Algorithmic Lie Theory for Solving Ordinary Differential Equations

Algorithmic Lie Theory for Solving Ordinary Differential Equations

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Description

Despite the fact that Sophus Lie's theory was virtually the only systematic method for solving nonlinear ordinary differential equations (ODEs), it was rarely used for practical problems because of the massive amount of calculations involved. But with the advent of computer algebra programs, it became possible to apply Lie theory to concrete problems. Taking this approach, Algorithmic Lie Theory for Solving Ordinary Differential Equations serves as a valuable introduction for solving differential equations using Lie's theory and related results.After an introductory chapter, the book provides the mathematical foundation of linear differential equations, covering Loewy's theory and Janet bases. The following chapters present results from the theory of continuous groups of a 2-D manifold and discuss the close relation between Lie's symmetry analysis and the equivalence problem. The core chapters of the book identify the symmetry classes to which quasilinear equations of order two or three belong and transform these equations to canonical form. The final chapters solve the canonical equations and produce the general solutions whenever possible as well as provide concluding remarks. The appendices contain solutions to selected exercises, useful formulae, properties of ideals of monomials, Loewy decompositions, symmetries for equations from Kamke's collection, and a brief description of the software system ALLTYPES for solving concrete algebraic problems.show more

Product details

  • Electronic book text | 448 pages
  • Taylor & Francis Ltd
  • Chapman & Hall/CRC
  • London, United Kingdom
  • 8 Tables, black and white; 7 Illustrations, black and white
  • 1584888903
  • 9781584888901

Table of contents

INTRODUCTIONLINEAR DIFFERENTIAL EQUATIONS Linear Ordinary Differential Equations Janet's AlgorithmProperties of Janet BasesSolving Partial Differential Equations LIE TRANSFORMATION GROUPS Lie Groups and Transformation Groups Algebraic Properties of Vector FieldsGroup Actions in the Plane Classification of Lie Algebras and Lie GroupsLie SystemsEQUIVALENCE AND INVARIANTS OF DIFFERENTIAL EQUATIONSLinear Equations Nonlinear First-Order EquationsNonlinear Equations of Second and Higher Order SYMMETRIES OF DIFFERENTIAL EQUATIONS Transformation of Differential Equations Symmetries of First-Order Equations Symmetries of Second-Order EquationsSymmetries of Nonlinear Third-Order EquationsSymmetries of Linearizable Equations TRANSFORMATION TO CANONICAL FORM First-Order Equations Second-Order Equations Nonlinear Third-Order Equations Linearizable Third-Order EquationsSOLUTION ALGORITHMSFirst-Order EquationsSecond-Order Equations Nonlinear Equations of Third OrderLinearizable Third-Order EquationsCONCLUDING REMARKS APPENDIX A: Solutions to Selected Problems APPENDIX B: Collection of Useful Formulas APPENDIX C: Algebra of Monomials APPENDIX D: Loewy Decompositions of Kamke's Collection APPENDIX E: Symmetries of Kamke's Collection APPENDIX F: ALLTYPES Userinterface REFERENCESINDEXshow more