9. Consider the beam PDE for the transverse deflection u(x, t) of an elastic beam Utt...
solve the PDE +u= at2 on 3 € (0,L), t > 0, with boundary conditions au 2x2 u(0,t) = 0, u(L, t) = 0 au and initial condition u(x,0) = f(x), at (x,0) = g(x) following the steps below. (a) Separate the variables and write differential equations for the functions (x) and h(t); pick the separation constant so that we recover a problem already studied. (b) Find the eigenfunctions and eigenvalues. (c) Write the general solution for this problem. (d)...
Consider the following second order PDE Uit – 9Uxx = 0, 0<x< < t > 0, (A) and the following boundary value/initial conditions: Ux(t,0) = uſt, 5) = 0, t>0, u(0, x) = 44(0, x) = 4 cos’ x, 0<x< (BC) (IC) for the function u= u(t, x). a. (5 points) Find ordinary differential equations for functions T = T(t) and X = X(x) such that the function u(t, x) = T(t)X(x) satisfies the PDE (A). b. (5 points) Find...
1. Consider the following inhomogeneous wave equation on (0,7) : utt - 4uxx = (1 - x) cost Uz(0,t) = cost-1, uz(7,t) = cost u(3,0) = 2(7,0) = cos 3x (a) Convert the PDE to an equation with homogeneous boundary conditions by making an appropriate substitution u(x, t) = u(x, t) - p(x, t), implying u(,t) = v(x, t) + p(2,t) for an appropriate function p(x, t). (b) Finish solving the PDE using the Method of Eigenfunction expansion.
Problem 2. Solve the following wave equation. Utt = Ucx + x for t > 0 and 0 < x < 1 Boundary Conditions: u(0,t) = 0 AND u(1,t) = 1 Inital Condition: u(x,0) = $(x) AND u1(x,0) = 0
Problem 1 (20 points) Consider the PDE for the function u(x, t) e 0<x<T, t> 0 with the boundary conditions n(0, t) 0, u(T, t) 0, t> 0 and the initial condition 0 u(x, 0) 1+cos(2a), (a) Give a one-sentence physical interpretation of this problem. (b) Find the solution u(x, t) using a Fourier cosine series representation An (t) cos(nax) u(x,t)= Ao(t) + n=1
Please detail Please detail PDE Utt = Uzx + 2a sin(at) sin(1x) 0 < x <1 0<t< oo BCS S u(0,t) = 0 | u(1,t) = 0 0<t< oo ICs u(x,0) = 0 | u4(,0) = sin(nx) 0 < x <1 u (0,t) = f (t) u (L,t) = g(t) S Use sine transform Uz (0,t) = f(t) uz (L,t) = g(t)) Use cosine transform 2 L S [u (x,t)] = Sn (t) = 1 | u(x, t) sin (ntx/L)...
=T 20 marks) Consider the following PDE with boundary and initial conditions: U = Upx + ur, for 0<x< 1 and to with u(0,t) = 1, u(1,t) = 0, u(1,0) = (a) Find the steady state solution, us(1), for the PDE. (b) Let Uſz,t) = u(?, t) – us(T). Derive a PDE plus boundary and initial conditions for U(2,t). Show your working. (c) Use separation of variables to solve the resulting problem for U. You may leave the inner products...
1. Solve the vibrating string problem PDE BC BC IC IC utt T.T Uz(0,t) = 0 u(1,t0 u(x,0cos(3T) 14(2.0) = x
9. Use a suitable Fourier Transform to find the solution of the IVP utt (x, t) Uz(0, t) u(x, t) , uz (z, t) 4uzz (x, t) + q (x, t), 0, t> 0, 0as x → 00, x > 0, t > 0, = = t>0. → = 0, ut (2,0)-( = { t, 0 0-x-2, -1, 0, > 2, u(x, 0) q(a, t) Leave your answer in the form of an integral. 9. Use a suitable Fourier Transform...