论文标题
使用上和下溶液的时间分数半线性抛物线方程的错误分析
Error analysis for time-fractional semilinear parabolic equations using upper and lower solutions
论文作者
论文摘要
考虑到（0,1）$的分数顺序$α\ caputo时间派生的半连接初始值问题，该解决方案通常在最初的时间表现出单数行为。为了使该问题的L1型离散化，我们采用上和下溶液的方法，以任意分级度获得准时分级的时间网格上的尖锐的时间误差界限。特别是，这些结果表明，较温和的（与最佳）分级在正时产生了最佳的收敛速率$ 2-α$，而准均匀的时间网格在正时产生一阶收敛。此外，在适当的条件下，上下溶液的方法立即暗示，与确切的解决方案相似，计算的溶液位于一定范围内。使用有限的差异和空间中有限元素的时间进行半散制和全面离散化。理论发现通过数值实验说明。
A semilinear initial-boundary value problem with a Caputo time derivative of fractional order $α\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. For L1-type discretizations of this problem, we employ the method of upper and lower solutions to obtain sharp pointwise-in-time error bounds on quasi-graded temporal meshes with arbitrary degree of grading. In particular, those results imply that milder (compared to the optimal) grading yields the optimal convergence rate $2-α$ in positive time, while quasi-uniform temporal meshes yield first-order convergence in positive time. Furthermore, under appropriate conditions on the nonlinearity, the method of upper and lower solutions immediately implies that, similarly to the exact solutions, the computed solutions lie within a certain range. Semi-discretizations in time and full discretizations using finite differences and finite elements in space are addressed. The theoretical findings are illustrated by numerical experiments.
