**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:**

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

**Reference Books:**

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