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Partial Differential Equations, 6 credits
Partiella differentialekvationer, 6 hp
TATA27
Main field of study
Mathematics Applied MathematicsCourse level
Second cycleCourse type
Programme courseExaminer
David RuleDirector of studies or equivalent
Jesper ThorénEducation components
Preliminary scheduled hours: 48 hRecommended self-study hours: 112 h
Available for exchange students
YesECV = Elective / Compulsory / Voluntary
Main field of study
Mathematics, Applied MathematicsCourse level
Second cycleAdvancement level
A1XCourse offered for
- Mathematics, Master's programme
- Mathematics
- Applied Physics and Electrical Engineering, M Sc in Engineering
- Applied Physics and Electrical Engineering - International, M Sc in Engineering
Entry requirements
Note: Admission requirements for non-programme students usually also include admission requirements for the programme and threshold requirements for progression within the programme, or corresponding.
Prerequisites
Linear algebra, single and multi-variable calculus, vector analysis, Fourier analysis
Intended learning outcomes
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.
Course content
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.
Teaching and working methods
Teaching is done with combined lectures/exercises.
The course runs over the entire spring semester.
Examination
| TEN1 | Written examination | 6 credits | U, 3, 4, 5 |
Grades
Four-grade scale, LiU, U, 3, 4, 5Department
Matematiska institutionenDirector of Studies or equivalent
Jesper ThorénExaminer
David RuleCourse website and other links
Education components
Preliminary scheduled hours: 48 hRecommended self-study hours: 112 h
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.| Code | Name | Scope | Grading scale |
|---|---|---|---|
| TEN1 | Written examination | 6 credits | U, 3, 4, 5 |
Regulations (apply to LiU in its entirety)
The university is a government agency whose operations are regulated by legislation and ordinances, which include the Higher Education Act and the Higher Education Ordinance. In addition to legislation and ordinances, operations are subject to several policy documents. The Linköping University rule book collects currently valid decisions of a regulatory nature taken by the university board, the vice-chancellor and faculty/department boards.
LiU’s rule book for education at first-cycle and second-cycle levels is available at http://styrdokument.liu.se/Regelsamling/Innehall/Utbildning_pa_grund-_och_avancerad_niva.
Course literature is preliminary.
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.
Note: The course matrix might contain more information in Swedish.
I = Introduce, U = Teach, A = Utilize
| I | U | A | Modules | Comment | ||
|---|---|---|---|---|---|---|
| 1. DISCIPLINARY KNOWLEDGE AND REASONING | ||||||
| 1.1 Knowledge of underlying mathematics and science (G1X level) |
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X
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X
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| 1.2 Fundamental engineering knowledge (G1X level) |
X
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| 1.3 Further knowledge, methods, and tools in one or several subjects in engineering or natural science (G2X level) |
X
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X
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| 1.4 Advanced knowledge, methods, and tools in one or several subjects in engineering or natural sciences (A1X level) |
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| 1.5 Insight into current research and development work |
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| 2. PERSONAL AND PROFESSIONAL SKILLS AND ATTRIBUTES | ||||||
| 2.1 Analytical reasoning and problem solving |
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X
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X
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| 2.2 Experimentation, investigation, and knowledge discovery |
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X
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X
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| 2.3 System thinking |
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| 2.4 Attitudes, thought, and learning |
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X
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X
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| 2.5 Ethics, equity, and other responsibilities |
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X
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| 3. INTERPERSONAL SKILLS: TEAMWORK AND COMMUNICATION | ||||||
| 3.1 Teamwork |
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| 3.2 Communications |
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X
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| 3.3 Communication in foreign languages |
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| 4. CONCEIVING, DESIGNING, IMPLEMENTING AND OPERATING SYSTEMS IN THE ENTERPRISE, SOCIETAL AND ENVIRONMENTAL CONTEXT | ||||||
| 4.1 External, societal, and environmental context |
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| 4.2 Enterprise and business context |
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| 4.3 Conceiving, system engineering and management |
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| 4.4 Designing |
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| 4.5 Implementing |
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| 4.6 Operating |
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| 5. PLANNING, EXECUTION AND PRESENTATION OF RESEARCH DEVELOPMENT PROJECTS WITH RESPECT TO SCIENTIFIC AND SOCIETAL NEEDS AND REQUIREMENTS | ||||||
| 5.1 Societal conditions, including economic, social, and ecological aspects of sustainable development for knowledge development |
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| 5.2 Economic conditions for knowledge development |
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| 5.3 Identification of needs, structuring and planning of research or development projects |
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| 5.4 Execution of research or development projects |
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| 5.5 Presentation and evaluation of research or development projects |
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