

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: 
MIDTERM EXAM II and solutions 
Week 7: 
Equations of the hyperbolic type. Method of the propagating waves. D’Alembert’s formula for the wave equation and its generalization for KleinGordon equation. 
Week 8: 
Integral equation for oscillations. Examples of solutions. Propagation of the discontinuities along the characteristics. Problem with data on the characteristics. 
Week 9: 
Method of successive approximations. Separation of variables method. 
Week 10: 
Adjoint differential operators. Integral form of solution. Physical interpretation of the Riemann function. 
Week 11: 
Equations of the parabolic type. Formulation of the boundary value problems. Principle of maximal value. Theorem of uniqueness. 
Week 12: 
MIDTERM EXAM II and solutions 
Week 13: 
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. Problems without initial conditions.

Week 14: 
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. Principle of radiation. Principle of limiting absorption. Principle of limiting amplitude. 
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.

