This paper presents space-time continuous and time discontinuous Galerkin schemes for solving nonlinear time-fractional partial differential equations based on B-splines in time and non-uniform rational B-splines (NUR...This paper presents space-time continuous and time discontinuous Galerkin schemes for solving nonlinear time-fractional partial differential equations based on B-splines in time and non-uniform rational B-splines (NURBS) in space within the framework of Iso-geometric Analysis. The first approach uses the space-time continuous Petrov-Galerkin technique for a class of nonlinear time-fractional Sobolev-type equations and the optimal error estimates are obtained through a concise equivalence analysis. The second approach employs a generalizable time discontinuous Galerkin scheme for the time-fractional Allen-Cahn equation. It first transforms the equation into a time integral equation and then uses the discontinuous Galerkin method in time and the NURBS discretization in space. The optimal error estimates are provided for the approach. The convergence analysis under time graded meshes is also carried out, taking into account the initial singularity of the solution for two models. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed methods.展开更多
基金supported in part by the National Natural Science Foundation of China(Grant No.12101509)by the Undergraduate Research and Learning Program of Southwestern University of Finance and Economics+1 种基金L.Yi was supported in part by the National Natural Science Foundation of China(Grant No.12171322)by the Natural Science Foundation of Shanghai(Grant No.21ZR1447200).
文摘This paper presents space-time continuous and time discontinuous Galerkin schemes for solving nonlinear time-fractional partial differential equations based on B-splines in time and non-uniform rational B-splines (NURBS) in space within the framework of Iso-geometric Analysis. The first approach uses the space-time continuous Petrov-Galerkin technique for a class of nonlinear time-fractional Sobolev-type equations and the optimal error estimates are obtained through a concise equivalence analysis. The second approach employs a generalizable time discontinuous Galerkin scheme for the time-fractional Allen-Cahn equation. It first transforms the equation into a time integral equation and then uses the discontinuous Galerkin method in time and the NURBS discretization in space. The optimal error estimates are provided for the approach. The convergence analysis under time graded meshes is also carried out, taking into account the initial singularity of the solution for two models. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed methods.
