Conditional symmetry of a porous medium equation

Zhdanov, Renat; Lahno, V.

doi:10.1016/s0167-2789(98)00191-2

Cited by 65 publications

(63 citation statements)

References 10 publications

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“…The natural reason to avoid examination of case (b) (so-called no-go case) follows from the well-known fact (firstly proved in [32]) that a complete description of Q-conditional symmetries of the form (44) for scalar evolution equations is equivalent to solving the equation in question.…”

Section: Q-conditional Symmetries Of An Rdc Equation With Exponentialmentioning

confidence: 99%

Lie and Q-Conditional Symmetries of Reaction-Diffusion-Convection Equations with Exponential Nonlinearities and Their Application for Finding Exact Solutions

2018

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This review is devoted to search for Lie and Q-conditional (nonclassical) symmetries and exact solutions of a class of reaction-diffusion-convection equations with exponential nonlinearities. A complete Lie symmetry classification of the class is derived via two different algorithms in order to show that the result depends essentially on the type of equivalence transformations used for the classification. Moreover, a complete description of Q-conditional symmetries for PDEs from the class in question is also presented. It is shown that all the well-known results for reaction-diffusion equations with exponential nonlinearities follow as particular cases from the results derived for this class of reaction-diffusion-convection equations. The symmetries obtained for constructing exact solutions of the relevant equations are successfully applied. The exact solutions are compared with those found by means of different techniques. Finally, an application of the exact solutions for solving boundary-value problems arising in population dynamics is presented.

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Section: Q-conditional Symmetries Of An Rdc Equation With Exponentialmentioning

confidence: 99%

Lie and Q-Conditional Symmetries of Reaction-Diffusion-Convection Equations with Exponential Nonlinearities and Their Application for Finding Exact Solutions

2018

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“…Since in this case ξ = 0, up to the equivalence of reduction operators we can assume ξ = 1 that implies Q = ∂ x + η∂ u . The conditional invariance criterion results in only a single determining equation on the coefficient η, which is reduced with a non-point transformation to the initial equation, where η becomes a parameter [18,54]. This is why the case τ = 0 is called "no-go"; and it has to be excluded from consideration under the classification of nonclassical symmetries.…”

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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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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“…It is well known that the operators with vanishing coefficient of ∂ t give the so-called 'no-go' case in the study of conditional symmetries of an arbitrary (1 + 1)-dimensional evolution equation since the problem of finding them is reduced to that of solving a single equation which is equivalent to the initial one (see e.g. [18]). Since the determining equation has more independent variables and, therefore, more degrees of freedom, it is more convenient often to guess a simple solution or a simple ansatz for the determining equation, which can give a parametric set of complicated solutions of the initial equation.…”

Section: On Nonclassical Symmetriesmentioning

confidence: 99%

Group analysis of nonlinear fin equations

Vaneeva

Johnpillai²,

Popovych

et al. 2008

Applied Mathematics Letters

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Group classification of a class of nonlinear fin equations is carried out exhaustively. Additional equivalence transformations and conditional equivalence groups are also found. These enable us to simplify the classification results and their further applications. The derived Lie symmetries are used to construct exact solutions of truly nonlinear equations for the class under consideration. Nonclassical symmetries of the fin equations are discussed. Adduced results complete and essentially generalize recent works on the subject [M. Pakdemirli and AA note on a symmetry analysis and exact solutions of a nonlinear fin equation, Appl.

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Conditional symmetry of a porous medium equation

Cited by 65 publications

References 10 publications

Lie and Q-Conditional Symmetries of Reaction-Diffusion-Convection Equations with Exponential Nonlinearities and Their Application for Finding Exact Solutions

Lie and Q-Conditional Symmetries of Reaction-Diffusion-Convection Equations with Exponential Nonlinearities and Their Application for Finding Exact Solutions

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

Group analysis of nonlinear fin equations

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