Continuous symmetries, lie algebras and differential equations

**Authors:**Euler, Norbert**Date:**2014-02-11**Subjects:**Differential equations, Nonlinear , Lie algebras**Type:**Thesis**Identifier:**uj:3754 , http://hdl.handle.net/10210/9131**Description:**D.Sc. (Mathematics) , In this thesis aspects of continuous symmetries of differential equations are studied. In particular the following aspects are studied in detail: Lie algebras, the Lie derivative, the jet bundle formalism for differential equations, Lie point and Lie-Backlund symmetry vector fields, recursion operators, conservation laws, Lax pairs, the Painlcve test, Lie algebra valued differenmtial forms and Dose operators as a representation of differential operators. The purpose of the study is to gain a better understanding of complicated nonlinear dirrerential equations that describe nature and to construct solutions. The differential equations under consideration were derived [rom physics and engineering. They are the following: the Kortcweg-dc Vries equation, Burgers' cquation , the sine-Gordon equation, nonlinear diffusion equations, the Klein Gordon equation, the Schrodinger equation, nonlinear Dirac equations, Yang-Mills equations, the Lorentz model, the Lotka-Volterra model, damped unharrnonic oscillators, and others. The newly found results and insights are discussed in chapters 8 to 17. Details on the COli tents of each chapter and rcfernces to some of my articles arc given in chapter 1.**Full Text:**

**Authors:**Euler, Norbert**Date:**2014-02-11**Subjects:**Differential equations, Nonlinear , Lie algebras**Type:**Thesis**Identifier:**uj:3754 , http://hdl.handle.net/10210/9131**Description:**D.Sc. (Mathematics) , In this thesis aspects of continuous symmetries of differential equations are studied. In particular the following aspects are studied in detail: Lie algebras, the Lie derivative, the jet bundle formalism for differential equations, Lie point and Lie-Backlund symmetry vector fields, recursion operators, conservation laws, Lax pairs, the Painlcve test, Lie algebra valued differenmtial forms and Dose operators as a representation of differential operators. The purpose of the study is to gain a better understanding of complicated nonlinear dirrerential equations that describe nature and to construct solutions. The differential equations under consideration were derived [rom physics and engineering. They are the following: the Kortcweg-dc Vries equation, Burgers' cquation , the sine-Gordon equation, nonlinear diffusion equations, the Klein Gordon equation, the Schrodinger equation, nonlinear Dirac equations, Yang-Mills equations, the Lorentz model, the Lotka-Volterra model, damped unharrnonic oscillators, and others. The newly found results and insights are discussed in chapters 8 to 17. Details on the COli tents of each chapter and rcfernces to some of my articles arc given in chapter 1.**Full Text:**

Conditional and approximate symmetries for nonlinear partial differential equations

**Authors:**Kohler, Astri**Date:**2014-07-21**Subjects:**Lie algebras , Symmetry , Differential equations, Nonlinear**Type:**Thesis**Identifier:**uj:11724 , http://hdl.handle.net/10210/11449**Description:**M.Sc. , In this work we concentrate on two generalizations of Lie symmetries namely conditional symmetries in the form of Q-symmetries and approximate symmetries. The theorems and definitions presented can be used to obtain exact and approximate solutions for nonlinear partial differential equations. These are then applied to various nonlinear heat and wave equations and many interesting solutions are given. Chapters 1 and 2 gives an introduction to the classical Lie approach. Chapters 3, 4 and 5 deals with conditional -, approximate -, and approximate conditional symmetries respectively. In chapter 6 we give a review of symbolic algebra computer packages available to aid in the search for symmetries, as well as useful REDUCE programs which were written to obtain the results given in chapters 2 to 5.**Full Text:**

**Authors:**Kohler, Astri**Date:**2014-07-21**Subjects:**Lie algebras , Symmetry , Differential equations, Nonlinear**Type:**Thesis**Identifier:**uj:11724 , http://hdl.handle.net/10210/11449**Description:**M.Sc. , In this work we concentrate on two generalizations of Lie symmetries namely conditional symmetries in the form of Q-symmetries and approximate symmetries. The theorems and definitions presented can be used to obtain exact and approximate solutions for nonlinear partial differential equations. These are then applied to various nonlinear heat and wave equations and many interesting solutions are given. Chapters 1 and 2 gives an introduction to the classical Lie approach. Chapters 3, 4 and 5 deals with conditional -, approximate -, and approximate conditional symmetries respectively. In chapter 6 we give a review of symbolic algebra computer packages available to aid in the search for symmetries, as well as useful REDUCE programs which were written to obtain the results given in chapters 2 to 5.**Full Text:**

An eigenvalue problem for a fermi system and lie algebras

- Steeb, Willi-Hans, Hardy, Yorick

**Authors:**Steeb, Willi-Hans , Hardy, Yorick**Date:**2013-04-15**Subjects:**Schrodinger equation , Eigenvalues , Lie algebras , Hamilton operators , Fermi Hamilton operators**Type:**Article**Identifier:**uj:5977 , http://hdl.handle.net/10210/8442**Description:**Please refer to full text to view abstract**Full Text:**

**Authors:**Steeb, Willi-Hans , Hardy, Yorick**Date:**2013-04-15**Subjects:**Schrodinger equation , Eigenvalues , Lie algebras , Hamilton operators , Fermi Hamilton operators**Type:**Article**Identifier:**uj:5977 , http://hdl.handle.net/10210/8442**Description:**Please refer to full text to view abstract**Full Text:**

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