Too much regularity may force too much uniqueness

By Martin Stynes. 2016

137 citations recorded by the source corpus.

Abstract Time-dependent fractional-derivative problems D t α u + A u = f $D_t^\alpha u + Au = f$ are considered, where D t α $D_t^\alpha$ is a Caputo fractional derivative of order α ∈ (0, 1)∪(1, 2) and A is a classical elliptic operator, and appropriate boundary and initial conditions are applied. The regularity of solutions to this class of problems is discussed, and it is shown that assuming more regularity than is generally true—as many researchers do—places a surprisingly severe restriction on the problem.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/d7b116546cd4ebec54a144ed1c1a7a61e73b2a67

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In Applied Mathematics

1. William McLean
2. Zhibo Wang
3. Pin Lyu
4. Raytcho D. Lazarov
5. Yubin Yan
6. Bangti Jin
7. Zhi Zhou
8. Hong-lin Liao

In Modeling and Simulation

1. Raytcho D. Lazarov
2. Zhi Zhou
3. Bangti Jin
4. Yubin Yan
5. William McLean
6. Zhibo Wang
7. Pin Lyu
8. Hong-lin Liao

Citations in the corpus

• A Discrete Grönwall Inequality with Applications to Numerical Schemes for Subdiffusion Problems (2018, background): “In fact, u can only be a smooth function of t if the initial data and source term satisfy some restrictive compatibility conditions [23].”
• An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data (2017, background): “[45] M. Stynes, Too much regularity may force too much uniqueness.”
• Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview (2018, background): “By now, it is well known that smooth solutions are produced by a small class of data [95].”
• A fast linearized finite difference method for the nonlinear multi-term time-fractional wave equation (2019, background): “It is worth to mention that the solution of many time-fractional differential equations typically displays a weak singularity near the initial time [8, 9, 10, 19, 20, 21, 26, 29, 30], which leads to the loss of time accuracy for many related high-order numerical methods.”