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Syllabus:

# Partial Differential Equations, 7.5 Credits

Swedish name: Partiella differentialekvationer

This syllabus is valid: 2023-08-28 and until further notice

Course code: 5MA211

Credit points: 7.5

Education level: Second cycle

Main Field of Study and progress level: Mathematics: Second cycle, has only first-cycle course/s as entry requirements
Computational Science and Engineering: Second cycle, has only first-cycle course/s as entry requirements

Established by: Faculty Board of Science and Technology, 2023-02-15

## Contents

The course gives an overview of the theory of partial differential equations (PDEs).

The course can be divided into two main parts. The first part treats the Laplace-, heat- and wave- equation which represent model examples of linear elliptic, parabolic and hyperbolic PDEs. Furthermore, first-order non-linear problems and explicit solution techniques (e.g. transform methods, fundamental solutions, Green's function, scale invariance) are considered. The second part of the course covers weak solutions for second-order equations. Sobolev spaces are introduced and studied. The existence, uniqueness and regularity in these spaces is discussed and the properties of solutions is studied.

## Expected learning outcomes

For a passing grade, the student must be able to

Knowledge and understanding

• account for the basic theory of first-order non-linear problems
• account for the basic theory of Sobolev spaces
• formulate, prove and apply central theorems

Skills and abilities

• use techniques to derive solutions and representation formulas for PDEs
• prove and apply maximum principles for PDEs
• discuss and prove properties of solutions to PDEs

Judgement and approach

• classify PDEs and choose suitable methods for solutions
• formulate and prove existence and uniqueness theorems for PDEs

## 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 the level required for basic eligibility for higher studies. 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.

## Literature

### Valid from: 2023 week 35

Evans Lawrence C.
Partial differential equations
2nd ed. : Providence, R.I. : American Mathematical Society : 2010. : 749 s. :
ISBN: 978-0-8218-4974-3 (alk. paper)
Mandatory
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