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Partial Differential Equations

General data

Course ID: 360-MS2-2RRC2a
Erasmus code / ISCED: 11.105 Kod klasyfikacyjny przedmiotu składa się z trzech do pięciu cyfr, przy czym trzy pierwsze oznaczają klasyfikację dziedziny wg. Listy kodów dziedzin obowiązującej w programie Socrates/Erasmus, czwarta (dotąd na ogół 0) – ewentualne uszczegółowienie informacji o dyscyplinie, piąta – stopień zaawansowania przedmiotu ustalony na podstawie roku studiów, dla którego przedmiot jest przeznaczony. / (0541) Mathematics The ISCED (International Standard Classification of Education) code has been designed by UNESCO.
Course title: Partial Differential Equations
Name in Polish: Partial Differential Equations
Organizational unit: Faculty of Mathematics
Course groups:
ECTS credit allocation (and other scores): 5.00 Basic information on ECTS credits allocation principles:
  • the annual hourly workload of the student’s work required to achieve the expected learning outcomes for a given stage is 1500-1800h, corresponding to 60 ECTS;
  • the student’s weekly hourly workload is 45 h;
  • 1 ECTS point corresponds to 25-30 hours of student work needed to achieve the assumed learning outcomes;
  • weekly student workload necessary to achieve the assumed learning outcomes allows to obtain 1.5 ECTS;
  • work required to pass the course, which has been assigned 3 ECTS, constitutes 10% of the semester student load.
view allocation of credits
Language: English
Type of course:

obligatory courses
Prerequisites (description):

Student should know basic methods from one-dimensional and multidimensional mathematical analysis and ordinary differential equations.
Short description:

Course profile: academic

Form of study: stationary

Academic discipline: Mathematics, field of study in science: mathematics

Prerequisities: Mathematical Analysis, Ordinary Differential Equations

lecture 30 h. exercise class 30 h.

ECTS credits: 5
Full description:

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
Bibliography:

1. L. Evans Partial Differential Equations, Springer 1998

2. M. Renardy, R.C.Rogers An introduction to partial differential equations, Springer 1993

3. E.C. Zachmanoglou, D. Thoe Introduction to partial differential equations with applications 1967

4. P. Olver Introduction to partial differential equations, Springer 2013
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 methods and assessment criteria:

The overall form of credit for the course: final exam

Classes in period "Academic year 2022/2023" (past)

Time span: 2022-10-01 - 2023-06-30
Selected timetable range:
Navigate to timetable
Type of class:
Class, 30 hours more information
Lecture, 30 hours more information
Coordinators: Tomasz Czyżycki, Aneta Sliżewska
Group instructors: (unknown)
Students list: (inaccessible to you)
Examination: Course - Examination
Class - Grading
Full description:

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.
Bibliography:

1. L. Evans Partial Differential Equations, Springer 1998

2. M. Renardy, R.C.Rogers An introduction to partial differential equations, Springer 1993

3. E.C. Zachmanoglou, D. Thoe Introduction to partial differential equations with applications 1967

4. P. Olver Introduction to partial differential equations, Springer 2013
Course descriptions are protected by copyright.
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