Course Catalog 2012-2013
Basic

Basic Pori International Postgraduate Open University

|Degrees|     |Study blocks|     |Courses|    

Course Catalog 2012-2013

MAT-51316 Partial Differential Equations, 5 cr

Additional information

Self-study course, contact Risto Silvennoinen for instructions. Pre-recorded lectures in English are available on-line for self-study.
Suitable for postgraduate studies
Will not be lectured year 2012-2013

Person responsible

Robert Piche, Risto Silvennoinen

Requirements

Exam. Homework bonus.
Completion parts must belong to the same implementation

Principles and baselines related to teaching and learning

-

Learning outcomes

Partial differential equations (PDEs) are used to model physical phenomena involving continua, such as fluid dynamics, electromagnetic fields, acoustics, gravitation, and quantum mechanics. They also arise as mathematical models of phenomena involving multivariate functions, for example in mathematical finance. After studying this course, the student is able to derive the basic linear PDEs (transport, heat/diffusion, wave, Laplace) and explain how they model physical phenomena. He/she can prove and apply theorems about solution properties, can solve simple PDE problems using standard analytical solution methods (separation of variables, Dirichlet's principle, Green's functions), and is acquainted with numerical PDE solution packages in Matlab and Maple.

Content

Content Core content Complementary knowledge Specialist knowledge
1. one-dimensional transport equation: derivation, solution using method of characteristics   classification of PDEs (order, linearity, homogeneity)  Maple PDEplot 
2. one-dimensional wave equation: derivation, boundary conditions (Dirichlet, Neumann, Robin), initial value problem solution using d'Alembert's formula, conservation of energy and uniqueness, reflection method for problem on half-line   derivation of d'Alembert's formula, range of influence, domain of dependence  ill-posed problems 
3. one-dimensional diffusion (or heat) equation: derivation, boundary conditions (Dirichlet, Neumann, Robin), initial value problem solution using diffusion kernel, solution stability and uniqueness, erf(x), Duhamel's principle, reflection method for problem on half-line  solution smoothness, infinite speed of propagation, Duhamel's principle for ordinary differential equations  derivation of diffusion kernel using Fourier transform 
4. Separation of variables: solution of constant-coefficient one-dimensional heat and wave PDEs  Sturm-Liouville theory (eigenvalue realness, positivity, orthogonality, eigenfunction uniqueness), Fourier series theory (Besssel's inequality, completeness, pointwise convergence)  Matlab pdepe 
5. Laplace equation: derivation as equilibrium state of heat equation or membrane vibration equation, maximum principle, solution uniqueness, mean value property, Dirichlet's principle  rotational invariance, Rayleigh-Ritz method   
6. Separation of variables for two-dimensional Laplace equation boundary value problem on a disk (Poisson integral formula) and on a rectangle  Laplace equation on a wedge and on the outside of a circle  streamlines 
7. Green's functions in one and two dimensions: derivation, reciprocity principle, method of images  nonhomogeneous boundary conditions, Green's functions from eigenfunctions  Matlab PDE toolbox 

Evaluation criteria for the course

The grade is based on a 3-hour closed-book final exam.

Assessment scale:

Numerical evaluation scale (1-5) will be used on the course

Partial passing:

Completion parts must belong to the same implementation

Study material

Type Name Author ISBN URL Edition, availability, ... Examination material Language
Book   Partial Differential Equations   Robert Piché            English  
Other online content   Home page   Robert Piche       Recorded lectures (flash), exercises (pdf)      English  

Additional information about prerequisites
The first-year engineering mathematics curriculum (IMA/LAMA 1-4) is prerequisite.

Prerequisite relations (Requires logging in to POP)

Correspondence of content

There is no equivalence with any other courses

Last modified07.03.2012