Through lectures and assignments, explore a variety of PDEs with emphasis on the theoretical aspects and related techniques to find exact solutions and understand their analytic properties. Learn both basic concepts and modern techniques for the formulation and solution of various PDE problems. Main topics include the method of characteristics for first order PDE, formulation and solutions to the wave equation, heat equation and Laplace equation, and classical tools to study properties of these PDEs.
Basic concepts about PDEs
First-order PDEs and method of characteristics
General first-order PDEs
Wave equation and D’Alembert’s formula
Conservation of energy and Duhamel’s principle
Spherical means, Kirchhoff’s and Poisson’s formulae
Heat equation and its fundamental solution
Energy method
Maximum principle for diffusion equations
Laplace equation and harmonic functions
Mean value property and maximum principle
Green’s function
Fourier series
Separation of variables
Exam: 50%, Homework: 50%
Partial Differential Equations, An Introduction to Theory and Applications, Michael Shearer, Rachel Levy, Princeton University Press, 2015.
• Partial Differential Equations, Lawrence C. Evans, 2nd edition, American Mathematical Society, 2022.
• Applied Partial Differential Equations: An Introduction, Alan Jeffrey, Academic Press, 2002.
• Partial Differential Equations: A First Course, Rustum Choksi, American Mathematical Society, 2022.