Partial Differential Equations, 7.5 Credits

Contents

In the course, the theory of hyperbolic, parabolic and elliptic partial differential equations, is developed.

The course mainly consists of two parts. The first part treats classical solutions for boundary- and initial value problems for the Laplace-, heat- and wave operators. Furthermore, nonlinear problems of the first order, and several theoretical tools and theorems are studied: e.g. Fourier transform methods, special onsets (variable separation, scale invariant solutions) and Cauchy-Kovalevskaya's theorem. The second part of the course covers weak solutions for initial value- and boundary problems for elliptic, parabolic and hyperbolic operators of the second order. Sobolev spaces are introduced and studied. Thereafter existence, uniqueness and regularity problems in terms of Sobolev spaces are treated. The properties of the solutions and different methods for solutions are studied.

Expected learning outcomes

For a passing grade, the student must be able to

Knowledge and understanding

• thoroughly describe the basic theory of first-order nonlinear problems.
• thoroughly describe the basic theory of distributions, Sobolev spaces and embedding theorems
• thoroughly describe existence, uniqueness and regularity problems in scales of Sobolev spaces
• formulate and discuss existence theorems for general parabolic and hyperbolic PDEs
• independently formulate and prove Cauchy-Kovalevskaya's theorem and Lax-Milgram's lemma

Skills and abilities

• use fundamental solutions and Green functions for different kinds of equations
• use Fourier- and Laplace transform methods, variable separation and scale invariant solutions
• use the maximum principle for elliptic PDEs
• apply some of the methods for transmission of non-linear equations to linear PDEs

Judgment and approach

• classify PDEs and choose suitable methods for solution

Required Knowledge

The course requires 90 ECTS of which 22,5 ECTS is within Mathematical Analysis including a course in Multivariable Calculus and Differential Equations minimum 7,5 ECTS and a course in Linear Algebra minimum 7,5 ECTS. Proficiency in English equivalent to Swedish upper secondary course English 5/A. Where the language of instruction is Swedish, applicants must prove proficiency in Swedish to the level required for basic eligibility for higher studies.

Form of instruction

The teaching is mainly through lectures and problem solving sessions.

Examination modes

The course is examined by a written exam. For the course, one of the following grades is assigned: Fail (U), Pass (G), Pass with distinction (VG).

A student who has received a passing grade on a test is not allowed to retake the test in order to receive a higher grade. A student who has not received a passing grade after participating in two tests has the right to be assigned another examiner, unless there are certain circumstances prohibiting this (see the Higher Education Ordinance, chapter 6, 22§). A request to be assigned another examiner should be addressed to the head of department for the department of mathematics and mathematical statistics. The possibility of being examined based on the current version of the syllabus is guaranteed for at least two years following the student's first participation in the course.

Credit transfer
All students have the right to have their previous education or equivalent, and  their working life experience evaluated for possible consideration in the corresponding education at Umeå university. Application forms should be adressed to Student services/Degree evaluation office. More information regarding credit transfer can be found on the student web pages of Umeå university, http://www.student.umu.se, and in the Higher Education Ordinance (chapter 6). If denied, the application can be appealed (as per the Higher Education Ordinance, chapter 12) to Överklagandenämnden för högskolan. This includes partially denied applications

Other regulations

In a degree, this course may not be included together with another course with a similar content. If unsure, students should ask the Director of Studies in Mathematics and Mathematical Statistics. The course can also be included in the subject area of computational science and engineering.