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PDE question Consider the one dimensional wave equation on the half line: Ut(x,0) = g(x) Utt...
Problem 6: Consider the wave equation with a dumping term r > 0, Ut - c?Uzx + rut = 0, (t, x) € R2. This corresponds to the vibrations of an infinite string in a medium that resists its motion (e.g., air or water). Let the energy of the string be given by 1 E(t) = } } [u? (1, 2) + uș(t, 2)] dr. Show that E(t) decreases but E(t)e2rt increases, i.e., the string loses energy due to resistance...
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)...
#4.2.3 (e, f, g only) & #4.3.11 (all of it) 42.3) Write down the solutions to the follig inboundary value problems for the wave equation in the form of a Fourier series: (a) utt = uzz , u(t, 0) = u(t, π) = 0, a(0,2) 1, ut(0,x) = 0; (d) ut4u (e) ut.-uzz , u(t, 0)u1) 0, u(0,), u, (0,x; u(t, 0)=ux(t, 1)=0, a(0,2)=1, ut(0,2)=0; (g) utt = uzx , ux(t, 0)-u, (t, 1) = 0, u(0,x)-x(1-x), ut(0,2 )-0. Explain...
4. Consider the semi-infinite string problem given by Utt = cʻuza, 0<x< 0,> 0 u(x,0) = f(x), 0<x< ~ ut(2,0) = g(2), 0 < x < 0 u(0,t) = 0, t> 0 Suppose that c=1, f(0) = (x - 1) - h(2 – 3) and g(C) = 0. (a) Write out the appropriate semi-infinite d'Alembert's solution for this problem and simplify. (b) Plot the solution surface and enough time snapshots to demostrate the dynam- ics of the solution.
PDE: Ut = Uxx, -00 < x < 0, t> 0 IC: u(x,0) = 38(x) + 28(x – 6) where is the Dirac delta function (impulse). u(x, t) =
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.
2. Consider the following 1-D wave equation with initial condition u (x, 0)- F (x) where F(x) is a given function. a) Show that u (x, t)-F (x - t) is a solution to the given PDE. b) If the function F is given as 1; x< 10 x > 10 u(x, 0) = F(x) = use part (a) to write the solution u(x, t) c) Sketch u(x,0) and u(x,1) on the same u-versus-x graph d) Explain in your own...
9. Consider the beam PDE for the transverse deflection u(x, t) of an elastic beam Utt + Kurz = 0 for 0 < x <L (30) where K > 0 is a constant. Suppose the boundary conditions are given by (31) u(0, t) = uz(0,t) = 0 Uwx (L, t) = Uzzz(L, t) = 0 (32) and the initial conditions are (33) u(x,0) = (x) u1(x,0) = V(x) (34) Use separation of variables to find the general solution to the...
PROBLEM 4.3. The one-dimensional wave equation is ə?u - 20u = 0, ət2 or where c> 0 is constant. Show that any function of the form u(x, t) = f(x - ct)+9(2+ct), where f,g: RR are twice continuously differentiable, satisfies this equation. Explain why we call c the wave speed.
Problem 2 Consider the one dimensional version of the heat PDE in Problem1 2 0x2 a(0, z) = uo(z) = e-r2. (a) Write down the Fourier transformed version of (2). Then, find the solution of this transformed version u(t,)-((,) (b) Invert the solution in part (a) to get the solution, u(t, x)-F-(u)(t, x), to (2) Problem 2 Consider the one dimensional version of the heat PDE in Problem1 2 0x2 a(0, z) = uo(z) = e-r2. (a) Write down the...