- Carrying out elementary transformations;
- Classification of PDEs;
- Perform linearization;
- Recognize which additional conditions make sense;
- Determine simple solution formulas by separation of variables, method of characteristic, or fundamental solutions;
- Proving the well-posedness of elliptic boundary value problems using the Lax-Milgram theorem;
- Computation with distributions and understanding their calculus;
- Determine qualitative properties of solutions using the maximum principle or from the characteristics.
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Definition of PDEs and classification scheme according to order, type, linearity;
Boundary value problems, initial value problem, initial boundary value problems;
Laplace equation, Heat equation, Wave equation and their essential properties;
Fundamental solution;
Characteristics;
Maximum principle;
Definition and elementary operations with distributions;
Definition and elementary results of Sobolev spaces;
Cauchy-Kovalevskaja theorem;
Holmgren's theorem;
Lax-Milgram lemma;
Definition of well- and ill-posedness;
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