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Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations. (English) Zbl 0932.58038
For $\alpha \ge 1$, $D > 0$, the authors consider the linear fractional order differential equation $\partial^\alpha u/\partial t^\alpha = D \partial^2 u/\partial x^2$, in the sense of the Riemann-Liouville fractional calculus. Similarity solutions with respect to the scaling transformations are found to be functions of the invariant $z = x t^{-\alpha/2}$. For them an ordinary differential equation in the Erdelyi-Kober derivative is obtained. As the final result, the general scale-invariant solution is computed in terms of Wright and generalized Wright functions.

##### MSC:
 58J72 Correspondences and other transformation methods (PDE on manifolds) 26A33 Fractional derivatives and integrals (real functions)
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##### References:
 [1] Engler, H.: Similarity solutions for a class of hyperbolic integrodifferential equations. Differential integral equations 10, 815-840 (1997) · Zbl 0892.45005 [2] Fujita, Y.: Integrodifferential equation which interpolates the heat and the wave equations. Osaka J. Math. 27, 309-321 (1990) · Zbl 0790.45009 [3] Grigoriev, Yu.N.; Meleshko, S. V.: Group analysis of kinetic equations. Russian J. Numer. anal. Math. modeling 10, 425-447 (1995) · Zbl 0838.35101 [4] Ibragimov, N. H.: Symmetries, exact solutions and conservation laws. (1994) · Zbl 0864.35001 [5] Ibragimov, N. H.: Applications in engineering and physical sciences. (1995) · Zbl 0864.35002 [6] Ibragimov, N. H.: New trends in theoretical developments and computational methods. (1996) · Zbl 0864.35003 [7] Kiryakova, V.: Generalized fractional calculus and applications. Pitman res. Notes in math. 301 (1994) · Zbl 0882.26003 [8] Logan, J. D.: An introduction to nonlinear partial differential equations. (1994) · Zbl 0834.35001 [9] Luchko, Yu.F.; Yakubovich, S. B.: Operational method of solution of some classes of integro-differential equations. Differentsial’nye uravneniya 30, 269-280 (1994) · Zbl 0827.44004 [10] Mainardi, F.: Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos solitons fractals 7, 1461-1477 (1996) · Zbl 1080.26505 [11] Mainardi, F.: The fundamental solutions for the fractional diffusion-wave equation. Appl. math. Lett. 9, 23-28 (1996) · Zbl 0879.35036 [12] Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics. Fractals and fractional calculus in continuum mechanics, 291-348 (1997) · Zbl 0917.73004 [13] Marichev, O. I.: Handbook of integral transforms of higher transcendental functions, theory and algorithmic tables. (1983) · Zbl 0494.33001 [14] Olver, P. J.: Applications of Lie groups to differential equations. (1986) · Zbl 0588.22001 [15] Srivastava, H. M.; Gupta, K. C.; Goyal, S. P.: The H-functions of one and two variables with applications. (1982) · Zbl 0506.33007 [16] Prüss, J.: Evolutionary integral equations and applications. (1993) · Zbl 0784.45006 [17] Saichev, A.; Zaslavsky, G.: Fractional kinetic equations: solutions and applications. Chaos 7, 753-764 (1997) · Zbl 0933.37029 [18] Schneider, W. R.; Wyss, W.: Fractional diffusion and wave equations. J. math. Phys. 30, 134-144 (1989) · Zbl 0692.45004 [19] Wright, E. M.: The generalized Bessel functions of order greater than one. Quart. J. Math. Oxford ser. (2) 11, 36-48 (1940) · Zbl 0023.14101 [20] Wyss, W.: Fractional diffusion equation. J. math. Phys. 27, 2782-2785 (1986) · Zbl 0632.35031 [21] Yakubovich, S. B.; Luchko, Yu.F.: The hypergeometric approach to integral transforms and convolutions. (1994) · Zbl 0803.44001