Course detail
Partial Differential Equations
FSI-SPD Acad. year: 2024/2025 Winter semester
The course deals with the following topics:
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
Czech
Number of ECTS credits
5
Supervisor
Department
Entry knowledge
Solution of algebraic equations and system of linear equations, differential and integral calculus of functions of one and more variables, ordinary differential equations.
Aims
The aim of the subject is to provide students with the basic knowledge of the partial differential equations, their basic properties, methods of solving them, and their application in mathematical modelling. Another goal is to teach the students to formulate and solve simple problems for mathematical physics equations.
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.
The study programmes with the given course
Programme B-MAI-P: Mathematical Engineering, Bachelor's, compulsory
Programme C-AKR-P: , Lifelong learning
specialization CZS: , elective
Type of course unit
Exercise
26 hours, compulsory
Teacher / Lecturer
Syllabus
1 O.D.E., solution of the 1st order equations and higher order linear equations.
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.