## Wright functions as scale-invariant solutions of the diffusion-wave equation.(English)Zbl 0973.35012

The authors obtain the time-fractional diffusion-wave equation from the classical diffusion or wave equation by replacing the first- or second-order time derivative by a fractional derivative of order $$\alpha$$ $$(0<\alpha\leq 2)$$.
They show by using the similarity method and the method of the Laplace transform that the scale-invariant solutions of the mixed problem of signaling type for time-fractional diffusion-wave equation are given in terms of the Wright function in the case $$0<\alpha< 1$$ and in terms of the generalized Wright function in the case $$1<\alpha< z$$.
The authors give the reduced equation for the scale-invariant solutions in terms of the Caputo-type modification of the Erdélyi-Kober fractional differential operator.

### MSC:

 35A25 Other special methods applied to PDEs 26A33 Fractional derivatives and integrals 33E20 Other functions defined by series and integrals 45J05 Integro-ordinary differential equations 45K05 Integro-partial differential equations
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### References:

 [1] Buckwar, E.; Luchko, Yu, Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations, J. math. anal. appl., 227, 81-97, (1998) · Zbl 0932.58038 [2] Caputo, M., Linear models of dissipation whose Q is almost frequency independent, part II, Geophys. J. roy. astron. soc., 13, 529-539, (1967) [3] M. Caputo, Elasticità e Dissipazione, Zanichelli, Bologna, 1969 (in Italian). [4] Caputo, M., The Green function of the diffusion of fluids in porous media with memory, Rend. fis. acc. lincei (ser. 9), 7, 243-250, (1996) · Zbl 0879.76098 [5] Caputo, M.; Mainardi, F., Linear models of dissipation in anelastic solids, Riv. nuovo cimento (ser. II), 1, 161-198, (1971) [6] Djrbashian, M.M., Harmonic analysis and boundary value problems in the complex domain, (1993), Birkhäuser Basel · Zbl 0816.30023 [7] Engler, H., Similarity solutions for a class of hyperbolic integrodifferential equations, Differential integral equations, 10, 815-840, (1997) · Zbl 0892.45005 [8] Y. Fujita, Integrodifferential equation which interpolates the heat and the wave equations, Osaka J. Math. 27 (1990) 309-321, 797-804. · Zbl 0790.45009 [9] Gajic̀, Lj.; Stankovic̀, B., Some properties of Wright’s function, Publ. del’institut mathèmatique, beograd, nouvelle Sèr, 20, 91-98, (1976) · Zbl 0343.33011 [10] Giona, M.; Roman, H.E., A theory of transport phenomena in disordered systems, Chem. eng. J., 49, 1-10, (1992) [11] Gorenflo, R.; Luchko, Yu.; Mainardi, F., Analytical properties and applications of the wright function, Fractional calculus appl. anal., 2, 383-414, (1999) · Zbl 1027.33006 [12] Gorenflo, R.; Mainardi, F., Fractional calculus: integral and differential equations of fractional order, (), 223-276 [13] Gorenflo, R.; Mainardi, F., Fractional calculus and stable probability distributions, Arch. mech., 50, 377-388, (1998) · Zbl 0934.35008 [14] R. Gorenflo, F. Mainardi, H.M. Srivastava, Special functions in fractional relaxation-oscillation and fractional diffusion-wave phenomena, in: D. Bainov (Ed.), Proceedings VIII International Colloquium on Differential Equations, Plovdiv 1997, VSP, Utrecht, 1998, pp. 195-202. · Zbl 0921.33009 [15] Hilfer, R., Exact solutions for a class of fractal time random walks, Fractals, 3, 211-216, (1995) · Zbl 0881.60066 [16] Kiryakova, V., Generalized fractional calculus and applications, (1994), Longman Harlow · Zbl 0882.26003 [17] Luchko, Yu.; Gorenflo, R., Scale-invariant solutions of a partial differential equation of fractional order, Fractional calculus appl. anal., 1, 63-78, (1998) · Zbl 0940.45001 [18] Yu. Luchko, R. Gorenflo, The initial value problem for some fractional differential equations with the Caputo derivatives, preprint A-08-98, Fachbereich Mathematik und Informatik, Freie Universität Berlin, 1998. [http://www.math.fu-berlin.de/publ/index.html] [19] Mainardi, F., On the initial value problem for the fractional diffusion-wave equation, (), 246-251 [20] Mainardi, F., Fractional diffusive waves in viscoelastic solids, (), 93-97 [21] Mainardi, F., The fundamental solutions for the fractional diffusion-wave equation, Appl. math. lett., 9, 6, 23-28, (1996) · Zbl 0879.35036 [22] Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, (), 291-348 · Zbl 0917.73004 [23] F. Mainardi, R. Gorenflo, On Mittag-Leffler type functions in fractional evolution processes, this issue, J. Comput. Appl. Math. 118 (2000) 283-299. · Zbl 0970.45005 [24] Mainardi, F.; Tomirotti, M., On a special function arising in the time fractional diffusion-wave equation, (), 171-183 · Zbl 0921.33010 [25] Mainardi, F.; Tomirotti, M., Seismic pulse propagation with constant Q and stable probability distributions, Ann. geofis., 40, 1311-1328, (1997) [26] Marichev, O.I., Handbook of integral transforms of higher transcendental functions, theory and algorithmic tables, (1983), Ellis Horwood Chichester · Zbl 0494.33001 [27] D. Matignon, G. Montseny (Eds.), Fractional Differential Systems: Models, Methods and Applications, Proceedings of the Colloquium FDS ’98, ESAIM (European Ser. Appl. Ind. Math.) Proceedings, Vol. 5, 1996. [http://www.emath.fr/Maths/Proc/Vol.5/index.htm] [28] Metzler, R.; Glöckle, W.G.; Nonnenmacher, T.F., Fractional model equation for anomalous diffusion, Physica A, 211, 13-24, (1994) [29] Nigmatullin, R.R., The realization of the generalized transfer equation in a medium with fractal geometry, Phys. stat. sol., B 133, 425-430, (1986) [30] Olver, P.J., Applications of Lie groups to differential equations, (1986), Springer New York · Zbl 0656.58039 [31] Pathak, R.S., A general differential equation satisfied by a special function, Progr. math., 6, 46-50, (1972) · Zbl 0269.33010 [32] Pipkin, A.C., Lectures on viscoelastic theory, (1986), Springer New York · Zbl 0625.73037 [33] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010 [34] Prüss, J., Evolutionary integral equations and applications, (1993), Birkhäuser Basel · Zbl 0793.45014 [35] Saichev, A.; Zaslavsky, G., Fractional kinetic equations: solutions and applications, Chaos, 7, 753-764, (1997) · Zbl 0933.37029 [36] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives: theory and applications, (1993), Gordon and Breach New York · Zbl 0818.26003 [37] Schneider, W.R.; Wyss, W., Fractional diffusion and wave equations, J. math. phys., 30, 134-144, (1989) · Zbl 0692.45004 [38] Stankovic̀, B., On the function of E.M. wright, Publ. l’inst. math. beograd, nouvelle Sèr., 10, 113-124, (1970) · Zbl 0204.08404 [39] Srivastava, H.M.; Gupta, K.C.; Goyal, S.P., The H-functions of one and two variables with applications, (1982), South Asian Publishers New Delhi · Zbl 0506.33007 [40] Wright, E.M., On the coefficients of power series having exponential singularities, J. London math. soc., 8, 71-79, (1933) · JFM 59.0383.01 [41] Wright, E.M., The asymptotic expansion of the generalized Bessel function, Proc. London math. soc. ser. II, 38, 257-270, (1935) · Zbl 0010.21103 [42] Wright, E.M., The asymptotic expansion of the generalized hypergeometric function, J. London math. soc., 10, 287-293, (1935) · Zbl 0013.02104 [43] Wright, E.M., The generalized Bessel function of order greater than one, Quart. J. math. Oxford ser., 11, 36-48, (1940) · Zbl 0023.14101 [44] Wyss, W., Fractional diffusion equation, J. math. phys., 27, 2782-2785, (1986) · Zbl 0632.35031
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