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Acta Applicandae Mathematicae

, Volume 106, Issue 1, pp 1–46 | Cite as

Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source

  • O. O. Vaneeva
  • R. O. Popovych
  • C. Sophocleous
Article

Abstract

A new approach to group classification problems and more general investigations on transformational properties of classes of differential equations is proposed. It is based on mappings between classes of differential equations, generated by families of point transformations. A class of variable coefficient (1+1)-dimensional semilinear reaction–diffusion equations of the general form f(x)u t =(g(x)u x ) x +h(x)u m (m≠0,1) is studied from the symmetry point of view in the framework of the approach proposed. The singular subclass of the equations with m=2 is singled out. The group classifications of the entire class, the singular subclass and their images are performed with respect to both the corresponding (generalized extended) equivalence groups and all point transformations. The set of admissible transformations of the imaged class is exhaustively described in the general case m≠2. The procedure of classification of nonclassical symmetries, which involves mappings between classes of differential equations, is discussed. Wide families of new exact solutions are also constructed for equations from the classes under consideration by the classical method of Lie reductions and by generation of new solutions from known ones for other equations with point transformations of different kinds (such as additional equivalence transformations and mappings between classes of equations).

Keywords

Group classification of differential equations Group analysis of differential equations Equivalence group Admissible transformations Lie symmetry Conditional symmetry Exact solutions Reaction–diffusion equation 

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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  • O. O. Vaneeva
    • 1
  • R. O. Popovych
    • 1
    • 2
  • C. Sophocleous
    • 3
  1. 1.Institute of Mathematics of National Academy of Sciences of UkraineKyiv-4Ukraine
  2. 2.Fakultät für MathematikUniversität WienWienAustria
  3. 3.Department of Mathematics and StatisticsUniversity of CyprusNicosiaCyprus

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