New conditional symmetries and exact solutions of reaction–diffusion–convection equations with exponential nonlinearities

Cherniha, Roman; Pliukhin, Oleksii

doi:10.1016/j.jmaa.2013.02.010

Cited by 32 publications

(8 citation statements)

References 23 publications

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“…It was established much later [38] (comparison with the solutions obtained in [42] is presented above) that these solutions are also obtainable via the relevant Q-conditional symmetries.…”

Section: Non-lie Solutionsmentioning

confidence: 76%

“…The proof can be found in [38]. It is worth noting that Equations (56) and (59) are invariant only under the principal Lie algebra generated by the operators ∂ t and ∂ x provided the lambda-s are arbitrary (all special cases leading to a nontrivial Lie symmetry are listed in Table 1).…”

Section: Remarkmentioning

confidence: 99%

“…Solving Equation (38) and using the ansatz from Case 2 of Table 5, we arrive at the solution in the implicit form:…”

Section: Ansatzmentioning

confidence: 99%

“…Cases 3-6 of Table 6 can be treated in a similar way to Case 2; however, new solutions cannot be derived. Moreover, it was shown in [38] that all the differential equations, which are produced by the operators listed in Cases 3-6 of Table 6, are reducible to Equations (87) and (88).…”

Section: Non-lie Solutionsmentioning

confidence: 99%

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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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“…It was established much later [38] (comparison with the solutions obtained in [42] is presented above) that these solutions are also obtainable via the relevant Q-conditional symmetries.…”

Section: Non-lie Solutionsmentioning

confidence: 76%

Section: Remarkmentioning

confidence: 99%

“…Solving Equation (38) and using the ansatz from Case 2 of Table 5, we arrive at the solution in the implicit form:…”

Section: Ansatzmentioning

confidence: 99%

Section: Non-lie Solutionsmentioning

confidence: 99%

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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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“…Remark 4. Some exact solutions to nonlinear reaction-diffusion equations of Form (6) were obtained, for example, in [3,13,16,22,[35][36][37][38][39][40][41][42][43][44][45][46]; for exact solutions to more complex, nonlinear delay reaction-diffusion equations, see [32,[47][48][49][50][51][52][53][54][55].…”

Section: Using the Methods Of Differential Constraintsmentioning

confidence: 99%

Comparison of the Effectiveness of Different Methods for Constructing Exact Solutions to Nonlinear PDEs. Generalizations and New Solutions

Polyanin

2019

Mathematics

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The paper shows that, in looking for exact solutions to nonlinear PDEs, the direct method of functional separation of variables can, in certain cases, be more effective than the method of differential constraints based on the compatibility analysis of PDEs with a single constraint (or the nonclassical method of symmetry reductions based on an invariant surface condition). This fact is illustrated by examples of nonlinear reaction-diffusion and convection-diffusion equations with variable coefficients, and nonlinear Klein–Gordon-type equations. Hydrodynamic boundary layer equations, nonlinear Schrödinger type equations, and a few third-order PDEs are also investigated. Several new exact functional separable solutions are given. A possibility of increasing the efficiency of the Clarkson–Kruskal direct method is discussed. A generalization of the direct method of the functional separation of variables is also described. Note that all nonlinear PDEs considered in the paper include one or several arbitrary functions.

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Modeling of Shell-Type Spatial Structural Forms by Superpositions of Support Nodes Coordinates

Vorontsov

Tulupova

Vorontsova

2020

Lecture Notes in Civil Engineering

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New conditional symmetries and exact solutions of reaction–diffusion–convection equations with exponential nonlinearities

Cited by 32 publications

References 23 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

Comparison of the Effectiveness of Different Methods for Constructing Exact Solutions to Nonlinear PDEs. Generalizations and New Solutions

Modeling of Shell-Type Spatial Structural Forms by Superpositions of Support Nodes Coordinates

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