### Schedule

9.9. Course practices and grading. Introduction to PDEs, periodic functions and the Fourier series. Pages 1-15 in the lecture notes.

16.9. The least square approximation, properties and convergence of the Fourier series, the Dirichlet kernel and convolution. Pages 15-36 in the lecture notes.

23.9. Solution to the Dirichlet problem on the unit circle by separation of variables and Fourier series. Pages 36-47 in the lecture notes.

30.9. Solution of the heat and wave equations by separation of variables and convolution approximation. Pages 47-64 in the lecture notes.

7.10. Properties of the Fourier transform, Fourier inverse theorem and convolution approximation. Pages 65-83 in the lecture notes.

14.10. Solution of the Laplace, heat and wave equations in the upper half space using the Fourier transform. Pages 83-94 in the lecture notes.

29.10. Gauss' theorem, Green's formulas, the fundamental solution of the Laplace equation, solution to the Poisson equation in the whole space. Pages 95-113 in the lecture notes.

5.11. Solution to the Dirichlet problem by Green's functions, the mean value property and maximum principle for harmonic functions. Pages 114-131 in the lecture notes.

12.11. Consequences of the maximum principle and variational methods for the Laplace equation. The fundamental solution of the heat equation and Duhamel's principle for the nonhomogeneous Cauchy problem. Pages 131-152 in the lecture notes.

19.11. Separation of variables and eigenvalue problems in the higher dimensional case, the maximum principle and its consequences for the heat equation. Pages 152-163 in the lecture notes.

26.11. Solution to the wave equation in the one- and three-dimensional cases. Pages 164-180 in the lecture notes.

3.12. Solution to the wave equation in the two-dimensional case, Duhamel's principle and energy methods for the wave equation. Pages 180-189. in the lecture notes. The course exam will be discussed at the end of the last lecture.

**Further reading**

L.C. Evans, Partial Differential Equations, American Mathematical Society, Second Edition, 2010.

E. DiBenedetto, Partial Differential Equations, Birkhäuser 1995.

Q. Han, A Basic Course in Partial Differential Equations, American Mathematical Society, 2011.

R. McOwen, Partial Differential Equations. Methods and Applications. Prentice-Hall, 1996.

E.M. Stein and R. Sakarchi, Fourier Analysis. An Introduction. Prenceton University Press, 2003.

A. Vasy, Partial Differential Equations. An Accessible Route through Theory and Applications. American Mathematical Society, 2015.