| Course title | Partial differential equations |
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| Course code | KMA/P529 |
| Organizational form of instruction | Lecture |
| Level of course | Bachelor |
| Year of study | not specified |
| Semester | Summer |
| Number of ECTS credits | 4 |
| Language of instruction | Czech |
| Status of course | unspecified |
| Form of instruction | unspecified |
| Work placements | unspecified |
| Recommended optional programme components | None |
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| Course content |
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1. Basic definitions - partial differential equations ( PDE ), linear and nonlinear equations, classical solution to PDE, basic problems for PDE. 2. The first order PDE, linear equation in two dimensions, characteristic system, fundamental system of solutions, initial-value problem, existence of the solution, homogeneous equation in higher dimension. 3. The second order PDE in two dimensions - Cauchy problem, existence and uniquennes of the solutin, generalized Cauchy problem in two dimensions . 4. Classification of the second order linear PDE in two dimensions, equations with constant coefficients. 5. Derivation of the main equations of mathematical physics -the Laplace equation, the heat equation, the diffusion equation, the wave equation . 6. The fundamental methods of solving equations, mentioned above - the wave equation - uniqueness result, fundamental solutions, classical solution of Cauchy problem, D`Alambert, Poissson and Kirchoff formula , Fourier method ; the heat equation - fundamental solutions, Poisson formula for classical solution of Cauchy problem, maximum principle for Dirichlet and Cauchy problems, uniqueness results ; the Laplace equation in two and three dimensions - fundamental solutions, harmonic functions, mean-value formula, maximum principle, Dirichlet and Neumann problems, existence of classical solution of Dirichlet problem, solution for sphere.
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| Learning activities and teaching methods |
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| Learning outcomes |
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The course is devoted to an elementary survey of types of the first and the second order partial differential equations ( PDE ) and to methods applied to solution ( in the classical sence ) of initial and boundary value problems for PDE. The classes of the first-order partial differential equations and elliptic, parabolic and hyperbolic second-order equations are defined. The properties and methods of treatment of several standart equations of mathematical physics - the Laplace equation, the heat equation , the diffusion equation and the wave equation are studied. Some of fundamental methods of solution are presented including characteristic, separation of variables and Fourier methods . Attention is paid also to the creation of the mathematical models of physical and technical problems.
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| Prerequisites |
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unspecified
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| Assessment methods and criteria |
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unspecified
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| Faculty | Study plan (Version) | Category of Branch/Specialization | Recommended semester |
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