The fundamental solutions for the fractional diffusion-wave equation. (English) Zbl 0879.35036

The author provides the fundamental solutions of the Cauchy and signalling problems for the fractional evolution equation \(\partial^{2\beta}u/\partial t^{2\beta}=D\partial^2u/\partial x^2\), where \(0<\beta\leq1\), \(D>0\), \(x\) and \(t\) are the space-time variables and \(u(x,t;\beta)\) is the field variable, which is assumed to be causal, that is, vanishing for \(t<0\). These solutions are expressed by the corresponding Green’s functions and are analyzed by the Laplace transform and are expressed in terms of an entire auxiliary function \(M(z;\beta)\) of Wright type, where \(z=|x|/t^\beta\) is the similarity variable.


35G10 Initial value problems for linear higher-order PDEs
26A33 Fractional derivatives and integrals
35A08 Fundamental solutions to PDEs
Full Text: DOI


[1] Davis, H. T., The Theory of Linear Operators (1936), The Principia Press: The Principia Press Bloomington · JFM 62.0457.02
[2] (Erdélyi, A., Tables of Integral Transforms, Volume 2 (1954), McGraw-Hill: McGraw-Hill New York), Chapter 13 · Zbl 1378.37001
[3] Gel’fand, I. M.; Shilov, G. E., (Generalized Functions, Volume 1 (1964), Academic Press: Academic Press New York) · Zbl 0115.33101
[4] Caputo, M., Elasticità e Dissipazione (1969), Zanichelli: Zanichelli Bologna, (in Italian)
[5] Oldham, K. B.; Spanier, J., The Fractional Calculus (1974), Academic Press: Academic Press New York · Zbl 0428.26004
[6] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives, Theory and Applications, ((1993), Gordon and Breach: Gordon and Breach Amsterdam). (Nauka i Tekhnika (1987), Gordon and Breach: Gordon and Breach Amsterdam), Minsk · Zbl 0617.26004
[7] Gorenflo, R.; Vessella, S., Abel integral equations: Analysis and applications, (Lecture Notes in Mathematics (1991), Springer-Verlag: Springer-Verlag Berlin), #1461 · Zbl 0717.45002
[8] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), Wiley: Wiley New York · Zbl 0789.26002
[9] Nigmatullin, R. R., The realization of the generalized transfer equation in a medium with fractal geometry, Phys. Stat. Sol. B, 133, 425-430 (1986)
[10] Appl. Mech. Rev., 46, 549 (1993)
[11] Wyss, W., Fractional diffusion equation, J. Math. Phys., 27, 2782-2785 (1986) · Zbl 0632.35031
[12] Schneider, W. R.; Wyss, W., Fractional diffusion and wave equations, J. Math. Phys., 30, 134-144 (1989) · Zbl 0692.45004
[13] Kochubei, A. N., A Cauchy problem for evolution equations of fractional order, J. Diff. Eqs., 25, 8, 967-974 (1989) · Zbl 0696.34047
[14] Kochubei, A. N., Fractional order diffusion, J. Diff. Eqs., 26, 4, 485-492 (1990) · Zbl 0729.35064
[15] Mainardi, F., On the initial value problem for the fractional diffusion-wave equation, (Rionero, S.; Ruggeri, T., Waves and Stability in Continuous Media (1994), World Scientific: World Scientific Singapore), 246-251
[16] Mainardi, F.; Tomirotti, M., On a special function arising in the time fractional diffusion-wave equation, (Rusev, P.; Dimovski, I.; Kiryakova, V., Transform Methods and Special Functions, Sofia 1994 (1995), Science Culture Technology: Science Culture Technology Singapore), 171-183 · Zbl 0921.33010
[17] (Erdélyi, A., Higher Transcendental Functions, Volume 3 (1955), McGraw-Hill: McGraw-Hill New York), Chapter 18 · Zbl 1378.37001
[18] Bender, C. M.; Orszag, S. A., Advanced Mathematical Methods for Scientists and Engineers (1987), McGraw-Hill: McGraw-Hill Singapore, Chapter 3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.