A quasilinear heat equation with a source: Peaking, localization, symmetry exact solutions, asymptotics, structures

Галактионов, В. А.; Дородницын, В. А.; Elenin, G. G.; Kurdyumov, S. P.; Samarskiî, A. A.

doi:10.1007/bf01098785

Cited by 95 publications

(88 citation statements)

References 39 publications

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“…In particular, for reaction-diffusion equations such as (1.1), exact solutions arising from symmetry methods can often be effectively used to study properties such as "blow-up" [38,39,40]. Furthermore, explicit solutions (such as those found by symmetry methods) can play an important role in the design and testing of numerical integrators; these solutions provide an important practical check on the accuracy and reliability of such integrators (cf.…”

Section: ) Is Invariant Under This Transformation Yields An Overdetementioning

confidence: 99%

Symmetry reductions and exact solutions of a class of nonlinear heat equations

Clarkson

Mansfield

1994

Physica D: Nonlinear Phenomena

255

290

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Classical and nonclassical symmetries of the nonlinear heat equationare considered. The method of differential Gröbner bases is used both to find the conditions on f (u) under which symmetries other than the trivial spatial and temporal translational symmetries exist, and to solve the determining equations for the infinitesimals. A catalogue of symmetry reductions is given including some new reductions for the linear heat equation and a catalogue of exact solutions of (1) for cubic f (u) in terms of the roots of f (u) = 0. 0

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Section: ) Is Invariant Under This Transformation Yields An Overdetementioning

confidence: 99%

Symmetry reductions and exact solutions of a class of nonlinear heat equations

Clarkson

Mansfield

1994

Physica D: Nonlinear Phenomena

255

290

View full text Add to dashboard Cite

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“…A prominent feature when compared to those in the works [20,21,53,78,93,138,254,267,268], for instance, is that no more continuity on c is required than that stated in Hypothesis 1. are exceptional in the group-theoretical classification of second-order parabolic equations [12,104]. Regarding the first, we easily deduce the following from Theorem 33.…”

Section: Fixed Signmentioning

confidence: 77%

Travelling Waves in Nonlinear Diffusion-Convection Reaction

Gilding¹,

Kersner²

2004

170

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The study of travelling waves or fronts has become an essential part of the mathematical analysis of nonlinear diffusion-convection-reaction processes. Whether or not a nonlinear second-order scalar reaction-convection-diffusion equation admits a travelling-wave solution can be determined by the study of a singular nonlinear integral equation. This article is devoted to demonstrating how this correspondence unifies and generalizes previous results on the occurrence of travelling-wave solutions of such partial differential equations. The detailed comparison with earlier results simultaneously provides a survey of the topic. It covers travelling-wave solutions of generalizations of the Fisher, Newell-Whitehead, Zeldovich, KPP and Nagumo equations, the Burgers and nonlinear Fokker-Planck equations, and extensions of the porous media equation.

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“…It is worth mentioning that the exact solution (6.4) is not invariant under any Lie or Lie-Bäcklund group of transformations. See a survey on the results of a general group classification of quasi-linear heat equations in [16] and for this equation in [33]. On the other hand, it is a solution of the equation (6.1) on a two-dimensional subspace Span{1, x 2 }, which is invariant under the quadratic operator A and can be constructed by a non-linear method of separation of variables.…”

Section: Example: the First Turning Patternmentioning

confidence: 99%

“…where the perturbation Y (ξ, τ ) → 0 as τ → ∞ solves the quasi-linear equation 13) and D is the linear operator 15) we obtain the equation 16) where D 1 is the hypergeometric operator…”

Section: Asymptotic Analysis Of Patterns In the Inner Regionmentioning

confidence: 99%

Behaviour of interfaces in a diffusion-absorption equation with critical exponents

Galaktionov¹,

Shmarev²,

Vázquez³

2000

Interfaces Free Bound.

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We consider the Cauchy problem for the porous medium equation with strong absorptionwith continuous compactly supported initial data u(x, 0) = u 0 (x) 0 in the critical case m + p = 2 of the range of parameters m > 1, p < 1. We study the regularity of solutions and interfaces and compare the results with the purely diffusive case u t = (u m ) x x . Important differences are found in the interface behaviour and equations, in the occurrence of turning points and inflection points of the interfaces, and in the fact that bounded solutions extinguish in finite time. All these phenomena are examined and described. The critical case studied here allows for a comparatively simpler and richer analysis of the qualitative behaviour of non-linear parabolic equations involving finite propagation (m > 1) and strong absorption ( p < 1).

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A quasilinear heat equation with a source: Peaking, localization, symmetry exact solutions, asymptotics, structures

Cited by 95 publications

References 39 publications

Symmetry reductions and exact solutions of a class of nonlinear heat equations

Symmetry reductions and exact solutions of a class of nonlinear heat equations

Travelling Waves in Nonlinear Diffusion-Convection Reaction

Behaviour of interfaces in a diffusion-absorption equation with critical exponents

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