Numerical Solution of Ordinary Differential Equations

Lecturer:

Luděk Beneš (gar.)

Tutor:

Luděk Beneš (gar.)

Supervisor:

Department of Technical Mathematics

Synopsis:

Initial value problems for higher-order differential equations and for system of differential equations (one-step and multistep methods). Boundary value problems of second-order differential equations The numerical solution of initial-value probléms in differential-algebraic equations. Applications.

Requirements:

Syllabus of lectures:

1. and 2. lecture:

The formulation of the principal problems : I. Initial value problems for system of differential equations, II. Initial value problems for higher-order differential equations, III. Boundary value problems of second-order differential equations, IV. Initial-value problems in differential-algebraic equations. The transfer II. to I. The existence and uniqueness of a solution I. and II. The autonomous systems, properties.

3. lecture:

The principle of numerical solution for problems I and II. One-step methods , methods of Runge-Kutta type, convergence and consistency. Using methods, physical interpretation. Constructing phase plane diagrams, applications. Variable-step methods.

4. and 5. lecture:

Implicite methods. Absolute stability of the methods. Intervals of absolute stability for Runge-Kutta methods. Effect of rounding error on higher-order methods. Stiff systems.

6. and 7.lecture:

Multistep methods. Adams implicite and explicite methods. The predictor-corrector formulas. Convergence and consistency. Intervals of the absolute stability. Comparison of one-step and multistep methods. Using a natural cubic spline interpolation.

8.,9. and 10. lecture:

Problem III :

a) Linear case. The existence and uniqueness of a solution .Shooting methods, problems on instability. Finite- difference methods. Existence of a solution of the diference schema, convergence and stability.

b) Nonlinear case. The existence and uniqueness of a solution . Shooting metods, problems on instability. Existence of a solution of the diference schema. Newton?s method for systems of nonlinear equations, convergence.

11.,12. and 13. lecture:

Problem IV :

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.

14. lecture:

Summary. Preparations for exam.

Syllabus of tutorials:

1. and 2. lecture:

The formulation of the principal problems : I. Initial value problems for system of differential equations, II. Initial value problems for higher-order differential equations, III. Boundary value problems of second-order differential equations, IV. Initial-value problems in differential-algebraic equations. The transfer II. to I. The existence and uniqueness of a solution I. and II. The autonomous systems, properties.

3. lecture:

The principle of numerical solution for problems I and II. One-step methods , methods of Runge-Kutta type, convergence and consistency. Using methods, physical interpretation. Constructing phase plane diagrams, applications. Variable-step methods.

4. and 5. lecture:

Implicite methods. Absolute stability of the methods. Intervals of absolute stability for Runge-Kutta methods. Effect of rounding error on higher-order methods. Stiff systems.

6. and 7.lecture:

Multistep methods. Adams implicite and explicite methods. The predictor-corrector formulas. Convergence and consistency. Intervals of the absolute stability. Comparison of one-step and multistep methods. Using a natural cubic spline interpolation.

8.,9. and 10. lecture:

Problem III :

a) Linear case. The existence and uniqueness of a solution .Shooting methods, problems on instability. Finite- difference methods. Existence of a solution of the diference schema, convergence and stability.

b) Nonlinear case. The existence and uniqueness of a solution . Shooting metods, problems on instability. Existence of a solution of the diference schema. Newton?s method for systems of nonlinear equations, convergence.

11.,12. and 13. lecture:

Problem IV :

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.

14. lecture:

Summary. Preparations for exam.

Study Objective:

Study materials:

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:

Time-table for winter semester 2011/2012:

Time-table is not available yet

Time-table for summer semester 2011/2012:

	06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon
Tue	roomKN:A-446 Beneš L. 09:00–12:15 (lecture parallel1) Karlovo nám. Poc.ucebna A446
Fri
Thu
Fri

The course is a part of the following study plans: