Topic outline

  • Lecture notes

    The course will follow the lecture notes. There is no need to buy books, but in the end of the page, we list some further reading material.

    The course uses flipped classroom method which means that

    • the students are expected to read the lecture notes before the lectures, and
    • answer to pre-lecture quizzes before the lectures (Monday 14:15) starting from week 2.

    Then it is possible to ask the lecturer about things that are still unclear. We will try both zoom-meeting and chat to see which method works best.

    • File icon
      Lecture Notes File
      PDF document

      These lecture notes will be updated during the course and the page numbers may also change. Please avoid unnecessary printing.

      There are mistakes in Remark 6.14. The problems start when, the right-hand-side is not multiplied by \(r^2\).

    • Weekly topics and lecture note pages

      1st period:

      (1) 7.9.  Introduction to PDEs, periodic functions and the Fourier series. Pages 1-18 in the lecture notes.

      (2) 14.9. The least square approximation, properties and convergence of the Fourier series, the Dirichlet kernel and convolution. Pages 15-34 in the lecture notes.

      (3) 21.9. Solution to the Dirichlet problem on the unit circle by separation of variables and Fourier series. Pages 34-45 in the lecture notes.

      (4) 28.9. Solution of the heat and wave equations by separation of variables and convolution approximation. Pages 46-64 in the lecture notes.

      (5) 5.10. Properties of the Fourier transform, Fourier inverse theorem and convolution approximation. Pages 65-83 in the lecture notes.

      (6) 12.10. Solution of the Laplace, heat and wave equations in the upper half space using the Fourier transform. Pages 83-94 in the lecture notes.

      2nd period:

      (7) 27.10.  Gauss' theorem, Green's formulas, the fundamental solution of the Laplace equation, solution to the Poisson equation in the whole space. Pages 95-115 in the lecture notes.

      (8) 3.11. Solution to the Dirichlet problem by Green's functions, the mean value property and maximum principle for harmonic functions. Consequences of the maximum principle. Pages 116-134(135) in the lecture notes.

      (9) 10.11. Variational methods for the Laplace equation. The fundamental solution of the heat equation and Duhamel's principle for the nonhomogeneous Cauchy problem. Pages 136-153 in the lecture notes.

      (10) 17.11. Separation of variables and eigenvalue problems in the higher dimensional case, the maximum principle and its consequences for the heat equation. Pages 153-163 in the lecture notes.

      (11) 24.11. Solution to the wave equation in the one- and three-dimensional cases. Pages 165-181 in the lecture notes.

      (12) 1.12. Solution to the wave equation in the two-dimensional case, Duhamel's principle and energy methods for the wave equation. Pages 181-190.  in the lecture notes. 


    • 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.



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Pre-lecture assignments