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Unique continuation principle for the one-dimensional time-fractional diffusion equation

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Published/Copyright: July 30, 2019
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 22 Issue 3

Abstract

This paper deals with the unique continuation of solutions for a one-dimensional anomalous diffusion equation with Caputo derivative of order α ∈ (0, 1). Firstly, the uniqueness of solutions to a lateral Cauchy problem for the anomalous diffusion equation is given via the Theta function method, from which we further verify the unique continuation principle.

MSC 2010: Primary 35R11; Secondary 26A33; 35B53; 44A10
Key Words and Phrases: fractional diffusion equation; unique continuation principle; Laplace transform; fractional Theta function

Acknowledgements

The first author thanks Grant-in-Aid for Research Activity Start-up 16H06712, JSPS, and National Natural Science Foundation of China 11801326. The second author is supported by Grant-in-Aid for Scientific Research (S) 15H05740 of Japan Society for the Promotion of Science, NSFC (No. 11771270, 91730303) and the “RUDN University Program 5-100”. This work was also supported by A3 Foresight Program “Modeling and Computation of Applied Inverse Problems” of Japan Society for the Promotion of Science.

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Received: 2018-08-08
Published Online: 2019-07-30
Published in Print: 2019-06-26

© 2019 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–Volume 22–3–2019)
  4. Survey Paper
  5. The probabilistic point of view on the generalized fractional partial differential equations
  6. Research Paper
  7. A system of coupled multi-term fractional differential equations with three-point coupled boundary conditions
  8. Fractal convolution: A new operation between functions
  9. Unique continuation principle for the one-dimensional time-fractional diffusion equation
  10. Asymptotics of eigenvalues for differential operators of fractional order
  11. The asymptotic behaviour of fractional lattice systems with variable delay
  12. On fractional asymptotical regularization of linear ill-posed problems in hilbert spaces
  13. Weighted Hölder continuity of Riemann-Liouville fractional integrals – Application to regularity of solutions to fractional cauchy problems with Carathéodory dynamics
  14. Green’s functions, positive solutions, and a Lyapunov inequality for a caputo fractional-derivative boundary value problem
  15. Finite element approximations for fractional evolution problems
  16. Stochastic diffusion equation with fractional Laplacian on the first quadrant
  17. Stability of fractional variable order difference systems
  18. Chaotic dynamics of fractional Vallis system for El-Niño
https://doi.org/10.1515/fca-2019-0036
Keywords for this article
fractional diffusion equation; unique continuation principle; Laplace transform; fractional Theta function

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–Volume 22–3–2019)
  4. Survey Paper
  5. The probabilistic point of view on the generalized fractional partial differential equations
  6. Research Paper
  7. A system of coupled multi-term fractional differential equations with three-point coupled boundary conditions
  8. Fractal convolution: A new operation between functions
  9. Unique continuation principle for the one-dimensional time-fractional diffusion equation
  10. Asymptotics of eigenvalues for differential operators of fractional order
  11. The asymptotic behaviour of fractional lattice systems with variable delay
  12. On fractional asymptotical regularization of linear ill-posed problems in hilbert spaces
  13. Weighted Hölder continuity of Riemann-Liouville fractional integrals – Application to regularity of solutions to fractional cauchy problems with Carathéodory dynamics
  14. Green’s functions, positive solutions, and a Lyapunov inequality for a caputo fractional-derivative boundary value problem
  15. Finite element approximations for fractional evolution problems
  16. Stochastic diffusion equation with fractional Laplacian on the first quadrant
  17. Stability of fractional variable order difference systems
  18. Chaotic dynamics of fractional Vallis system for El-Niño
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