \alpha <1$ for any fixed $t_{n}>0$ for smooth and nonsmooth data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the theoretical results are consistent with the numerical results.
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