1. Solve the initial value problem  ut + cos(kt)ux = ax2 u(x, 0) = 0 (1) Plot the solution for k = π/3 and a = 1, as well as the projected characteristic curves for this PDE, finding a way to plot...

1. Solve the initial value problem  ut + cos(kt)ux = ax2 u(x, 0) = 0 (1) Plot the solution for k = π/3 and a = 1, as well as the projected characteristic curves for this PDE, finding a way to plot them both on the same graph. 2. Show that, for functions obeying homogeneous Dirichlet conditions on the boundary of a domain D, the differential operator L L = a(x) d 2 dx2 + b(x) d dx + c(x) (2) is self-adjoint if a 0 (x) = b(x). 3. Find all solutions to the Sturm-Liouville problem on the domain D = {|x| ≤ d}: −  d 2 dx2 − β 2  u = λu on D du dx = 0 on ∂D for β < π/d.="" plot="" the="" 5="" solutions="" with="" the="" smallest="" values="" of="" λ.="" prove="" that="" all="" solutions="" are="" orthogonal.1.="" solve="" the="" initial="" value="" problem="" ="" ut="" +="" cos(kt)ux="ax2" u(x,="" 0)="0" (1)="" plot="" the="" solution="" for="" k="π/3" and="" a="1," as="" well="" as="" the="" projected="" characteristic="" curves="" for="" this="" pde,="" finding="" a="" way="" to="" plot="" them="" both="" on="" the="" same="" graph.="" 2.="" show="" that,="" for="" functions="" obeying="" homogeneous="" dirichlet="" conditions="" on="" the="" boundary="" of="" a="" domain="" d,="" the="" differential="" operator="" l="" l="a(x)" d="" 2="" dx2="" +="" b(x)="" d="" dx="" +="" c(x)="" (2)="" is="" self-adjoint="" if="" a="" 0="" (x)="b(x)." 3.="" find="" all="" solutions="" to="" the="" sturm-liouville="" problem="" on="" the="" domain="" d="{|x|" ≤="" d}:="" −="" ="" d="" 2="" dx2="" −="" β="" 2="" ="" u="λu" on="" d="" du="" dx="0" on="" ∂d="" for="" β="">< π/d.="" plot="" the="" 5="" solutions="" with="" the="" smallest="" values="" of="" λ.="" prove="" that="" all="" solutions="" are="">


37336 Mathematical Methods: Assignment 1 Due: Friday August 31st, 2018. IMPORTANT: Please complete and sign the cover sheet for the assignment. This assignment must be completed by yourself. 1. Solve the initial value problem{ ut + cos(kt)ux = ax 2 u(x, 0) = 0 (1) Plot the solution for k = π/3 and a = 1, as well as the projected charac- teristic curves for this PDE, finding a way to plot them both on the same graph. 2. Show that, for functions obeying homogeneous Dirichlet conditions on the boundary of a domain D, the differential operator L L = a(x) d 2 dx2 + b(x) d dx + c(x) (2) is self-adjoint if a′(x) = b(x). 3. Find all solutions to the Sturm-Liouville problem on the domain D = {|x| ≤ d}: − [ d2 dx2 − β2 ] u = λu on D du dx = 0 on ∂D for β < π/d. plot the 5 solutions with the smallest values of λ. prove that all solutions are orthogonal. π/d.="" plot="" the="" 5="" solutions="" with="" the="" smallest="" values="" of="" λ.="" prove="" that="" all="" solutions="" are="">