Ordinary differential equations I

obligatory courses

(in Polish) Oczekuje się dobrej znajomości zagadnień ujętych w sylabusach przedmiotów Analiza matematyczna I.2 oraz Analiza matematyczna II.1.

Ordinary differential equations (ODEs), their properties and applications. Solution methods for ODEs: using paper and pencil, and using numerical schemes. Computer lab experiments: numerical and symbolic ODE packages.

Differential equation and its solution, first order and higher order equations, systems of differential equations, direction field, solution methods for simple types equations.

Simple numerical one- and multistep schemes. Runge-Kutta methods. Explicit and implicit schemes. Ways to derive numerical methods for ODEs.

Local existence and uniqueness theorems. Prolongation of the solution. Dependence on a parameter or on the initial condition; differentiability with respect to the parameter.

Systems of linear ODEs, the basis of the solutions. The fundamental matrix. Wronskian, Liouville's theorem. Systems with constant coefficients. Exponential of a matrix, nonhomogeneous systems. Higher order linear ODEs with constant coefficients.

Difference equations and their properties. Convergence theory for one-step methods. Consistency and stability. Stability and strong stability of multistep methods.

Nonlinear ODEs and stability. Lyapunov function. Phase plane and taxonomy of phase curves of autonomous systems. Singular points on a plane.

Absolute stability and the region of absolute stability. Stiffness and how to cope with it.

Computer lab experiments: numerical and symbolic ODE packages.

E. Hairer, S. P. Norsett, G. Wanner "Solving Ordinary Differential Equations", Springer

V.I.Arnold, R.Crooke "Ordinary differential equations", Springer

Boyce, DiPrima, "Elementary differential equations", Wiley

Knowledge and skills:

The students:

1. know the concepts of differential equation, the solutions of initial value problem (IVP), can verify whether the specified function is the solution of ODE or IVP;
2. can solve: separable, homogeneous, Bernoulli ODEs;
3. know the sufficient conditions of existence and uniqueness of solution of IVP;
4. can give an example of IVP with infinite number of solutions;
5. know the theorem about extending solutions of ODEs and can give an example of IVP which cannot be extended beyond some finite interval;
6. can solve the linear ODEs;
7. can convert higher order ODE to a system of the first order ODEs;
8. can find the fundamental matrices for systems of linear ODEs;
9. know the concept of vector field;
10. know the concept of equilibrium points and know the definitions of asymptotic and Lyapunov stabilities of equilibrium points;
11. can verify the stability of an equilibrium point;
12. know examples of applications of ODEs in sciences and real life.

Competence:

1. The students understand the role of ODEs in modelling natural processes.