New conditional symmetries and exact solutions of nonlinear reaction–diffusion–convection equations

Cherniha, Roman; Pliukhin, Oleksii

doi:10.1088/1751-8113/40/33/009

Cited by 29 publications

(50 citation statements)

References 37 publications

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“…and (2) with the convective term B(U)U x (here B(U), D(U) and F (U) are arbitrary smooth functions) [17,18,19]. It is well-known that conditional symmetries can be applied for finding exact solutions of the relevant equations, which are not obtainable by the classical Lie method.…”

Section: Introductionmentioning

confidence: 99%

New conditional symmetries and exact solutions of reaction–diffusion systems with power diffusivities

Cherniha¹,

Pliukhin²

2008

J. Phys. A: Math. Theor.

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A wide range of new Q-conditional symmetries for reaction-diffusion systems with power diffusivities is constructed. The relevant non-Lie ansätze to reduce the reactiondiffusion systems to ODE systems and examples of exact solutions are obtained. Relation of the solutions obtained with the development of spatially inhomogeneous structures is discussed. In 1952 A.C. Turing published the remarkable paper [1], in which a revolutionary idea about mechanism of morphogenesis (the development of structures in an organism during the life) has been proposed. Roughly speaking, his idea says that the diffusion process can interact with the chemical reaction process in such a way that this can stimulate the development and growth of different forms and structures in an organism. Moreover this differentiation is often impossible without diffusion, i.e., only one may destabilize the spatially homogeneous structures. This effect is known as the Turing instability and mathematically leads to systems of reaction-diffusion equations with nonlinearities of the special form [2,3]. Since 1952 the reaction-diffusion systems have been extensively studied by means of different mathematical methods, including group-theoretical methods. The main attention was paid to investigation of the two-component RD systems of the formwhere U = U(t, x) and V = V (t, x) are two unknown functions representing the densities of cells and chemicals, F (U, V ) and G(U, V ) are two given functions describing interaction between them and environment, the functions D 1 (U) and D 2 (V ) are the relevant diffusivities (usually they are assumed to be constants or power functions), and the subscripts t and x denote differentiation with respect to these variables. At the present time one can claim that all possible Lie symmetries of (1) with the constant diffusivities are completely described in [4,5,6], while in [7,8,9] it has been done for the non-constant diffusivities.The problem of construction of conditional symmetries for (1) is still not solved even in the case of Q-conditional symmetries (non-classical symmetries). Moreover, to our best knowledge, 1 to appear in J.Phys.A: Math.Theor.

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Section: Introductionmentioning

confidence: 99%

New conditional symmetries and exact solutions of reaction–diffusion systems with power diffusivities

Cherniha¹,

Pliukhin²

2008

J. Phys. A: Math. Theor.

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“…Therefore, (18)- (20) are nonlinear BVPs, which is the standard object for investigation. In [17], it was proven that (18) admits the Q-conditional symmetry:…”

Section: To Check Items (D)-(f)mentioning

confidence: 99%

Lie and Conditional Symmetries of a Class of Nonlinear (1 + 2)-Dimensional Boundary Value Problems

Cherniha

King

2015

Symmetry

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A new definition of conditional invariance for boundary value problems involving a wide range of boundary conditions (including initial value problems as a special case) is proposed. It is shown that other definitions worked out in order to find Lie symmetries of boundary value problems with standard boundary conditions, followed as particular cases from our definition. Simple examples of direct applicability to the nonlinear problems arising in applications are demonstrated. Moreover, the successful application of the definition for the Lie and conditional symmetry classification of a class of (1 + 2)-dimensional nonlinear boundary value problems governed by the nonlinear diffusion equation in a semi-infinite domain is realised. In particular, it is proven that there is a special exponent, k = −2, for the power diffusivity u k when the problem in question with non-vanishing flux on the boundary admits additional Lie symmetry operators compared to the case k = −2. In order to demonstrate the applicability of the symmetries derived, they are used for reducing the nonlinear problems with power diffusivity u k and a constant non-zero flux on the boundary (such problems are common in applications and describing a wide range of phenomena) to (1 + 1)-dimensional problems. The structure and properties of the problems obtained are briefly analysed. Finally, some results demonstrating how Lie invariance of the boundary value problem in question depends on the geometry of the domain are presented.

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“…The family of transformations (11) which is parameterized by the arbitrary element f generates a peculiar gauge of the arbitrary elements of class (7). Namely, every fixed pair (F, H ) is an image of a multitude of pairs (f, h).…”

Section: Theorem 5 the Class Of Equationsmentioning

confidence: 99%

“…Namely, every fixed pair (F, H ) is an image of a multitude of pairs (f, h). Moreover, all results on Lie symmetries and exact solutions of class (12) can be extended to class (7) by the inversion of transformation (11).…”

Section: Theorem 5 the Class Of Equationsmentioning

confidence: 99%

“…Similarly to the group classification, at first we gauge class (1) to subclass (7) constrained by the condition f = g. Further the cases m = 2 and m = 2 should be considered separately. If m = 2, class (7) is mapped to the imaged class (12) by transformation (11). If m = 2, class (14) is mapped to the double-imaged class (16) by the transformation…”

Section: Proposition 8 the Set Of Reduction Operators Of The Initial mentioning

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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New conditional symmetries and exact solutions of nonlinear reaction–diffusion–convection equations

Cited by 29 publications

References 37 publications

New conditional symmetries and exact solutions of reaction–diffusion systems with power diffusivities

New conditional symmetries and exact solutions of reaction–diffusion systems with power diffusivities

Lie and Conditional Symmetries of a Class of Nonlinear (1 + 2)-Dimensional Boundary Value Problems

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

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