Differential and differences equations


Aim of the course

Presentation of the main results of the qualitative theory of ordinary and partial differential equations to help understand the principles of mathematical modeling of reality, the conditions for the correctness of these models and the principles of numerical methods as well these of the design and analysis of algorithms based on mapping the exact solutions onto finite-dimensional subspaces of solutions.

Lecture programme

Projective methods of functional analysis, Fourier series in Hilbert spaces, ordinary differential equations, general integral and particular integral, Cauche task, 1st order linear equations, systems of ordinary equations, Piccard-Lindelof theorem and Peanao theorem, the method of variation of constants, fundamental matrix solutions, case of constant coefficients, stability of solutions according to Lyapunov, simple patterns of differences, partial 2nd order equations, classification of linear equations, strong initial-boundary problems, Sobolev spaces and weak solutions, regular distributions, elliptic model problem, residual formulation and classical variational formulation, Lax-Milgrahm lemma, Galerkin method, Cea's lemma, convergence of Galernik method, Galernik equation in the space of spline - finite element method, simple mixed diagrams for parabolic and hyperbolic equations

Overview of the course elements

As part of the laboratory there are presented and implemented (Matlab) simple diagrams of differential equations, ordinary. Analytical methods for integrating ordinary equations introduced during lectures are focused on, expanded and trained on simple examples. There are solved simple problems for the boundary conditions of second order partial differential equations using differential schemes and the finite element method. Students are given the practice of creating optimal approximation spaces and the choice of linear systems solwerów that represent approximate boundary problems.

Reading list

1. Pelczar A.; Wstęp do teorii równań różniczkowych. Część I. PWN, Warszawa 1987
2. Żakowski W., Leksiński W.; Matematyka, Część IV, WNT, Warszawa 1973
3. Hartman P.; Ordinary Differential Equations. Birkhauser, June 1982
4. Ciarlet P., G.; The Finite Element Method for Elliptic Problems. SIAM. April, 2002
5. Marcinkowska H.; Wstęp do teorii równań różniczkowych cząstkowych. PWN, Warszawa 1972

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