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.
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.
- 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.
- Partial differential equations as models of various phenomena. Well-posed and ill-posed problem. Change of variables.
- First order PDEs. Characteristics. Cauchy problem. Conservation laws. Classical and generalized solutions. Regularization. Hamilton-Jacobi equations
- Normal form of partial differential equations. Cauchy-Kovalevskaya theorem.
- Hyperbolic equations in two-dimensional space. Cauchy problems. Initial-boundary value problems.
- Generalized functions (distributions). Definitions and elementary theorems. Convolution. Fundamental solutions.
- Linear differential equations with constant coefficients and Fourier transform. Examples of explicit solutions.
- Laplace operator and harmonic functions.
- The heat equation. Maximum principles.
- The wave equation. Energy inequality. Cauchy problems.
- Boundary value problems for elliptic equations. Sobolev spaces. Weak solutions. Variational approach. Non-linear elliptic problems.
- Eigenbasis. Separation of variables for PDE in symmetrical domains.
- M.A.Shubin, Partial differential equations.