# Syllabus for Introduction to Partial Differential Equations

Partiella differentialekvationer, introduktionskurs

## Syllabus

• 5 credits
• Course code: 1MA053
• Education cycle: Second cycle
• Main field(s) of study and in-depth level: Mathematics A1N
• Grading system: Fail (U), Pass (3), Pass with credit (4), Pass with distinction (5)
• Established: 2007-03-15
• Established by:
• Revised: 2018-08-30
• Revised by: The Faculty Board of Science and Technology
• Applies from: week 24, 2019
• Entry requirements: 120 credits including 60 credits in mathematics, including Several Variable Calculus, Linear Algebra II, and Transform Methods.
English language proficiency that corresponds to English studies at upper secondary (high school) level in Sweden ("English 6").
• Responsible department: Department of Mathematics

## Learning outcomes

On completion of the course, the student should be able to:

• describe the most common partial differential equations that appear in problems concerning e.g. heat conduction, flow, elasticity and wave propagation;
• give an account of basic questions concerning the existence and uniqueness of solutions, and continuous dependence of initial and boundary data;
• solve simple first order equations using the method of characteristics;
• classify second order equations;
• solve simple initial and boundary value problems using e.g. d'Alembert's solution formula, separation of variables, Fourier series expansion or expansion in other orthogonal systems;
• describe, compute and analyse wave propagation and heat conduction in mathematical terms;
• formulate maximum principles for various equations and derive consequences.

## Content

Introduction of some common partial differential equations, physical background and derivation from physical principles.
First order partial differential equations: characteristics, linear, quasilinear and general nonlinear equations.
Classification of second order partial differential equations in two variables.
The one-dimensional wave equation, Cauchy's problem, d'Alembert's formula, the nonhomogeneous wave equation.
Separation of variables, the heat and wave equations. The energy method, uniqueness.
Sturm-Liouville problems and eigenfunction expansions.
Elliptic equations. Dirichlet's problem, harmonic functions, the maximum principle. Poisson's formula.
Green functions and integral representations. The heat kernel.
Partial differential equations in higher dimensions.

## Instruction

Lectures and problem solving sessions.

## Assessment

Written examination at the end of the course combined with assignments given during the course.

If there are special reasons for doing so, an examiner may make an exception from the method of assessment indicated and allow a student to be assessed by another method. An example of special reasons might be a certificate regarding special pedagogical support from the disability coordinator of the university.