Subordination principles for the multi-dimensional space-time-fractional diffusion-wave equation
Author:
Yu. Luchko
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
MathSciNet review:
3824683
Full-text PDF
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Additional Information
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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References
- E. Bazhlekova, Subordination principle for fractional evolution equations, Fract. Calc. Appl. Anal. 3 (2000), 213–230. MR 1788162
- E. Bajlekova, Fractional Evolution Equations in Banach Spaces, Ph.D. thesis, Eindhoven, The Netherlands, 2001. MR 1868564
- E. Bazhlekova, Completely monotone functions and some classes of fractional evolution equations, Integral Transforms and Special Functions 26 (2015), no. 9, 737–752. MR 3354052
- E. Bazhlekova and I. B. Bazhlekov, Subordination approach to multi-term time-fractional diffusion-wave equations, J. Comput. Appl. Math. (in press); DOI: 10.1016/j.cam.2017.11.003. MR 3787686
- L. Boyadjiev and Yu. Luchko, Mellin integral transform approach to analyze the multidimensional diffusion-wave equations, Chaos Solitons Fractals 102 (2017), 127–134. MR 3672003
- L. Boyadjiev and Yu. Luchko, Multi-dimensional $\alpha$-fractional diffusion-wave equation and some properties of its fundamental solution, Comput. Math. Appl. 73 (2017), 2561–2572. MR 3649996
- S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, J. Differential Equations 199 (2004), 211–255. MR 2047909
- A. Erdélyi, Higher Transcendental Functions, vol. 2, McGraw-Hill, New York, 1953.
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- M. Ferreira and N. Vieira, Fundamental solutions of the time fractional diffusion-wave and parabolic Dirac operators, J. Math. Anal. Appl. 447 (2016), 329–353. MR 3566475
- C. Fox, The $G$- and $H$-functions as symmetrical Fourier kernels, Trans. Amer. Math. Soc. 98 (1961), 395–429. MR 131578
- R. Gorenflo, J. Loutchko, and Yu. Luchko, Computation of the Mittag-Leffler function and its derivatives, Fract. Calc. Appl. Anal. 5 (2002), 491–518. MR 1967847
- R. Gorenflo, A. A. Kilbas, F. Mainardi, and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer, Berlin, 2014. MR 3244285
- A. Hanyga, Multi-dimensional solutions of space-time-fractional diffusion equations, Proc. R. Soc. Lond. A 458 (2002), 429–450. MR 1889936
- A. Hanyga, Multidimensional solutions of time-fractional diffusion-wave equations, Proc. R. Soc. Lond. A 458 (2002), 933–957. MR 1898095
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- J. Kemppainen, J. Siljander, and R. Zacher, Representation of solutions and large-time behavior for fully nonlocal diffusion equations, J. Diff. Equat. 263 (2017), 149–201. MR 3631303
- V. Kiryakova, Generalized Fractional Calculus and Applications, Longman, Harlow, 1994. MR 1265940
- A. N. Kochubei, Fractional-order diffusion, Differ. Equ. 26 (1990), 485–492. MR 1061448
- A. N. Kochubei, Cauchy problem for fractional diffusion-wave equations with variable coefficients, Appl. Anal. 93 (2014), 2211–2242. MR 3240385
- M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal. 20 (2017), 7–51. MR 3613319
- K.-H. Kim and S. Lim, Asymptotic behaviors of fundamental solution and its derivatives related to space-time fractional differential equations, J. Korean Math. Soc. 53 (2016), 929–967. MR 3521245
- Yu. Luchko, Operational method in fractional calculus, Fract. Calc. Appl. Anal. 2 (1999), 463–489. MR 1752383
- Yu. Luchko, Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation, Comput. Math. Appl. 59 (2010), 1766–1772. MR 2595950
- Yu. Luchko, Fractional wave equation and damped waves, J. Math. Phys. 54 (2013), 031505. MR 3059427
- Yu. Luchko, Multi-dimensional fractional wave equation and some properties of its fundamental solution, Commun. Appl. Ind. Math. 6 (2014), e-485. MR 3277315
- Yu. Luchko, Wave-diffusion dualism of the neutral-fractional processes, J. Comput. Phys. 293 (2015), 40–52. MR 3342455
- Yu. Luchko, A new fractional calculus model for the two-dimensional anomalous diffusion and its analysis, Math. Model. Nat. Phenom. 11 (2016), 1–17. MR 3530396
- Yu. Luchko, On some new properties of the fundamental solution to the multi-dimensional space- and time-fractional diffusion-wave equation, Mathematics 5 (2017), no. 4, 1–16.
- Yu. Luchko and R. Gorenflo, Scale-invariant solutions of a partial differential equation of fractional order, Fract. Calc. Appl. Anal. 1 (1998), 63–78. MR 1662409
- Yu. Luchko and V. Kiryakova, The Mellin integral transform in fractional calculus, Fract. Calc. Appl. Anal. 16 (2013), 405–430. MR 3033696
- F. Mainardi and G. Pagnini, Salvatore Pincherle: The pioneer of the Mellin–Barnes integrals, J. Comput. Appl. Math. 153 (2003), 331–342. MR 1985704
- F. Mainardi, Yu. Luchko, and G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal. 4 (2001), 153–192. MR 1829592
- O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions, Theory and Algorithmic Tables, Ellis Horwood, Chichester, 1983. MR 689711
- A. M. Mathai and R. K. Saxena, The $H$-functions with Applications in Statistics and Other Disciplines, John Wiley, New York, 1978. MR 513025
- K. S. Miller and S. G. Samko, Completely monotonic functions, Integral Transforms and Special Functions 12 (2001), 389–402. MR 1872377
- J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser, Basel, 1993. MR 2964432
- A. Saichev and G. Zaslavsky, Fractional kinetic equations: Solutions and applications, Chaos 7 (1997), 753–764. MR 1604710
- S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, 1993. MR 1347689
- R. L. Schilling, R. Song, and Z. Vondraček, Bernstein Functions. Theory and Applications, De Gruyter, Berlin, 2010. MR 2978140
- W. R. Schneider and W. Wyss, Fractional diffusion and wave equations, J. Math. Phys. 30 (1989), 134–144. MR 974464
- E. M. Wright, The asymptotic expansion of the generalized hypergeometric function, J. Lond. Math. Soc. 10 (1935), 287–293. MR 0003876
- S. Yakubovich and Yu. Luchko, The Hypergeometric Approach to Integral Transforms and Convolutions, Kluwer Acad. Publ., Dordrecht, 1994. MR 1304259
- K. Yosida, Functional Analysis, Springer-Verlag, Berlin, 1965. MR 0180824
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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
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