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 semester “Advanced Calculus”, 2 semester “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.
  2. Qualitative and quantative analysis. Well-posed and ill-posed problems. Existence and uniqueness for the Cauchy problem. Boundary problem. Double-sweep method. Sturm-Liouville problem.
  3. Dimensional analysis of physical problems.
  4. Laplace transform for linear differential equations
  5. Partial differential equations as models of various phenomena. First integrals. Self-similar solutions.
  6. Normal form of partial differential equations. Cauchy-Kovalevskaya theorem.
  7. Linear differential equations with constant coefficients and Fourier transform. Examples of explicit solutions.
  8. Fourier transform as a unitary operator.
  9. Generalized functions (distributions). Definitions and elementary theorems. Convolution. Foundamental solutions.
  10. Evolutionary equations, well-posedness according to I. G. Petrovski. Finite-difference approximations of equations and Pade approximations. Hyperbolic and parabolic systems. Examples of ill-posed problems. Small parameter and regularizations. Quasi-linear equations and conservation laws. Characteristics and bicharacteristics. Hamilton-Jacobi equations. Blow-up. Solitons.
  11. Boundary problem for elliptic equations.
  12. Mixed problem and Shapiro-Lopatinsky theory of well-posed boundary conditions.
  13. Separation of variables for PDE in symmetrical domains.
  14. Additional directions of study.

Textbooks

  • M.A.Shubin, Partial differential equations.