Wright functions as scale-invariant solutions of the diffusion-wave equation

Journal of Computational and Applied Mathematics

Volumn 118, Issue 1-2, 2000, Pages 175-191

  Gorenflo, Rudolf ^a   Luchko, Yuri ^a   Mainardi, Francesco ^b  

^a FREIE UNIVERSITÄT BERLIN   (Germany)

^b UNIVERSITY OF BOLOGNA   (Italy)

Author keywords

26A33; 33E20; 45J05; 45K05; Diffusion wave equation; Erd lyi Kober operators; Scale invariant solutions; Wright functions

Indexed keywords

EID: 0000103589     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-0427(00)00288-0     Document Type: Article

References (44)

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. 227:1998;81-97.
2. Caputo M. Linear models of dissipation whose. Q is almost frequency independent, Part II Geophys. J. Roy. Astron. Soc. 13:1967;529-539.
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). 7:1996;243-250.
5. Caputo M., Mainardi F. Linear models of dissipation in anelastic solids. Riv. Nuovo Cimento (Ser. II). 1:1971;161-198.
6. Djrbashian M.M. Harmonic Analysis and Boundary Value Problems in the Complex Domain. 1993;Birkhäuser, Basel.
7. Engler H. Similarity solutions for a class of hyperbolic integrodifferential equations. Differential Integral Equations. 10:1997;815-840.
8. Y. Fujita, Integrodifferential equation which interpolates the heat and the wave equations, Osaka J. Math. 27 (1990) 309-321, 797-804.
9. Gajic̀ Lj., Stankovic̀ B. Some properties of Wright's function. Publ. del'Institut Mathèmatique, Beograd, Nouvelle Sèr. 20:1976;91-98.
10. Giona M., Roman H.E. A theory of transport phenomena in disordered systems. Chem. Eng. J. 49:1992;1-10.
11. Gorenflo R., Luchko Yu., Mainardi F. Analytical properties and applications of the Wright function. Fractional Calculus Appl. Anal. 2:1999;383-414.
12. Gorenflo R., Mainardi F. Fractional calculus: integral and differential equations of fractional order. Carpinteri A., Mainardi F. Fractals and Fractional Calculus in Continuum Mechanics. 1997;223-276 Springer, Wien.
13. Gorenflo R., Mainardi F. Fractional calculus and stable probability distributions. Arch. Mech. 50:1998;377-388.
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.
15. Hilfer R. Exact solutions for a class of fractal time random walks. Fractals. 3:1995;211-216.
16. Kiryakova V. Generalized Fractional Calculus and Applications. 1994;Longman, Harlow.
17. Luchko Yu., Gorenflo R. Scale-invariant solutions of a partial differential equation of fractional order. Fractional Calculus Appl. Anal. 1:1998;63-78.
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. Rionero S., Ruggeri T. Waves and Stability in Continuous Media. 1994;246-251 World Scientific, Singapore.
20. Mainardi F. Fractional diffusive waves in viscoelastic solids. Wagner J.L., Norwood F.R. IUTAM Symposium - Nonlinear Waves in Solids. 1995;93-97 ASME, / AMR Fairfield NJ.
21. Mainardi F. The fundamental solutions for the fractional diffusion-wave equation. Appl. Math. Lett. 9(6):1996;23-28.
22. Mainardi F. Fractional calculus: some basic problems in continuum and statistical mechanics. Carpinteri A., Mainardi F. Fractals and Fractional Calculus in Continuum Mechanics. 1997;291-348 Springer, Wien.
23. F. Mainardi, R. Gorenflo, On Mittag-Leffler type functions in fractional evolution processes, this issue, J. Comput. Appl. Math. 118 (2000) 283-299.
24. 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. 1995;171-183 Science Culture Technology, Singapore.
25. Mainardi F., Tomirotti M. Seismic pulse propagation with constant. Q and stable probability distributions Ann. Geofis. 40:1997;1311-1328.
26. Marichev O.I. Handbook of Integral Transforms of Higher Transcendental Functions, Theory and Algorithmic Tables. 1983;Ellis Horwood, Chichester.
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. 211:1994;13-24.
29. Nigmatullin R.R. The realization of the generalized transfer equation in a medium with fractal geometry. Phys. Stat. Sol. B 133:1986;425-430.
30. Olver P.J. Applications of Lie Groups to Differential Equations. 1986;Springer, New York.
31. Pathak R.S. A general differential equation satisfied by a special function. Progr. Math. 6:1972;46-50.
32. Pipkin A.C. Lectures on Viscoelastic Theory. 1986;Springer, New York.
33. Podlubny I. Fractional Differential Equations. 1999;Academic Press, San Diego.
34. Prüss J. Evolutionary Integral Equations and Applications. 1993;Birkhäuser, Basel.
35. Saichev A., Zaslavsky G. Fractional kinetic equations: solutions and applications. Chaos. 7:1997;753-764.
36. Samko S.G., Kilbas A.A., Marichev O.I. Fractional Integrals and Derivatives: Theory and Applications. 1993;Gordon and Breach, New York.
37. Schneider W.R., Wyss W. Fractional diffusion and wave equations. J. Math. Phys. 30:1989;134-144.
38. Stankovic̀ B. On the function of E.M. Wright. Publ. l'Inst. Math. Beograd, Nouvelle Sèr. 10:1970;113-124.
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.
40. Wright E.M. On the coefficients of power series having exponential singularities. J. London Math. Soc. 8:1933;71-79.
41. Wright E.M. The asymptotic expansion of the generalized Bessel function. Proc. London Math. Soc. Ser. II. 38:1935;257-270.
42. Wright E.M. The asymptotic expansion of the generalized hypergeometric function. J. London Math. Soc. 10:1935;287-293.
43. Wright E.M. The generalized Bessel function of order greater than one. Quart. J. Math. Oxford Ser. 11:1940;36-48.
44. Wyss W. Fractional diffusion equation. J. Math. Phys. 27:1986;2782-2785.