Space-time methods based on isogeometric analysis for nonlinear time-fractional PDEs

主讲人：沈金叶 西南财经大学博士

时间：2023年12月14日15：00

地点：三号楼332室

举办单位：数理学院

主讲人介绍：沈金叶博士，西南财经大学数学学院硕士生导师。研究兴趣：金融期权定价模型的数值算法，Bernoulli 自由边界问题的自适应算法，非线性发展方程的差分方法，分数阶模型的数值算法。主持国家自然科学基金青年基金项目1项，参与完成国家自然科学基金面上项目2项。近5年在国际主流计算数学，金融数学杂志上发表学术论文20余篇。长期从事《数值分析》、《偏微分方程数值解》等课程的教学工作。

内容介绍：In this work, we propose a time discontinuous Galerkin scheme for solving time-fractional Allen-Cahn equation, nonlinear time-fractional Sobolev equation, the nonlinear time-fractional Schr\{o}dinger equation using B-splines in time and Non-Uniform Rational B-splines (NURBS) in space. The technique of comparing real and imaginary parts is utilized to obtain optimal $L^2([0,T];L^2(\Omega))$ norm error estimate. Specifically, we have achieved $r+1$ accuracy in time and $p+1$ accuracy in space, where $r$ and $p$ represent the spline degrees in time and space, respectively. The convergence analysis is also provided on time graded mesh, taking into account solutions with initial singularity. Additionally, the space-time isogeometric analysis method is employed to solve the linear time-fractional Schr\{o}dinger equation. A new discrete norm is constructed, and the well-posedness analysis and error estimate are performed based on this norm. We can attain $\hat{p}$ accuracy concerning the new discrete norm error in space-time domain, where $\hat{p}$ denotes space-time spline degree. Theoretical results are validated through using numerical examples.