## Similarity solutions to nonlinear heat conduction and Burgers/Korteweg-deVries fractional equations.(English)Zbl 1157.35470

Summary: We analyze self-similar solutions to a nonlinear fractional diffusion equation and fractional Burgers/Korteweg-deVries equation in one spatial variable. By using Lie-group scaling transformation, we determined the similarity solutions. After the introduction of the similarity variables, both problems are reduced to ordinary nonlinear fractional differential equations. In two special cases exact solutions to the ordinary fractional differential equation, which is derived from the diffusion equation, are presented. In several other cases the ordinary fractional differential equations are solved numerically, for several values of governing parameters. In formulating the numerical procedure, we use special representation of a fractional derivative that is recently obtained.

### MSC:

 35Q53 KdV equations (Korteweg-de Vries equations) 35A30 Geometric theory, characteristics, transformations in context of PDEs 35C05 Solutions to PDEs in closed form 35C20 Asymptotic expansions of solutions to PDEs
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