Numerical Solution of Ordinary Differential Equations

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Department of Technical Mathematics

Synopsis:

Initial value problems for ordinary differential equations and for system of differential equations (one-step and multistep methods). Boundary value problems of second-order differential equations (finite differences method, shooting method, variational formulation -Galerkin) The numerical solution of initial-value problems in differential-algebraic equations. Applications.

Requirements:

Syllabus of lectures:

1. and 2. lecture:

Revision of M3 = analytical solving of the differential equations. Formulation of the initial value problem for ODE.

The existence and uniqueness of a solution.

3.-6. lectures:

The principle of numerical solution IVPI. One-step methods , methods. Errors, convergence, consistency, absolute stability. The methods of Taylor and Runge-Kutta type, stabilised R-K methods. Using methods, applications. Error estimation, variable-step, R-K-Fehlberg method.

7. - 9. lecture:

Linear Multistep Methods (LMM), Errors, convergence, consistency, zero and absolute stability. Numerical methods based on the integration and derivation (Adams-Bashforth, Adams-Moulton, BDF), Predictor-corrector methods. Comparison to the one-step methods. Stiff problems. A-stability.

10.-12. lectures

Boundary value problem of linear and nonlinear second order equations.

The existence and uniqueness of a solution .Shooting methods. Finite- difference methods. Existence of a solution of the difference schema, convergence, consistency and stability. Discretization of the boundary conditions. Compact differences.

13. lecture

Variational formulation of the boundary value problems. Generalized and week solutions. Ritz, Galerkin and Petrov-Galerkin methods, principle of the finite element method.

14. lecture:

Differential-algebraic equations (DAE). Basic type of DAE`s. Linear and nonlinear systems. Solvability and the index. Index reduction and constraint stabilization . One-step methods. Convergence for index 1 and 2 problems.

Syllabus of tutorials:

The working on the excercises is directly connected to the lectures.

1. and 2. lecture:

Revision of M3 = analytical solving of the differential equations. Formulation of the initial value problem for ODE.

The existence and uniqueness of a solution.

3.-6. lectures:

The principle of numerical solution IVPI. One-step methods , methods. Errors, convergence, consistency, absolute stability. The methods of Taylor and Runge-Kutta type, stabilised R-K methods. Using methods, applications. Error estimation, variable-step, R-K-Fehlberg method.

7. - 9. lecture:

Linear Multistep Methods (LMM), Errors, convergence, consistency, zero and absolute stability. Numerical methods based on the integration and derivation (Adams-Bashforth, Adams-Moulton, BDF), Predictor-corrector methods. Comparison to the one-step methods. Stiff problems. A-stability.

10.-12. lectures

Boundary value problem of linear and nonlinear second order equations.

The existence and uniqueness of a solution .Shooting methods. Finite- difference methods. Existence of a solution of the difference schema, convergence, consistency and stability. Discretization of the boundary conditions. Compact differences.

13. lecture

Variational formulation of the boundary value problems. Generalized and week solutions. Ritz, Galerkin and Petrov-Galerkin methods, principle of the finite element method.

14. lecture:

Differential-algebraic equations (DAE). Basic type of DAE`s. Linear and nonlinear systems. Solvability and the index. Index reduction and constraint stabilization . One-step methods. Convergence for index 1 and 2 problems.

Study Objective:

Study materials:

R.J. Le Vegue: Finite Difference Methods for Differential Equations.

E. Vitásek: Numerické metody.SNTL 1987

Buchanan J.L.: Numerical Methods and Analysis, McGraw-Hill, 1992

Nakamura S.: Applied Numerical Methods with Software, Prentice Hall, New York, 1991

Brenan K.E., Campb S.L.,Petzo L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, Elseview., New York, 1989

Haier E.,Wanner G.:Solving Ordinary Differential Equations II, Springer-Verlag, 1996

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans: