Exact solutions and invariant subspaces of nonlinear PDEs in by Victor A. Galaktionov, Sergey R. Svirshchevskii

By Victor A. Galaktionov, Sergey R. Svirshchevskii

Distinctive recommendations and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics is the 1st e-book to supply a scientific development of tangible ideas through linear invariant subspaces for nonlinear differential operators. performing as a advisor to nonlinear evolution equations and types from physics and mechanics, the ebook specializes in the lifestyles of latest specific recommendations on linear invariant subspaces for nonlinear operators and their an important new homes. This functional reference bargains with numerous partial differential equations (PDEs) and versions that show a few universal nonlinear invariant beneficial properties. It starts with classical in addition to newer examples of options on invariant subspaces. within the rest of the e-book, the authors strengthen numerous innovations for developing detailed suggestions of varied nonlinear PDEs, together with reaction-diffusion and fuel dynamics types, thin-film and Kuramoto-Sivashinsky equations, nonlinear dispersion (compacton) equations, KdV-type and Harry Dym versions, quasilinear magma equations, and Green-Naghdi equations. utilizing distinctive ideas, they describe the evolution homes of blow-up or extinction phenomena, finite interface propagation, and the oscillatory, altering signal habit of vulnerable strategies close to interfaces for nonlinear PDEs of assorted kinds and orders. The options surveyed in specified suggestions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics function a initial creation to the overall idea of nonlinear evolution PDEs of alternative orders and kinds.

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There are many models of this © 2007 by Taylor & Francis Group, LLC 1 Linear Invariant Subspaces: Examples 9 type. For instance, the PME with a nonlinear convection term u t = γ (uu x )x + βuu x , also known as the diffusion-convection Boussinesq equation, occurs in the various fields of petroleum technology and ground water hydrology. Let us begin with another example, where the interesting exact solutions on invariant subspaces arise. 8 (PME with absorption: Kersner’s solution) Consider the exact solution constructed by R.

There are many models of this © 2007 by Taylor & Francis Group, LLC 1 Linear Invariant Subspaces: Examples 9 type. For instance, the PME with a nonlinear convection term u t = γ (uu x )x + βuu x , also known as the diffusion-convection Boussinesq equation, occurs in the various fields of petroleum technology and ground water hydrology. Let us begin with another example, where the interesting exact solutions on invariant subspaces arise. 8 (PME with absorption: Kersner’s solution) Consider the exact solution constructed by R.

J Therefore, the invariance criterion of Wn is (i, j,l) ai, j,l j σµi σν σωl = 0 for all µ ≤ ν ≤ ω, µ + ν + ω − k > n − 1. 91) Similar conditions occur for quartic, quintic and higher-degree operators. 1 Some classification results and examples We next move to quadratic and cubic operators preserving W3 = L{1, cos x, sin x}. 92) This 3D subspace and its 5D extensions induce finite Fourier expansions of solutions of several nonlinear PDEs. For such subspaces, typical computations are more involved than those for the polynomial expansions.

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