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Partial Differential Equations, 6 ECTS credits.
/Partiella differentialekvationer/
For:
Mat
MMAT
Y
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Prel. scheduled
hours: 48
Rec. self-study hours: 112
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Area of Education: Science
Main field of studies: Mathematics, Applied Mathematics
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Advancement level
(G1, G2, A): A
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Aim:
The course deals mainly with second order linear partial differential equations. It will provide some familiarity with different types of equations occurring in physics, particularly in mechanics involving heat conduction. The course also discusses questions of existence and uniqueness of solutions to these equations. Students will gain an understanding of the properties of different solutions in general, as well as some knowledge of the practical dealing with different types of boundary-value problems and initial-value problems. The course will cover numerical methods for partial differential equations, eigenvalues problems, calculus of variations and distributions. During this course students will gain knowledge of modelling of diffusion and wave phenomenon and analysis of stability, existence and uniqueness properties of solutions. After the course students should:
- be able to solve heat and wave equations, elliptic equations and eigenvalue-problems associated with them, using transformations and separation variables.
- have knowledge about existence and uniqueness results and numerical methods for PDE.
- be able to use calculus of variations and distributions.
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Prerequisites: (valid for students admitted to programmes within which the course is offered)
Linear algebra, single and multi-variable calculus, vector analysis, Fourier analysis
Note: Admission requirements for non-programme students usually also include admission requirements for the programme and threshhold requirements for progression within the programme, or corresponding.
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Organisation:
Teaching is done with combined lectures/exercises.
The course runs over the entire spring semester.
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Course contents:
Origin of PDEs. Derivation of the heat equation, Laplace's equation and the wave equation, starting from physical balance laws. Classification of equations. Properties of harmonic functions. Connections with complex analysis. General properties of elliptic equations. Properties of solutions of time-dependent problems. Wave propagation. Integral transforms. Green's function. Distributions. The fundamental solution. Maximum principles. Weak solutions, weak formulation. Existence and uniqueness results. Numerical methods for PDE. Simple error analysis. Eigenvalue problems. Calculus of variations.
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Course literature:
Strauss, W.A: Partial Differential Equations. An introduction. John Wiley & Sons 2008.
Evans, L.W: Partial Differential Equations. American Mathematical Society, 1998.
Folland, G.B: Introduction to Partial Differential Equations, Princeton University Press 1995.
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Examination: |
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Written examination
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6 ECTS
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Course language is Swedish/English.
Department offering the course: MAI.
Director of Studies: Jesper Thorén
Examiner: David Rule
Link to the course homepage at the department
Course Syllabus in Swedish
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