There are many ordinary and partial differential equations and systems describing various phenomena in physics, mechanics, chemistry, biology, sociology, and other fields. We study how to formulate the equations and the corresponding boundary conditions. What does “well-posedness” mean? Various kinds of PDEs 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 solution are considered as well.
Prerequisites
Foundations of Analysis, Analysis I, Analysis II: Differentiation and Integration, Complex Analysis, Advanced Linear Algebra, Analysis on Manifolds, Ordinary Differential Equations.
Curriculum:
- Ordinary differential equations as models of various physical, chemical, biological, and other phenomena. First integrals. Existence and uniqueness for the Cauchy problem. Boundary value problems. Sturm-Liouville problem.
- Partial differential equations as models of various phenomena. Well-posed and ill-posed problems. 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. Nonlinear elliptic problems.
- Eigenbasis. Separation of variables for PDEs in symmetric domains.
Textbooks
- M. A. Shubin, Partial Differential Equations.