PHYSICS 3K03 Study Guide - Midterm Guide: Rayleigh Quotient, Heat Equation, Fourier Series

• School: McMaster University
• Department: Physics
• Course: PHYSICS 3K03
• Professor: Rajat Bhaduri

Document Summary

Let , where is the steady state solution, then satisfies homogeneous boundary conditions, with initial condition when. Use integrating factor to get or simply separate variables. Variable coefficient are eigenfunctions obtained in the non-forced case. Heat is transferred to reservoir of temperature at. Multiply both sides by or and integrate over one period. Same as the linear case, except may be functions of. Shocks occur when characteristics intersect path is given by. The solution satisfying all four boundary conditions is and has the form the sum of the above solutions. Decompose into odes initially; varies along data curve. Data curve is tangent to characteristic curve to , and expect to get. Changing to canonical form sometimes helps with finding an analytic solution. Eigenvalues are real set of eigenfunctions with exactly zeros on. Given a sturm-liouville operator , on with boundary conditions has the solution satisfies.