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Theory of Probability and Mathematical Statistics

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Subordination principles for the multi-dimensional space-time-fractional diffusion-wave equation


Author: Yu. Luchko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 98 (2018).
Journal: Theor. Probability and Math. Statist. 98 (2019), 127-147
MSC (2010): Primary 26A33, 35C05, 35E05, 35L05, 45K05, 60E99
DOI: https://doi.org/10.1090/tpms/1067
Published electronically: August 19, 2019
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Abstract: This paper is devoted to an in-depth investigation of the first fundamental solution to the linear multi-dimensional space-time-fractional diffusion-wave equation. This equation is obtained from the diffusion equation by replacing the first order time-derivative by the Caputo fractional derivative of order $ \beta $, $ 0 <\beta \leq 2$, and the Laplace operator by the fractional Laplacian $ (-\Delta )^{\alpha /2}$ with $ 0<\alpha \leq 2$. First, a representation of the fundamental solution in form of a Mellin-Barnes integral is deduced by employing the technique of the Mellin integral transform. This representation is then used for establishing several subordination formulas that connect the fundamental solutions for different values of the fractional derivatives $ \alpha $ and $ \beta $. We also discuss some new cases of completely monotone functions and probability density functions that are expressed in terms of the Mittag-Leffler function, the Wright function, and the generalized Wright function.


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Additional Information

Yu. Luchko
Affiliation: Beuth University of Applied Sciences Berlin
Address at time of publication: Beuth Hochschule für Technik Berlin, Fachbereich II Mathematik - Physik - Chemie, Luxemburger Str. 10, 13353 Berlin
Email: luchko@beuth-hochschule.de

DOI: https://doi.org/10.1090/tpms/1067
Keywords: Multi-dimensional diffusion-wave equation, fundamental solution, Mellin--Barnes integral, Mittag-Leffler function, Wright function, generalized Wright function, completely monotone functions, probability density functions
Received by editor(s): February 23, 2018
Published electronically: August 19, 2019
Dedicated: Dedicated to Julia Loutchko in recognition of her support and encouragement
Article copyright: © Copyright 2019 American Mathematical Society