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| TATM58 | Partial Differential Equations, 3 p (sw) /Partiella differentialekvationer och finita element/ Advancement level: D | |
| Aim: The course deals mainly with linear partial differential equations of second order. It gives familiarity with how the different types of equations occur in physics, particularly in mechanics with heat conduction included. Furthermore, it discusses questions of existence and uniqueness and deals with the mathematical principles which are the foundation of the finite-element method. It is also essential to gain an understanding of the properties of different soutions in general, as well as proficiency in how one, in practice, deals with different types of boundary-value problems and initial-value problems. The search for solutions by separation of variables as well as using transforms is another central point in the course. Prerequisites: TATM 72 Analys A, TATM 73 Analys B, TATM 74 Analys F or equivalent, ATM 18 Linear Algebra YD; TATM 41 Vector Analysis Y; TATM 50 Theory of Analytic Functions; TATM 51 Transform Theory.Course organization: Teaching is done with combined lectures/exercises.Course content: Derivation of the heat equation, Laplace's equation and the wave equation, beginning with physical balance laws. Classification of equations. Characteristics. Properties of harmonic functions. Connections with complex analysis. General properties of elliptic equations. Application of conformal mappings. Properties of solutions of time-dependent problems. Wave propagation. Eigenfunction expansions. Integral transforms. Green's function. The fundamental solution. Maximum principles. The Euler equation. Introduction to finite-element methods. Weak solutions, weak formulation. Natural and essential boundary conditions. The concept of Sobolev space. Simple error analysis.Course literature: Strauss, W.A: Partial Differential Equations. An introduction. John Wiley & Sons 1992. | ||
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