Algorithmic methods for ordinary differential equations / von Kai Frederik Gehrs. 2006
Inhalt
- Introduction
- 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
- Conclusions
- 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
- 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
- Non--local symmetries
- Non--local symmetries for 2nd order linear ODEs
- Overview: Nilpotent and recursive flows
- Overview: Basics in differential Galois theory
- An alternative algorithm for symmetric powers
- Conclusions
- Bibliography
- List of some notation
- Glossary of Algorithms
- Index