The exams are now graded and are in the process of double checked carefully. Grades will be posted by the registrar once available. A solution draft of the final is online. About PDE's: always write the initial part as a Fourier series sumn bn sin(n x): Sum bn exp(-n2 t) sin(n x) solves heat ut = A u = uxx for position Sum bn cos(n t) sin(n x) solves wave utt = A u = uxx for position Sum bn sin(n t)/n sin(n x) solves wave utt = A u = uxx for velocity Proof: sin(n x) is the eigenfunction of the eigenvalue L = -n^2 of A=D^2. With sin(n x) as initial heat we get u_t = L u solved by e^(-n^2 t). With sin(n x) as initial position, get u_tt +n^2 u = 0 solved by cos(nt). with sin(n x) as initial velocity, get u_tt +n^2 u = 0 solved by sin(nt)/n This can even fit into a song. (More generally, just replace -n^2 t with L t and n t with sqrt(-L) t, where L is the eigenvalue of A to the eigenfunction sin(nx). Some slides from the review with comments.

Please send questions and comments to knill@math.harvard.edu

Math21b Harvard College Class Number:16325 Course ID:110989| Oliver Knill | Spring 2016 | Department of Mathematics | Faculty of Art and Sciences | Harvard University, [Canvas, for admin], Twitter