# Equations of Mathematical Physics

There are many ordinary/partial differential equations/systems describing various phenomena in physics, mechanics, chemistry, biology, sociology, etc. We study how to write out the equations and the related boundary conditions. What does “well-posedeness” mean? Various kinds of PDE and the main properties of their solutions are considered. How does one determine the solutions and what does “determine” mean? What are the reasons behind blow-up phenomena? Some methods of analytical and numerical solutions will be considered as well.

Prerequisites:
1 semester “Advanced Calculus”, 1 semester “Linear Algebra”. Optional: 2 semesters “Advanced Calculus”, 2 semesters “Linear Algebra”, 1 semester “Functions of Complex Variables”, elements of the functional analysis and the ordinary differential equations theory.

### Curriculum:

1. Ordinary differential equations as models of various physical, chemical, biological etc. phenomena. First integrals. Existence and uniqueness for the Cauchy problem. Boundary value problem. Sturm-Liouville problem.
2. Partial differential equations as models of various phenomena. Well-posed and ill-posed problem. Change of variables.
3. First order PDEs. Characteristics. Cauchy problem. Conservation laws. Classical and generalized solutions. Regularization. Hamilton-Jacobi equations
4. Normal form of partial differential equations. Cauchy-Kovalevskaya theorem.
5. Hyperbolic equations in two-dimensional space. Cauchy problems. Initial-boundary value problems.
6. Generalized functions (distributions). Definitions and elementary theorems. Convolution. Fundamental solutions.
7. Linear differential equations with constant coefficients and Fourier transform. Examples of explicit solutions.
8. Laplace operator and harmonic functions.
9. The heat equation. Maximum principles.
10. The wave equation. Energy inequality. Cauchy problems.
11. Boundary value problems for elliptic equations. Sobolev spaces. Weak solutions. Variational approach. Non-linear elliptic problems.
12. Eigenbasis. Separation of variables for PDE in symmetrical domains.

### Textbooks

• M.A.Shubin, Partial differential equations.