**Preprint No.**
A-98-01

# Evelyn Buckwar, Yurii Luchko

## Invariance of a partial differential equation of fractional order under a certain Lie group

**Abstract:** In this article a symmetry group of
scaling transformations is determined for a partial differential equation of
fractional order $\alpha$, containing among particular cases the diffusion
equation, the wave equation, and the
fractional diffusion-wave equation. For its group-invariant solutions an
ordinary differential equation of fractional order with
the new independent variable $z=xt^{-\alpha/2}$ is derived. The
derivative then is an Erdelyi-Kober derivative depending on a parameter
$\alpha$. Its complete solution is given in terms of generalized Wright
functions.
**Keywords: **Fractional diffusion-wave equation, group-invariant solutions, Erdelyi-Kober derivative, generalized Wright function

**Mathematics Subject Classification (MSC91):** 45K05 Integro-partial differential equations
, 45J05 Integro-ordinary differential equations

**Language:** ENG
**Available:** Pr-A-98-01.ps

**Contact**: Evelyn Buckwar, Yurii Luchko, Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2-6, 14195 Berlin, Germany (luchko@math.fu-berlin.de)

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