Cauchy problem for fractional diffusion equations

By S. Eidelman, Anatoly N. Kochubei. 2003

426 citations recorded by the source corpus.

Abstract We consider an evolution equation with the regularized fractional derivative of an order α ∈(0,1) with respect to the time variable, and a uniformly elliptic operator with variable coefficients acting in the spatial variables. Such equations describe diffusion on inhomogeneous fractals. A fundamental solution of the Cauchy problem is constructed and investigated.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/af1030f18503aec05f6ef6b1da06983cf76f529b

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In Applied Mathematics

1. Yuri Luchko
2. V. Vijayakumar
3. Masahiro Yamamoto
4. Yong Zhou
5. Yongxiang Li
6. Francesco Mainardi
7. Pengyu Chen
8. Carlos Lizama

In Modeling and Simulation

1. Yuri Luchko
2. Masahiro Yamamoto
3. Rudolf Gorenflo
4. Francesco Mainardi
5. Yong Zhou
6. V. Vijayakumar
7. Carlos Lizama
8. Yongxiang Li

In Numerical Analysis

1. Yuri Luchko
2. Masahiro Yamamoto
3. Francesco Mainardi
4. Rudolf Gorenflo
5. Yongxiang Li
6. V. Vijayakumar
7. Carlos Lizama
8. Yong Zhou

Citations in the corpus

• The fundamental solution of a diffusion-wave equation of fractional order (2009, background): “The authors of [3]–[5] study the Cauchy problem for diffusion equations of fractional order (0 < α < 1) with a regularized fractional derivative (Caputo derivative) and with an elliptic operator whose coefficients depend on the space variables.”
• Maximum principle for the generalized time-fractional diffusion equation (2009, neutral): “[4,6,9,10,15,17,19].”
• Mild solutions for abstract fractional differential equations (2013, neutral): “[1–11] and the references therein).”
• An inverse problem for a one-dimensional time-fractional diffusion problem (2012, background): “We note that the mathematical and numerical analysis of the direct problem of the timefractional diffusion has gained much attention in recent years; see [24, 16, 5, 12, 22] for a rather incomplete list.”
• Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions (2013, result): “[5], Sabatier et al. [6], Kilbas et al. [7], and the papers by Eidelman and Kochubei [8], Lakshmikantham and Vatsala [9] and Agarwal et al. [10].”
• Controllability for fractional evolution inclusions without compactness (2015, background): “[14], Podlubny [20], Diethelm [6], Tarasov [23], Zhou [30, 31] and the papers [25, 21, 15, 8, 29, 32].”
• Hölder continuous solutions for fractional differential equations and maximal regularity (2013, background): “approaches to the existence and uniqueness of solutions to fractional differential equations can be found, for example, in [6, 9, 13, 17, 20, 32].”
• Numerical methods for time-fractional evolution equations with nonsmooth data: A concise overview (2018, background): “Eidelman and Kochubei [21] derived fundamental solutions to problem in the whole space using Fox H-functions, and derived various estimates, see also [92, 29];.”
• Distributed order calculus and equations of ultraslow diffusion (2007, methodology): “19) does not coincide with the fundamental solution Z—just as this happens for fractional diffusion equations [9,10].”
• Overview to mathematical analysis for fractional diffusion equations - new mathematical aspects motivated by industrial collaboration (2010, methodology): “Kochubei [21], [22] applied the semigroup theory in Banach spaces, and Eidelman and Kochubei [8] constructed the fundamental solution in Rn and proved the maximum principle for the Cauchy problem.”