 Second order linear equations with two variables: classification of the equations in the case of constant and variable coefficients, characteristics, canonical forms.
 SturmLiouville theory.
 String or wave equation. Initial and boundary value conditions (fixed and free boundary conditions). The d’Alembert method for an infinitely long string. Characteristics. Wave problems for halfinfinite and finite strings. A solution of a problem for a finite string with fixed and free boundary conditions by the method of separation of variables. The uniqueness proof by the energy method. Wellposedness of the vibrating string problem.
 Laplace and Poisson equations. Maximum principle. Wellposedness of the Dirichlet problem. Laplace equation in a rectangle. Laplace equation in a circle and Poisson formula. An illposed problem  the Cauchy problem. Uniqueness of a solution of the Dirichlet problem. Green formula in the plane and its application to Neumann problems.
 Heat equation. The method of separation of variables for the onedimensional heat equation. Maximum principle. Uniqueness for the onedimensional heat equation. The Cauchy problem for heat equations. Green?s function in one dimension. If time permits: Green?s function in the two dimensional case.
 Nonhomogeneous heat equations, Poisson equations in a circle and nonhomogeneous wave equations.
 If time permits: free vibrations in circular membranes. Bessel equations.

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