Partial Differential Equations

Lecturer:

Jiří Neustupa, Stanislav Kračmar

Tutor:

Jiří Neustupa, Stanislav Kračmar

Supervisor:

Department of Technical Mathematics

Synopsis:

: Principles of mathematical modelling by means of partial differential equations (PDE?s) and introduction to classical and modern theory of PDE?s. The modern theory is ilustrated on 2nd order elliptic PDE?s and on PDE?s appearing in mathematical models used in fluid mechanics.

Requirements:

Syllabus of lectures:

1. Principles of application of PDE?s to description of states and processes in continuum. The procedure of derivation of the transport equation, the heat equation and the equation of continuity.

2. The procedure of derivation of the Navier-Stokes equations, the wave equation, the Lame equations and the Maxwell equations.

3. Usage of the Laplace equation and the Poisson equation in mathematical models.

4. 1st order PDE?s - formulation of an initial or boundary-value problem, the principle of analytic solution.

5. Harmonic functions and their properties (the mean value theorem, the maximum principle).

5. Introduction to classical theory of PDE?s of the elliptic type, the Laplace and the Poisson equation, the role of the volume potential, the single-layer potential and the double-layer potential.

6. Introduction to classical theory of PDE?s of the parabolic type, the heat equation, the maximum principle, the Fourier method.

7. Introduction to classical theory of PDE?s of the hyperbolic type, the wave equation, characteristics, domain of dependence and domain of influence, the Fourier method.

8. Principles of modern theory of PDR?s. Distributions and their derivatives, generalized derivative.

9. Lebesgue?s space L{2} and Sobolev?s space W{1,2}. The scalar product and the norm in these spaces.

10. The theorem on traces in the space W{1,2}. The generalized boundary-value problem for the 2nd order elliptic equation, the weak solution.

11. Existence and uniqueness of the weak solution. Equivalence with the variational problem of finding the minimum of a quadratic functional.

12. The Galerkin and the Ritz metod of approximate solution.

13. The weak formulation of the initial-boundary value problem for the Navier-Stokes equations. Construction of approximations.

14. Convergence of approximations, existence of the weak solution, energy inequality.

Syllabus of tutorials:

1. Principles of application of PDE?s to description of states and processes in continuum. The procedure of derivation of the transport equation, the heat equation and the equation of continuity.

2. The procedure of derivation of the Navier-Stokes equations, the wave equation, the Lame equations and the Maxwell equations.

3. Usage of the Laplace equation and the Poisson equation in mathematical models.

4. 1st order PDE?s - formulation of an initial or boundary-value problem, the principle of analytic solution.

5. Harmonic functions and their properties (the mean value theorem, the maximum principle).

5. Introduction to classical theory of PDE?s of the elliptic type, the Laplace and the Poisson equation, the role of the volume potential, the single-layer potential and the double-layer potential.

6. Introduction to classical theory of PDE?s of the parabolic type, the heat equation, the maximum principle, the Fourier method.

7. Introduction to classical theory of PDE?s of the hyperbolic type, the wave equation, characteristics, domain of dependence and domain of influence, the Fourier method.

8. Principles of modern theory of PDR?s. Distributions and their derivatives, generalized derivative.

9. Lebesgue?s space L{2} and Sobolev?s space W{1,2}. The scalar product and the norm in these spaces.

10. The theorem on traces in the space W{1,2}. The generalized boundary-value problem for the 2nd order elliptic equation, the weak solution.

11. Existence and uniqueness of the weak solution. Equivalence with the variational problem of finding the minimum of a quadratic functional.

12. The Galerkin and the Ritz metod of approximate solution.

13. The weak formulation of the initial-boundary value problem for the Navier-Stokes equations. Construction of approximations.

14. Convergence of approximations, existence of the weak solution, energy inequality.

Study Objective:

Study materials:

- L.C.Evans: Partial Differential Equations. American Mathematical Society, series „Graduate Studies in Mathematics“, Vol. 19, New York 1997.

Note:

Time-table for winter semester 2011/2012:

Time-table is not available yet

Time-table for summer semester 2011/2012:

Time-table is not available yet

The course is a part of the following study plans: