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Classification of potential symmetries of generalised inhomogeneous nonlinear diffusion equations

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  • Sophocleous, Christodoulos

Abstract

We consider the class of generalised nonlinear diffusion equations f(x)ut=[g(x)unux]x which are of considerable interest in mathematical physics. We classify the nonlocal symmetries, which are known as potential symmetries, for these equations. It turns out that potential symmetries exist only if the parameter n takes the values −2 or −23. Also certain relations must be satisfied by the functions f(x) and g(x). For the cases where we obtain infinite-parameter potential symmetries, linearising mappings are constructed. Furthermore we employ the potential symmetries to derive similarity solutions.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:phsmap:v:320:y:2003:i:c:p:169-183
    DOI: 10.1016/S0378-4371(02)01591-1
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    References listed on IDEAS

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    1. Khater, A.H. & Moussa, M.H.M. & Abdul-Aziz, S.F., 2002. "Potential symmetries and invariant solutions for the inhomogeneous nonlinear diffusion equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 312(1), pages 99-108.
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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. Ivanova, N.M. & Popovych, R.O. & Sophocleous, C. & Vaneeva, O.O., 2009. "Conservation laws and hierarchies of potential symmetries for certain diffusion equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(4), pages 343-356.
    3. 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.
    4. Gandarias, M.L., 2008. "New potential symmetries for some evolution equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(10), pages 2234-2242.
    5. 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.
    6. 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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    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. 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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