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Im Werk suchen
Algorithmic methods for ordinary differential equations / von Kai Frederik Gehrs. 2006
Inhalt
Introduction
Two examples
This thesis
Outline of approach
What is new in this thesis?
Symmetries
Generalized symmetries and their generators
Curve functions
Groups of transformations and symmetry groups
Prolongations
Lie point symmetries and first order ODEs
The methods of Cheb-Terrab for first order ODEs
Symmetries of evolution equations
Lie point symmetries in phase space
Integration via commuting symmetries
Tensors, Lie derivatives and exterior derivatives
An integrability result
Examples and applications
Conclusions
Resumé
Open problems and perspectives
Integrating factors
Introduction
Basic terminology
Integrating factors and Associated ODEs
Integrating factors for second order ODEs
The Euler Operator: Exactness of an ODE
Integrating factors for third order ODEs
Integrating factors (x,y).
Integrating factors (x,y').
Integrating factors (y,y').
Integrating factors (y'').
Integrating factors f(x,y,y') (y'')m.
The case = f(x,y,y') (y'')m, m N
The case = f(x,y,y') 1(y'')m, m N, m 3
The case = f(x,y,y') 1y''
Generalizations to higher order ODEs
Integrating factors (y(i),y(n-2)), 0 i n-3.
Integrating factors (x,y(n-2)).
Integrating factors (y(i),y(j)), 0 i < j n-3.
Integrating factors (y(n-1)).
Integrating factors f(x,y,y',…,y(n-2)) (y(n-1))m.
The case = f(x,y,y',…,y(n-2)) (y(n-1))m, m N
The case = f(x,y,y',…,y(n-2)) 1(y(n-1))m, m N, m 3
The case = f(x,y,y',…,y(n-2)) 1y(n-1)
Conclusions
Resumé
Open problems and perspectives
Skew symmetric hierarchies
Basic terminology and general definitions
Canonical form of differential expressions
Computing integrating factors
Fundamental forms of skew symmetric operators
Recursion formulas for symbolic integration
Conclusions
Resumé
Back to the symmetries
Integrating factors and symmetry generators: the mixed case
Open problems and perspectives
Non--local symmetries
Non--local symmetries for 2nd order linear ODEs
Overview: Nilpotent and recursive flows
Nilpotent flows for second order linear ODEs
Overview: Basics in differential Galois theory
Symmetric powers
Applications of symmetric powers
Computation of symmetric powers
An alternative algorithm for symmetric powers
Conclusions
Resumé
Open problems and perspectives
Bibliography
List of some notation
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Glossary of Algorithms
Index
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