Content

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  Introduction
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   Two examples
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   This thesis
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   Outline of approach
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   What is new in this thesis?
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   Symmetries
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   Generalized symmetries and their generators
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   Curve functions
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   Groups of transformations and symmetry groups
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   Prolongations
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   Lie point symmetries and first order ODEs
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   The methods of Cheb-Terrab for first order ODEs
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   Symmetries of evolution equations
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   Lie point symmetries in phase space
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  Integration via commuting symmetries
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   Tensors, Lie derivatives and exterior derivatives
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   An integrability result
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   Examples and applications
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  Conclusions
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   Resumé
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   Open problems and perspectives
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  Integrating factors
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   Introduction
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   Basic terminology
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   Integrating factors and Associated ODEs
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   Integrating factors for second order ODEs
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   The Euler Operator: Exactness of an ODE
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   Integrating factors for third order ODEs
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   Integrating factors (x,y).
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   Integrating factors (x,y').
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   Integrating factors (y,y').
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   Integrating factors (y'').
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   Integrating factors f(x,y,y') (y'')m.
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   The case = f(x,y,y') (y'')m, m N
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   The case = f(x,y,y') 1(y'')m, m N, m 3
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   The case = f(x,y,y') 1y''
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   Generalizations to higher order ODEs
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   Integrating factors (y(i),y(n-2)), 0 i n-3.
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   Integrating factors (x,y(n-2)).
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   Integrating factors (y(i),y(j)), 0 i < j n-3.
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   Integrating factors (y(n-1)).
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   Integrating factors f(x,y,y',…,y(n-2)) (y(n-1))m.
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   The case = f(x,y,y',…,y(n-2)) (y(n-1))m, m N
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   The case = f(x,y,y',…,y(n-2)) 1(y(n-1))m, m N, m 3
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   The case = f(x,y,y',…,y(n-2)) 1y(n-1)
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  Conclusions
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   Resumé
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   Open problems and perspectives
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   Skew symmetric hierarchies
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   Basic terminology and general definitions
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   Canonical form of differential expressions
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   Computing integrating factors
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   Fundamental forms of skew symmetric operators
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   Recursion formulas for symbolic integration
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  Conclusions
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   Resumé
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   Back to the symmetries
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   Integrating factors and symmetry generators: the mixed case
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   Open problems and perspectives
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   Non--local symmetries
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   Non--local symmetries for 2nd order linear ODEs
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   Overview: Nilpotent and recursive flows
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   Nilpotent flows for second order linear ODEs
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   Overview: Basics in differential Galois theory
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   Symmetric powers
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   Applications of symmetric powers
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   Computation of symmetric powers
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   An alternative algorithm for symmetric powers
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  Conclusions
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   Resumé
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   Open problems and perspectives
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   Bibliography
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  List of some notation
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   Chapter 1
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   Chapter 2
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   Chapter 3
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   Chapter 4
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   Glossary of Algorithms
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   Index