Exact solutions and invariant subspaces of nonlinear partial differential equations in mechanics and physics. (©2007)

• Record Type: Book
• Further Information: Note: Victor A. Galaktionov, Sergey R. Svirshchevskii.
• Other Names: Galaktionov, Victor A; Svirshchevskii, Sergey R
• Contents: INTRODUCTION: NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS AND EXACT SOLUTIONS; Exact Solutions: History, Classical Symmetry Methods, Extensions; Examples: Classic Fundamental Solutions belong to Invariant Subspaces; Models, Targets, and Prerequisites; ; LINEAR INVARIANT SUBSPACES IN QUASILINEAR EQUATIONS: BASIC EXAMPLES AND MODELS; History: First Eexamples of Solutions on Invariant Subspaces; Basic Ideas: Invariant Subspaces and Generalized Separation of Variables; More Examples: Polynomial Subspaces; Examples: Trigonometric Subspaces; Examples: Exponential Subspaces; Remarks and Comments on the Literature; ; INVARIANT SUBSPACES AND MODULES: MATHEMATICS IN ONE DIMENSION; Main Theorem on Invariant Subspaces; The Optimal Estimate on Dimension of Invariant Subspaces; First-Order Operators with Subspaces of Maximal Dimension; Second-Order Operators with Subspaces of Maximal Dimension; First- and Second-Order Quadratic Operators with Subspaces of Lower Dimensions; Operators Preserving Polynomial Subspaces; Extensions to ?/?t-Dependent Operators; Summary: Basic Types of Equations and Solutions; Remarks and Comments on the Literature; Open Problems; ; PARABOLIC EQUATIONS IN ONE DIMENSION: THIN FILM, KURAMOTO-SIVASHINSKY, AND MAGMA MODELS; Thin Film Models and Polynomial Subspaces; Applications to Extinction, Blow-Up, Free-Boundary Problems, and Interface Equations; Exact Solutions with Zero Contact Angle; Extinction Behavior for Sixth-Order Thin Film Equations; Quadratic Models: Trigonometric and Exponential Subspaces; 2mth-Order Thin Film Operators and Equations; Oscillatory, Changing Sign Behavior in the Cauchy Problem; Invariant Subspaces in Kuramoto-Sivashinsky-Type Models; Quasilinear Pseudo-Parabolic Models: The Magma Equation; Remarks and Comments on the Literature; Open Problems; ; ODD-ORDER ONE-DIMENSIONAL EQUATIONS: KORTEWEG-DE VRIES, COMPACTON, NONLINEAR DISPERSION, AND HARRY DYM MODELS; Blow-Up and Localization for KdV-Type Equations; Compactons and Shocks Waves in Higher-Order Quadratic Nonlinear Dispersion Models; Higher-Order PDEs: Interface Equations and Oscillatory Solutions; Compactons and Interfaces for Singular mKdV-Type Equations; On Compactons in IRN for Nonlinear Dispersion Equations; "Tautological" Equations and Peakons; Subspaces, Singularities, and Oscillatory Solutions for Harry Dym-Type Equations; Remarks and Comments on the Literature; Open Problems; ; QUASILINEAR WAVE AND BOUSSINESQ MODELS IN ONE DIMENSION: SYSTEMS OF NONLINEAR EQUATIONS; Blow-Up in Nonlinear Wave Equations on Invariant Subspaces; Breathers in Quasilinear Wave Equations and Blow-Up Models; Quenching and Interface Phenomena, Compactons; Invariant Subspaces in Systems of Nonlinear Evolution Equations; Remarks and Comments on the Literature; Open Problems; ; APPLICATIONS TO NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS IN IRN; Second-Order Operators and Some Higher-Order Extensions; Extended Invariant Subspaces for Second-Order Operators; On the Remarkable Operator in IR2; On Second-Order p-Laplacian Operators; Invariant Subspaces for Operators of Monge-Ampère Type; Higher-Order Thin Film Operators; Moving Compact Structures in Nonlinear Dispersion Equations; From Invariant Polynomial Subspaces in IR N to Invariant Trigonometric Subspaces in IR N -1; Remarks and Comments on the Literature; Open Problems; ; PARTIALLY INVARIANT SUBSPACES, INVARIANT SETS, AND GENERALIZED SEPARATION OF VARIABLES; Partial Invariance for Polynomial Operators; Quadratic Kuramoto-Sivashinsky Equations; Method of Generalized Separation of Variables; Generalized Separation and Partially Invariant Modules; Evolutionary Invariant Sets for Higher-Order Equations; A Separation Technique for the Porous Medium Equation in IRN; Remarks and Comments on the Literature; Open Problems; ; SIGN-INVARIANTS FOR SECOND-ORDER PARABOLIC EQUATIONS AND EXACT SOLUTIONS; Quasilinear Models, Definitions, and First Examples; Sign-Invariants of the Form ut - ?(u); Stationary Sign-Invariants of the Form H (r, u, ur); Sign-Invariants of the Form ut - m(u)(ux)2 - M(u); General First-Order Hamilton-Jacobi Sign-Invariants; Remarks and Comments on the Literature; ; INVARIANT SUBSPACES FOR DISCRETE OPERATORS, MOVING MESH METHODS, AND LATTICES; Backward Problem of Invariant Subspaces for Discrete Operators; On the Forward Problem of Invariant Subspaces; Invariant Subspaces for Finite-Difference Operators; Invariant Properties of Moving Mesh Operators and Applications; Applications to Anharmonic Lattices; Remarks and Comments on the Literature; Open Problems; ; REFERENCES; LIST OF FREQUENTLY USED ABBREVIATIONS; INDEX
• Publisher Details: Boca Raton : Chapman & Hall/CRC
• Publication Date: 2007
• Copyright Date: 2007
• Extent: 1 online resource (xxx, 498 pages), illustrations
• Subjects: 518/.64; Differential equations, Partial -- Numerical solutions; Nonlinear theories -- Methodology; Invariant subspaces -- Methodology; Exact (Philosophy) -- Mathematics; Mathematical physics; Équations aux dérivées partielles -- Solutions numériques; Théories non linéaires -- Méthodologie; Sous-espaces invariants -- Méthodologie; Exact (Philosophie) -- Mathématiques; Physique mathématique; MATHEMATICS -- Numerical Analysis; Partiella differentialekvationer; Electronic books
• Languages: English
• ISBNs: 1420011626; 9781420011623
• Related ISBNs: 1584886633; 9781584886631
• Notes: Note: Includes bibliographical references (pages 467-492) and index.; Note: Print version record.
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.