Conditional symmetry and reduction of partial differential equations

Fushchich, V. L.; Zhdanov, Renat

doi:10.1007/bf01056141

Cited by 32 publications

(69 citation statements)

References 8 publications

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“…Every value of the arbitrary element G is the image of values of the arbitrary element F running through the two-parametric general solution of (17). Each two equations from class (14), associated with the pairs (F 1 , H ) and (F 2 , H ), where F 1 and F 2 are particular solutions of (17) H ) and (F, H ) have the same numbers.…”

Section: Group Classification Of the Double-imaged Classmentioning

confidence: 99%

“…Elements of Q f will be identified with their representatives in Q. If the equation L is conditionally invariant with respect to the operator Q, then it is conditionally invariant with respect to any operator equivalent to Q [17,58]. Therefore, the equivalence relation on Q induces a well-defined equivalence relation on Q(L); and the factorization of Q with respect to this equivalence relation can be naturally restricted on Q(L) that results in the subset Q f (L) of Q f .…”

Section: Definitionmentioning

confidence: 99%

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

2008

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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 coefficientis 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).

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Section: Group Classification Of the Double-imaged Classmentioning

confidence: 99%

Section: Definitionmentioning

confidence: 99%

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

2008

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“…So, we will also call operators of conditional symmetry by reduction operators of L. Lemma 1. [23,26] If the equation L is conditionally invariant with respect to the operator Q, then it is conditionally invariant with respect to any operator which is equivalent to Q.…”

Section: Equivalence Of Reduction Operators With Respect To Transformmentioning

confidence: 99%

Potential nonclassical symmetries and solutions of fast diffusion equation

2007

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The fast diffusion equation u t = (u −1 u x ) x is investigated from the symmetry point of view in development of the paper by Gandarias [M.L. Gandarias, Phys. Lett. A 286 (2001) 153]. After studying equivalence of nonclassical symmetries with respect to a transformation group, we completely classify the nonclassical symmetries of the corresponding potential equation. As a result, new wide classes of potential nonclassical symmetries of the fast diffusion equation are obtained. The set of known exact non-Lie solutions are supplemented with the similar ones. It is shown that all known non-Lie solutions of the fast diffusion equation are exhausted by ones which can be constructed in a regular way with the above potential nonclassical symmetries. Connection between classes of nonclassical and potential nonclassical symmetries of the fast diffusion equation is found.

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“…Another extension of the classical symmetry method is to require that both the considered partial differential equation (PDE) and the invariant surface are invariant under certain vector fields, while the classical symmetry method only requires the considered PDEs are invariant under the vector field. The idea of this extension is the key point of the conditional symmetry method called by Fuschych et al [8], which was first introduced as the nonclassical method by Bluman and Cole [9]. Olver and Rosenau [10] extended the nonclassical symmetry to the weak symmetry.…”

Section: Introductionmentioning

confidence: 98%

Conditional Lie‐Bäcklund Symmetry of Evolution System and Application for Reaction‐Diffusion System

Shen

2014

Stud Appl Math

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Conditional Lie-Bäcklund symmetry (CLBS) method is developed to study system of evolution equations. It is shown that reducibility of a system of evolution equations to a system of ordinary differential equations can be fully characterized by the CLBS of the considered system. As an application of the approach, a class of two-component nonlinear diffusion equations is studied. The governing system and the admitted CLBS can be identified. As a consequence, exact solutions defined on the polynomial, exponential, trigonometric, and mixed invariant subspaces are constructed due to the corresponding symmetry reductions.

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Conditional symmetry and reduction of partial differential equations

Cited by 32 publications

References 8 publications

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

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

Potential nonclassical symmetries and solutions of fast diffusion equation

Conditional Lie‐Bäcklund Symmetry of Evolution System and Application for Reaction‐Diffusion System

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