Partial Differential Equations 360-MS2-2RRC2a
Course is devoted to classical theory of partial differential equations.
After course students should know basic, classical methods of solving the first and the second order PDEs, especcialy equations of mathematical physics such as heat equation, wave equation and Laplace equation.
Course objectives:
Knowledge of fundamenta notions and theorems:
1. Cauchy-Kowalewska theorem.
2. Integration of first order quasi-linear and linear PDEs. First integrals. Hamiltonian systems.
3. Classification of second order PDEs.
4. Boundary value problems of different kinds. Well-posed boundary value problems.
5. Hiperbolic equations . Cauchy problem for wave equations. mixed boundary problem for wave equation.
6. Elliptic equations. Properties of harmonic functions. Green function and its properties. Solution of Dirichlet problem.
7. Parabolic equations. Heat equation. Maximum and minimum principles. existence and uniqueness theorem for solutions of Cauchy problem for heat equation.
8. Applications of knowledge to solving theoretical problems as well as to practical ones.
Verification methods: lectures, exercises, consultations, studying literature, home works, discussions in groups.
Balance of student workload:
attending lectures15x2h = 30h
attending exercise classes 15x2h = 30h
preparation for classes 7x3h = 21h
completing notes after exercises and lectures 7x2h = 14h
consultations 12x1h = 12h
the final examination: preparation.and take 12h + 3h = 15h
control works: repeating the material and preparation 3x4h = 12h
Quantitative description
Direct interaction with the teacher: 75 h., 3 ECTS
Practical exercises: 77 h., 3 ECTS
Type of course
Prerequisites (description)
Course coordinators
Learning outcomes
Learning outcomes:
Student knows classification of first order partial differential equations (PDE), understands theorem of existence and uniqueness of solutions of Cauchy problem for quasilinear first order PDE. Student knows a notion of first integral; is able to construct general solution of problem for quasilinear first order PDEs by using characteristics. KA7_WG02, KA7_WG03, KA7_UW02, KA7_UW06
Student knows classification of first order PDEs and boudary value problems of different kinds; knows a notion of well-posed problem for mathematical physics equations and understands connection between equations and physical processes, described by them. Student is able to determine the type of PDE with two independent variables.KA7_WG02, KA7_WG04, KA7_WG06, KA7_UW06, KA7_UW10
Student knows a canonical form of hiperbolic PDE, methods of wave propagation, d'Alembert formula and understand Kirchoff formula. Student is able to use these formulas in simple examples. KA7_WG02, KA7_WG06, KA7_UW06
Student knows fundamental solution of Laplace equation, properties of harmonic functions, notion of Green function and its applications.KA7_WG02, KA7_WG03, KA7_UW01, KA7_UW06
Student understands maximum pronciple and uniqueness of solution in boundary value problem for heat equation with two independent variables. Student knows fundamental solution and formula for solutions in Cauchy problem for heat equation.KA7_WG02, KA7_WG04, KA7_WG05, KA7_UW06, KA7_UW10
Student obtains basic practice in creative development of theory of differential equations. KA7_KK01, KA7_KK02, KA7_KK07, KA7_UU01
Assessment criteria
The overall form of credit for the course: final exam
Additional information
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: