Approximate Solutions and Error Bounds for Quasi-Linear Hyperbolic Initial Boundary Value Problems

1975 ◽  
Vol 12 (1) ◽  
pp. 37-45 ◽  
Author(s):  
To-Yat Cheung
Keyword(s):  
Boundary Value Problems ◽  
Error Bounds ◽  
Boundary Value ◽  
Initial Boundary ◽  
Approximate Solutions ◽  
Initial Boundary Value Problems ◽  
Initial Boundary Value

Blow-up of error estimates in time-fractional initial-boundary value problems

2020 ◽  
Author(s):  
Hu Chen ◽  
Martin Stynes
Keyword(s):  
Numerical Methods ◽  
Boundary Value Problems ◽  
Error Bounds ◽  
Time Domain ◽  
Blow Up ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value Problems ◽  
Initial Boundary Value

Abstract Time-fractional initial-boundary value problems of the form $D_t^\alpha u-p \varDelta u +cu=f$ are considered, where $D_t^\alpha u$ is a Caputo fractional derivative of order $\alpha \in (0,1)$ and the spatial domain lies in $\mathbb{R}^d$ for some $d\in \{1,2,3\}$. As $\alpha \to 1^-$ we prove that the solution $u$ converges, uniformly on the space-time domain, to the solution of the classical parabolic initial-boundary value problem where $D_t^\alpha u$ is replaced by $\partial u/\partial t$. Nevertheless, most of the rigorous analyses of numerical methods for this time-fractional problem have error bounds that blow up as $\alpha \to 1^-$, as we demonstrate. We show that in some cases these analyses can be modified to obtain robust error bounds that do not blow up as $\alpha \to 1^-$.

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