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.


  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.


  • M.A.Shubin, Partial differential equations.