MAT 335 Final, Spring 2002

NDU MAT 335 (PDE) Final Exam Spring 2002 Duration: 2 hours Ny ‘There ave 7 problems total. Do them all. _ 110%) Use the method of separation of variables to solve the nonlinear IVP PDE: u, -3uu, = 0 -am<x<w,0<1<o IC: u(x,0)= x4) -o<x<o Elec To Co " 2(15%) Solve the nonhomogeneous heat problem Onn, Cer, 97074, PDE. uw, = aru, + sin 3av C<x<lO0<i<m BCs: u(0,9) =u(l,O =0 O<t<a IC: u(x,0) = sin ax O0<x<l (a positive constant) 3(15%) The general solution of the 2" order PDE u, =c7u,, (c¢ positve constant) is given to be u(x,1) = 6(x-cl)+y(x+cr),where ¢ and: y are twice differentiable functions of their arguments. a) Find the solution of the 1-d wave problem PDE: u, =c7u,, 0<x<0,0</<a BC: u(0,.)=h( O0<1<a«, (A(?) agiven function) u(x,0) =0 0<xr<0 ICs: u,(x,0) = 0 0<x<0,0<t<a@ b) Deduce the solution of the 1-d wave problem PDE: u, =c7u,, 0<x<0,0<1<a BC: u,(0,0 =hO . 0<s<oo, (A(t) agiven function) fF ulx,0) =0 0<x <0 ICs: . u,(x,0) = 0 Q<x<0,0<1<w Hint Let v =u, and show that v satiafies the same problem as 10 (a) Uz Xs) Use the method of characteristics to solve the IVP PDE: te ~ xu, =u -~a<x<wo, 0O<1<oa IC u(v.0)= f(x) -a<x<m, (f(x) agiven function) What do the characteristics look like? Sketch, in particular, the characteristic (y) that passes through the point A(2,0) S(IS%) The Heaviside unit step function Af(¢—a) is defined to be 1 f2a W(t-a)= { (a is constant) 0 {<a a) Find the Laplace Transform of H(r-a) b) Prove thatif F(s) = L(/())(s), then Lf -a)H(t-a)f(s) =e" F'(s) c) Use the Laplace Transform to solve the IBVP PDE: xu,tu,=x O<x<w, 0<1<w BC: u(0,1) =0 O<l<w IC: u(x,0) =0 O<x<w d) Sketch the solution for ¢=1 6(1S%) The solution of Burger’s nonlinear IVP Y PDE: u,+g(uju,=0 -ao<x<w, 0<t<o Ic: u(x,0) = d(x) -oO<x<w is given implicitly by u = ¢(x - g(ui/) Solve the following system u,—v,+2u, =0 ~a<x<0,0<1<a PDEs: 2u,—3v,+u,+2v,=0 -w<x<m,0<1<m =F -O<xX<H ICs: {i =F) s (FG given functions) v(x,0) =G(x) -a<x<0 715%) Use the Fourier Transform to show that the solution of the following Dirichlet problem for a half pluie PDE: Au=u, tu, =0 -w<x<mo, O<y<w BC: u(x,0) = f(x) -a<x<o (/ isa given function) u is bounded —-OM<X<O, O<j<w is given by the Poisson integral formula for the half plane i? / us) = | LO a ni(x-ép +" Good Luck