On Exact Solutions of Nonlinear Diffusion Equations

Barannyk, A. F.; Yuryk, I. I.

doi:10.1007/s11253-005-0256-x

Cited by 7 publications

(11 citation statements)

References 5 publications

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“…h 3 = 0. Moreover, two essentially different cases, a 3 = 0 and a 3 = 0, follow from the first equation of (15).…”

Section: Remarkmentioning

confidence: 99%

“…As a result, scalar PDEs only were under study for a long time (see extensive reviews about this matter in Chapter 1 of [9]) because systems of DEs for systems of PDEs are much more complicated. To the best of our knowledge, there are only a few papers devoted to the search of Q-conditional symmetries for systems of PDEs published before 2010 [14][15][16][17][18]. A majority of such papers were published during the current decade [19][20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

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Conditional symmetries and exact solutions of a nonlinear three-component reaction-diffusion model

Cherniha,

Davydovych

2019

Preprint

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Q-conditional (nonclassical) symmetries of the known three-component reactiondiffusion system [K. Aoki et al Theor. Pop. Biol. 50(1) (1996)] modeling interaction between farmers and hunter-gatherers are constructed for the first time. A wide variety of Q-conditional symmetries are found in an explicit form and it is shown that these symmetries are not equivalent to the Lie symmetries. Some operators of Q-conditional (nonclassical) symmetry are applied for finding exact solutions of the reaction-diffusion system in question. Properties of the exact solutions (in particular, their asymptotic behaviour) are identified and possible biological interpretation is discussed.

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“…h 3 = 0. Moreover, two essentially different cases, a 3 = 0 and a 3 = 0, follow from the first equation of (15).…”

Section: Remarkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Conditional symmetries and exact solutions of a nonlinear three-component reaction-diffusion model

Cherniha,

Davydovych

2019

Preprint

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“…To the best of our knowledge, there are not many paper devoted to search of Q-conditional symmetries for the systems of PDEs [20][21][22][23][24]. One may easily check that Definition 2 was only used in all these papers.…”

Section: Remarkmentioning

confidence: 99%

Reaction-Diffusion Systems with Constant Diffusivities: Conditional Symmetries and Form-Preserving Transformations

Cherniha

Davydovych

2014

Springer Proceedings in Mathematics &Amp; Statistics

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Q-conditional symmetries (nonclassical symmetries) for a general class of two-component reaction-diffusion systems with constant diffusivities are studied. Using the recently introduced notion of Q-conditional symmetries of the first type (R. Cherniha J. Phys. A: Math. Theor., 2010. vol. 43., 405207), an exhaustive list of reaction-diffusion systems admitting such symmetry is derived. The formpreserving transformations for this class of systems are constructed and it is shown that this list contains only non-equivalent systems. The obtained symmetries permit to reduce the reaction-diffusion systems under study to two-dimensional systems of ordinary differential equations and to find exact solutions. As a non-trivial example, multiparameter families of exact solutions are explicitly constructed for two nonlinear reaction-diffusion systems. A possible interpretation to a biologically motivated model is presented.

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“…Remark 1 To the best of our knowledge, there are not many paper devoted to search of Qconditional symmetries for the systems of PDEs [14,[21][22][23][24][25]. One may easily check that Definition 2 was only used in all these papers.…”

Section: Conditional Symmetry For Rd Systems 21 Definitions and Prelmentioning

confidence: 99%

Conditional symmetries and exact solutions of nonlinear reaction–diffusion systems with non-constant diffusivities

Cherniha

Davydovych

2012

Communications in Nonlinear Science and Numerical Simulation

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Q-conditional symmetries (nonclassical symmetries) for the general class of twocomponent reaction-diffusion systems with non-constant diffusivities are studied. Using the recently introduced notion of Q-conditional symmetries of the first type, an exhausted list of reaction-diffusion systems admitting such symmetry is derived. The results obtained for the reaction-diffusion systems are compared with those for the scalar reactiondiffusion equations. The symmetries found for reducing reaction-diffusion systems to two-dimensional dynamical systems, i.e., ODE systems, and finding exact solutions are applied. As result, multiparameter families of exact solutions in the explicit form for a nonlinear reaction-diffusion system with an arbitrary diffusivity are constructed. Finally, the application of the exact solutions for solving a biologically and physically motivated system is presented.

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On Exact Solutions of Nonlinear Diffusion Equations

Cited by 7 publications

References 5 publications

Conditional symmetries and exact solutions of a nonlinear three-component reaction-diffusion model

Conditional symmetries and exact solutions of a nonlinear three-component reaction-diffusion model

Reaction-Diffusion Systems with Constant Diffusivities: Conditional Symmetries and Form-Preserving Transformations

Conditional symmetries and exact solutions of nonlinear reaction–diffusion systems with non-constant diffusivities

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