Similarity Methods for Differential Equations

Bluman, George W.; Cole, J. D.

doi:10.1007/978-1-4612-6394-4

Cited by 837 publications

(577 citation statements)

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“…[65]). The classical method for finding symmetry reductions of pdes is the Lie group method of infinitesimal transformations [3,10,11,49,73,78,86,93]. To apply the classical method to the nonlinear heat equation (1.1) we consider the one-parameter Lie group of infinitesimal transformations in (x, t, u) given byx = x + εξ(x, t, u) + O(ε 2 ),…”

Section: Introductionmentioning

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: Introductionmentioning

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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“…More insight on the behavior expected for general solutions can be obtained by using the symmetries of the equation to find special solutions and to examine their stability. We look for a scaling symmetry [4,15,16,59], by asking if there exist constants U, T such that the ansatz…”

Section: Self-similar Behaviors In An Odementioning

confidence: 99%

“…Finally, recall that an autonomous ODE of degree n with a scale invariance can be reduced by a change of both independent and dependent variables to a system of degree n−1 [15,16,59]. Equation (2.17) suggests two scale-invariant variables,…”

Section: Global Behaviormentioning

confidence: 99%

“…While similarity solutions for ODEs are often derived via symmetry methods, e.g. [16,15,59,30], the linear stability of these solutions is rarely studied. We describe a method for studying the linear stability of the similarity solutions to (1.1) and use it to argue that generic initial conditions yield solutions that exist only for a finite-time interval, blowing-up both forwards and backwards in time from generic initial data.…”

Section: Introductionmentioning

confidence: 99%

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Stability and dynamics of self-similarity in evolution equations

Bernoff

Witelski

2009

J Eng Math

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Abstract. We discuss a methodology for studying the linear stability of self-similar solutions. We will illustrate these fundamental ideas on three prototype problems: a simple ODE with finite-time blowup, a second-order semi-linear heat equation with infinite-time spreading solutions, and the fourth-order Sivashinsky equation with finite-time self-similar blow-up. These examples are used to show that self-similar dynamics can be studied using many of the ideas arising in the study of dynamical systems. In particular, we will discuss the use of dimensional analysis to derive scaling invariant similarity variables, and the role of symmetries in the context of stability of self-similar dynamics. The spectrum of the linear stability problem determines the rate at which the solution will approach a self-similar profile. For blow-up solutions we will demonstrate that the symmetries give rise to positive eigenvalues associated with the symmetries, and show how this stability analysis can identify a unique stable (and observable) attracting solution from a countable infinity of similarity solutions.

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“…This is fundamental to modelling (cf. [1][2][3][4][5][6][7][8][9][10][11]). More generally, similarity (symmetry) analysis has become a very sophisticated subject.…”

Section: Similaritymentioning

confidence: 99%