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Symmetries and form-preserving transformations of generalised inhomogeneous nonlinear diffusion equations

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  • Sophocleous, Christodoulos

Abstract

We consider the variable coefficient inhomogeneous nonlinear diffusion equations of the form f(x)ut=[g(x)unux]x. We present a complete classification of Lie symmetries and form-preserving point transformations in the case where f(x)=1 which is equivalent to the original equation. We also introduce certain nonlocal transformations. When f(x)=xp and g(x)=xq we have the most known form of this class of equations. If certain conditions are satisfied, then this latter equation can be transformed into a constant coefficient equation. It is also proved that the only equations from this class of partial differential equations that admit Lie–Bäcklund symmetries is the well-known nonlinear equation ut=[u−2ux]x and an equivalent equation. Finally, two examples of new exact solutions are given.

Suggested Citation

  • Sophocleous, Christodoulos, 2003. "Symmetries and form-preserving transformations of generalised inhomogeneous nonlinear diffusion equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 324(3), pages 509-529.
    • Handle: RePEc:eee:phsmap:v:324:y:2003:i:3:p:509-529
    DOI: 10.1016/S0378-4371(03)00063-3
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    References listed on IDEAS

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    1. Sophocleous, Christodoulos, 2003. "Classification of potential symmetries of generalised inhomogeneous nonlinear diffusion equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 320(C), pages 169-183.
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    Cited by:

    1. Feng, Wei & Ji, Lina, 2013. "Conditional Lie–Bäcklund symmetries and functionally separable solutions of the generalized inhomogeneous nonlinear diffusion equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(4), pages 618-627.
    2. Sophocleous, Christodoulos, 2005. "Further transformation properties of generalised inhomogeneous nonlinear diffusion equations with variable coefficients," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 345(3), pages 457-471.
    3. Ji, Lina, 2010. "Conditional Lie–Bäcklund symmetries and solutions of inhomogeneous nonlinear diffusion equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(24), pages 5655-5661.

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