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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 α1, D>0, the authors consider the linear fractional order differential equation α u/t α =D 2 u/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=xt -α/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:
58J72Correspondences and other transformation methods (PDE on manifolds)
26A33Fractional derivatives and integrals (real functions)