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

主講人：沈金葉 西南財經(jīng)大學博士

時間：2023年12月14日15：00

地點：三號樓332室

舉辦單位：數(shù)理學院

主講人介紹：沈金葉博士，西南財經(jīng)大學數(shù)學學院碩士生導師。研究興趣：金融期權定價模型的數(shù)值算法，Bernoulli 自由邊界問題的自適應算法，非線性發(fā)展方程的差分方法，分數(shù)階模型的數(shù)值算法。主持國家自然科學基金青年基金項目1項，參與完成國家自然科學基金面上項目2項。近5年在國際主流計算數(shù)學，金融數(shù)學雜志上發(fā)表學術論文20余篇。長期從事《數(shù)值分析》、《偏微分方程數(shù)值解》等課程的教學工作。

內容介紹：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.