Course Details:

Introduction, PDEs as mathematical models, first order PDEs: method of characteristics, quasilinear first order PDEs.

Distributions, distribution solutions and weak solutions.

Classification of second order PDEs, Laplace equation: fundamental solution, mean-value formulas, The maximum principle, Poisson equation, properties of harmonic functions, Green’s functions, energy methods.

Heat equation: diffusion and Brownian motion, Fourier transforms, fundamental solution, the maximum principle, energy methods

Wave equation: one-dimensional wave equation, d'Alembert’s formula, higher-dimensional wave equation, energy estimates

Fourier series, boundary value problems, separation of variables

Text Books:

  1. Walter A. Strauss,  Partial Differential equations : An Introduction, John Wiley & Sons; 2nd edition, 2008
  2. Lawrence C. Evans,  Partial Differential Equations, American Mathematical Society; 2nd Revised edition, 2010
  3. Qing Han,  A Basic Course in Partial Differential Equations, American Mathematical Society; 2011
  4. András, Vasy, Partial Differential Equations: An Accessible Route through Theory and Applications: American Mathematical Society, 2015

Reference Books:

  1. Gerald B. Folland, Introduction to partial differential equations, Princeton University Press, Princeton, NJ, second edition, 1995.
  2. Michael E. Taylor. Partial differential equations I: Basic theory, Texts in Applied Mathematics, Vol. 23, Springer-Verlag;  1996
  3. Yehuda Pinchover and Jacob Rubinstein, An Introduction to Partial Differential Equations, Cambridge University Press; 2005
  4. S. Kesavan,  Topics in Functional Analysis and Applications, New Age International Private Limited; 2015
  5. Fritz John,  Partial Differential Equations (Applied Mathematical Sciences), Springer; 4th ed., 1991
  6. Jurgen Jost, Partial Differential Equations, Graduate Texts in Mathematics 214,  Springer; 2007
  7. Hörmander, L. Lectures on Nonlinear Hyperbolic Differential Equations, Springer; 1997
  8. Courant, R. and Hilbert, D. Methods of mathematical physics. Vol. II: Partial differential equations, Wiley VCH;  1989