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Course Catalog 2010-2011
MAT-51316 Partial Differential Equations, 5 cr |
Person responsible
Robert Piche
Requirements
Exam, or exam and mid-term test.
Completion parts must belong to the same implementation
Principles and baselines related to teaching and learning
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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 final exam and a 2-hour midterm exam. Both exams are closed-book.
Assessment scale:
Numerical evaluation scale (1-5) will be used on the course
Partial passing:
Study material
| Type | Name | Author | ISBN | URL | Edition, availability, ... | Examination material | Language |
| Other online content | , | Robert Piche,Robert Piche | , | Course notes, exercises, solutions,Course notes, exercises, solutions | English |
Prerequisites
| Course | Mandatory/Advisable | Description |
| MAT-20451 Fourier'n menetelmät | Advisable | |
| MAT-33500 Differentiaaliyhtälöt | Mandatory |
Additional information about prerequisites
The first-year engineering mathematics material (IMA/LAMA 1-4) is also prerequisite.
Prerequisite relations (Requires logging in to POP)
Correspondence of content
There is no equivalence with any other courses
Additional information
Suitable for postgraduate studies