Partial Differential Equations

Course Aim

• State the definitions of basics notions and adopt the terminology in partial differential equations.
• Apply the method of characteristics to solve nonlinear first order partial differential equations.
• Use formulas to compute explicit solutions to second order equations including the wave equation, heat equation and Laplace equation.
• Apply analytic methods appropriately to investigate partial differential equations.
• Adopt proof techniques correctly to verify properties of solutions rigorously.

Course Description

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.

Course Contents

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

Assessment

Exam: 50%, Homework: 50%

Prerequisites or Prior Knowledge

Single-variable and multi-variable calculus, Linear algebra, ordinary differential equations, real analysis, or equivalent knowledge.

Textbooks

Partial Differential Equations, An Introduction to Theory and Applications, Michael Shearer, Rachel Levy, Princeton University Press, 2015.

Reference Books

• 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.