Monday **12.11.2018** at **12:15** in **Geb. E1.1, Zi. 3.23.2**

The purpose of this course is to provide to the students a realistic and practically oriented introduction to the typical models of physical phenomena (diusion-type problems, hyperbolic-type problems, elliptic-type problems) and for methods of solving these problems. It would be showed how to formulate a partial dierential equation from the physical problem (constructing the mathematical model) and how to solve the equation along with initial and boundary conditions by using analytical, numerical and approximate methods.

Calculus (one and several variables), basic knowledge of Ordinary Dierential Equations (desirable but not necessary).

- Introduction to PDEs. What are PDEs? Why are PDEs useful? Kinds of PDEs.
- Diffusion-type problems (parabolic equations). Simple heat-flow experiment and its mathematical model. Boundary conditions for diusion-type problems. Separation of variables. Eigenfunction expansions. Fourier and Laplace transforms. Solving initial boundary value problem with Maple.
- Hyperbolic-type problems. Vibrating-string problem and its mathematical model. The D'Alembert solution of the wave equation. The wave equations in 2D and in 3D. Vibrating drumhead. Solving hyperbolic problems with Maple.
- Elliptic-type problems. An intuitive description of the Laplacian. Nature of the elliptic problems. Interior and exterior Dirichlet problem for a circle.
- Numeric and approximate methods. Analytic versus numerical solutions. Numerical solutions of elliptic problems. An explicit nite-dierence method. The Crank-Nicolson Method. Monte-Carlo methods.

Please send e-mail to darya@math.uni-sb.de

Compact course, approximately 2 weeks. Exact dates would be fixed during the organisational meeting on 12.11.2018.

The course is suitable for students specializing in applied mathematics, physics, computer science, visual computing, bioinformatics. The course language is English.

The script in terms of transparencies will be available for download in the table below.

- L.C. Evans, Partial Differential Equations, Graduate Stud. Math.,
**19**, Amer. Math. Soc., Providence, Rhode Island, 1998. - M.A. Pinsky, Partial-Differential Equations and Boundary-Value Problems with Applications, Reprint of the third (1998) edition, Pure and Applied Undergraduate Texts,
**15**, Amer. Math. Soc., Providence, Rhode Island, 2011. - S.J. Farlow, Partial Differential Equations for Scientists and Engineers, Dover Publications, INC. New York, 1993.