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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:
58J72Correspondences and other transformation methods (PDE on manifolds)
26A33Fractional derivatives and integrals (real functions)
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References:
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