

Contents


Week 1: 
Derivation of the equations of mathematical physics. 
Week 2: 
Free oscillations of a string and the wave equation. Oscillations of a string in an elastic medium and KleinGordon equation. Kirchhoff lows for the currents in wires; the wave equation and KleinGordon equation. 
Week 3: 
Equations of acoustics. Thermal conductivity equation and diffusion equation. Schroedinger equation. 
Week 4: 
Possible physical reasons of the nonlinearities in the equations of mathematical physics. Steady states and Helmholtz equations. 
Week 5: 
Oscillations of the membranes. Initialvalue problems, boundaryvalue problems and mixed problems. Physical analogies. Wellposed problems in Hadamars terminology. The uniqueness theorem. 
Week 6: 
Equations of the hyperbolic type. Method of the propagating waves. D’Alembert’s formula for the wave equation and its generalization for KleinGordon equation. Midterm Exam I. 
Week 7: 
Integral equation for oscillations. Examples of solutions. Propagation of the discontinuities along the characteristics. Problem with data on the characteristics. 
Week 8: 
Method of successive approximations. Separation of variables method. 
Week 9: 
Adjoint differential operators. Integral form of solution. Physical interpretation of the Riemann function.

Week 10: 
Equations of the parabolic type. Formulation of the boundary value problems. Principle of maximal value. Theorem of uniqueness. 
Week 11: 
Method of separation of variables. Homogeneous boundary value problem. Function of a source. Boundary value problems with discontinuous initial conditions. Inhomogeneous heat equation and boundary value problems. Problems on the infinite line. 
Week 12: 
Problems without initial conditions. Midterm Exam II. 
Week 13: 
Equations of the elliptic type. Problems reducible to the Laplace equation. General properties of the harmonic functions. Solutions by separation of variables for the simplest domains. Source functions. Potential theory. Pivotal problems reducible to the Helmholtz equation. Problems for infinite domain. Limit absorption principle. Limited amplitude principle. Principle of radiation. 
Week 14: 
The problems of mathematical scattering theory. Formulation of the problem. The uniqueness of the solution of the scattering problem. Special functions. 
Week 15*: 


Week 16*: 
Final Exam. 
Textbooks and materials: 
Partial Differential Equations with Boundary Value Problems by Larry C. Andrews.

Recommended readings: 
Partial Differential Equations. Lawrence C. Evans Applied Partial Differential Equations Paul DuChateau, David Zachmann Applied Partial Differential Equations Richard Haberman Applied Partial Differential Equations John R. Ockendon, Sam Howison, John Ockendon, Andrew Lacey, Alexander Movchan


* Between 15th and 16th weeks is there a free week for students to prepare for final exam.

