Arbeitsgruppe Prof. S. Rjasanow Institut für angewandte Mathematik

#### PDEs and Boundary-Value Problems

##### Organisational meeting

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

##### Aims and objectives:

The purpose of this course is to provide to the students a realistic and practically oriented introduction to the typical models of physical phenomena (di usion-type problems, hyperbolic-type problems, elliptic-type problems) and for methods of solving these problems. It would be showed how to formulate a partial di erential 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.

##### Prerequisites:

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

##### Sillabus:
• 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 di usion-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-di erence method. The Crank-Nicolson Method. Monte-Carlo methods.

##### Lecture

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

##### Target Group

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

##### Script

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

##### Literature
• 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.
lehre/vorlesung/pdebvp1819.txt · Zuletzt geändert: 2018/10/15 12:04 von agrja 