Applications of Lie's Theory of Ordinary and Partial Differential Equations
By: Lawrence Dresner (author)Paperback
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Lie's group theory of differential equations unifies the many ad hoc methods known for solving differential equations and provides powerful new ways to find solutions. The theory has applications to both ordinary and partial differential equations and is not restricted to linear equations. Applications of Lie's Theory of Ordinary and Partial Differential Equations provides a concise, simple introduction to the application of Lie's theory to the solution of differential equations. The author emphasizes clarity and immediacy of understanding rather than encyclopedic completeness, rigor, and generality. This enables readers to quickly grasp the essentials and start applying the methods to find solutions. The book includes worked examples and problems from a wide range of scientific and engineering fields.
Conventions Used in This BookOne-Parameter GroupsGroups of transformationsInfinitesimal transformationsGroup invariantsInvariant curves and families of curvesTransformation of derivatives: the extended groupTransformation of derivatives (continued)Invariant differential equations of the first orderFirst-Order Ordinary Differential EquationsLie's integrating factorThe converse of Lie's theoremInvariant integral curvesSingular solutionsChange of variablesTabulation of differential equationsNotes to chapter twoSecond-Order Ordinary Differential EquationsInvariant differential equations of the second orderLie's reduction theoremStretching groupsStreching groups (continued)Stretching groups (continued)Other groupsEquations invariant to two groupsTwo-parameter groupsNoether's theoremNoether's theorem (continued)Similarity Solutions of Partial Differential EquationsOne-parameter families of stretching groupsSimilarity solutionsThe associated groupThe asymptotic behavior of similarity solutionsProof of the ordering theoremFunctions invariant to an entire family of stretching groupsA second exampleFurther use of the associated groupMore wave propagation problemsWave propagation problems (continued)ShocksTraveling-Wave SolutionsOne-parameter families of translation groupsThe diffusion equation with sourceDetermination of the propagation velocity aDetermination of the propagation volocity: role of the initial conditionThe approach to traveling wavesThe approach to traveling waves (continued)A final exampleConcluding remarksNotes of chapter fiveApproximate MethodsIntroductionSuperfluid diffusion equation with a slowly varying face temperatureOrdinary diffusion equation with a nonconstant diffusion coefficientCheck on the accuracy of the approximate formulaEpilogueAppendix 1: Linear, First-Order Partial Differential EquationsAppendix II: Riemann's Method of CharacteristicsAppendix III: The Calculus of Variations and the Euler-Lagrange EquationAppendix IV: Computation of Invariants and First Differential Invariants from the Transformation EquationsSolutions to the ProblemsReferencesSymbols and Their Definitions
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- ID: 9780750305310
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