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Course, academic year 2019/2020
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Partial Differential Equations I - NDIR044
Title in English: Parciální diferenciální rovnice I
Guaranteed by: Department of Mathematical Analysis (32-KMA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2015
Semester: winter
E-Credits: 6
Hours per week, examination: winter s.:2/2 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: not taught
Language: Czech
Teaching methods: full-time
Class: DS, matematické a počítačové modelování
Classification: Mathematics > Differential Equations, Potential Theory
In complex pre-requisite: NMMA349, NMNM349
Annotation -
Last update: T_KMA (22.05.2008)
Classical solvability of boundary and initial value problems for partial differential equations. Elliptic, parabolic and hyperbolic equations of the second order.
Literature -
Last update: T_KMA (22.05.2008)

L. C. Evans: Partial Differential Equations, AMS 1999

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

O. John, J. Nečas: Rovnice matematické fyziky, SPN 1972 (in Czech)

Syllabus -
Last update: T_KMA (22.05.2008)

I. PDEs of first order and their connection with the systems of ODEs. Fundamental systems of solutions. Cauchy problem for transport and Burgers' equations - examples of the non-existence of a global classical solution.

II. Theorem of Cauchy-Kowalevskaya. Higher order partial differential equations. Characteristics. Classification of PDEs of the second order.

III. Classical solutions of the basic types of PDEs

a) Laplace and Poisson equations. Fundamental solutions, Green representation formula. Poisson's formula. Properties of harmonic functions: Mean-value formula, strong maximum principle, Liouville's theorem, analyticity, theorem on removable singularity, Harnack's theorems, Uniqueness of the solution for external Dirichlet problem. Existence of a classical solution to Dirichlet problem.

b) Heat equation. Fundamental solution. Poisson formula for the classical solution of Cauchy problem. Duhamel's principle. Maximum principles the initial-boundary value problem and for Cauchy problem. Uniqueness results. Energy inequalities.

c) Wave equation. Uniqueness result. Fundamental solutions. Classical solution of Cauchy problem, D'Alembert, Poisson and Kirchhoff formula. Duhamel principle.

 
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