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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)