Course detail
Partial Differential Equations
FSI-SPDAcad. year: 2023/2024
The course deals with the following topics: Ordinary differential equations - a brief survay of material studied within the 3rd semester subject and extending of the subject matter (theorems on existence and uniqueness of the solution, stability of the solution, boundary value problems).
Partial differential equations - basic concepts. The first-order equations. The Cauchy problem for the k-th order equation. Transformation, classification and canonical form of the second-order equations.
Derivation of selected equations of mathematical physics (heat conduction, wave equation, variational prinsiple), formulation of initial and boundary value problems.
The classical methods: method of characteristics, The Fourier series method, integral transform method, the Green function method. Maximum principles. Properties of the solutions to the elliptic, parabolic and hyperbolic equations.
Language of instruction
Number of ECTS credits
Mode of study
Guarantor
Department
Entry knowledge
Rules for evaluation and completion of the course
Control test 1: O.D.E.: (a) solution of the 1st order equation, (b) solution of the 2nd order linear equation, (c) solution of a system of linear equations - stability, classification of trajectories.
Control test 2: P.D.E.: (a) solution of the 1st order equation, (b) classification, and transformation of the 2nd order equation to its canonical form, (c) formulation of an initial boundary value problem related to the physical setting and finding its solution by means of the Fourier series method.
The examination consists of a practical and a theoretical part. Practical part: solving examples of P.D.E., see Control test 2. Theoretical part: theory of O.D.E. and P.D.E. (1 + 3 questions).
Absence has to be made up by self-study using lecture notes. Passing the control tests is required, in cases of bad result or absence in additional term.
Aims
Revision and deepening of the knowledge of Ordinary Differential Equations. Elements of the theory of Partial Differential Equations and survey of their application to the mathematical modelling. Ability to formulate mathematical model of the selected problems of mathematical physics and to compute the solution or propose an algorithm for numerical solution.
Study aids
Prerequisites and corequisites
Basic literature
L. C. Evans: Partial Differential Equations, AMS, Providence 1998
V. J. Arsenin: Matematická fyzika, Alfa, Bratislava 1977
W. E. Williams: Partial differential equations,
Recommended reading
J. Škrášek, Z. Tichý: Základy aplikované matematiky II, SNTL, Praha 1986 (CS)
K. Rektorys: Přehled užité matematiky II., Prometheus 1995 (CS)
V. J. Arsenin: Matematická fyzika, Alfa, Bratislava 1977. (SK)
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Classification of course in study plans
- Programme B-MAI-P Bachelor's 3 year of study, winter semester, compulsory
Type of course unit
Lecture
Teacher / Lecturer
Syllabus
1 Revision of O.D.E. - 1st order and higher order linear equations.
2 Systems of linear O.D.E., existence and uniqueness of the solution.
3 Test 1: O.D.E. Elements of P.D.E., 4 The 1st order equations.
5 The Cauchy problem, classification of 2nd order equations.
6 Mathematical Physics Equations: derivation of the heat equation.
7 Derivation of the equation of string vibration and wave equations.
8 Derivation of membrane equation via variational principle.
9 Method of characteristics for 1D wave equation.
10 Fourier series method.
11 Integral transform method. Test 2 of P. D. E.
12 Green function method and the maximum principles.
13 Properties of the solutions, reserve.
Exercise
Teacher / Lecturer
Syllabus
2 Solution of systems of linear O.D.E., stability of the solution.
3 The phase portrait of solutions to autonomous system.
4 P.D.E., solving of the 1st order equations.
5 Written test 1, classification of 2nd order equations.
6 Formulation of problems related to the heat equation.
7 Formulation of problems related to the wave equation.
8 Derivation of membrane equation via variational principle.
9 Solving problems by the method of characteristics.
10 Solving problems by the Fourier series method.
11 Written test 2.
12 Using the Green function method, harmonic functions.
13 Properties of the solutions, course-credits.
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