1. Solve the initial-boundary value problem one = 4 for () <<3, t> 0, u(0,t) = u(3, 1) = 0 for t> 0, u(x,0) = 3x – 2” for 0 < x < 3. (30 pts.)
(1 point) Solve the nonhomogeneous heat problem Ut = uzz + 4 sin(5x), 0< I<T, u(0, t) = 0, u(T, t) = 0 u(x,0) = sin(3.c) u(x, t) = Steady State Solution lim, , u(x, t)
(1 point) Solve the nonhomogeneous heat problem Ut Uzz + 3 sin(3.c), 0<x<1, u(0,t) = 0, u(T,t) = 0 u(2,0) sin(52) u(x, t) = Steady State Solution lim oo u(a,t) =
(1 point) Solve the heat problem with non-homogeneous boundary conditions v (2,t) = (2,t), 0<=<4, t>0 u(0,t) =0, u(4,t) = 2, t>0, ulz,0) = , 0 <I<4. Recall that we find h(2), set u(2,t) = u(2,t) – h(2), solve a heat problem for v(, t) and write u(2,t) = v(2,t) +h(2) Find h() (2) = The solution (I, t) can be written as uz, t) =h(2) + (,t), where (2,t) = »=Ecseh (a) v2,t) = Finally, find limu,t) = t-o
(1 point) Solve the nonhomogeneous heat problem u; = Uxx + 4 sin(5x), 0 < x < t, u(0, t) = 0, u(1, t) = 0 u(x,0) = 2 sin(2x) u(x, t) = Steady State Solution limt700 u(x, t) =
** Lon u. that the solution to the heat conduction problem aug , 0<<L t > 0 u(0,t) - 0, u(L,t) = 0 (u(a,0) = f (3) is given by u(3,4) – È che+n*/2°' sin (182), – Ž Š 5(2) sin (%), vnen. Explicitly show by substitution that this function u(x, t) satisfies the equation aus = U, and all of the given boundary conditions. Note: You can interchange/swap sums and derivatives for this function (that doesn't always work!).
1. Consider the insulated heat equation up = cum, 0 <r<L, t > 0 u (0,t) = u (L, t) = 0, t > 0 u(x,0) = f(2). What is the steady-state solution? 2. Solve the two-dimensional wave equation (with c=1/) on the unit square (i.e., [0, 1] x [0,1) with homogeneous Dirichlet boundary conditions and initial conditions: (2, y,0) = sin(x) sin(y) (,y,0) = sin(x). 3. Solve the following PDE: Uzr + Uyy = 0, 0<<1,0 <y < 2...
+ 2y = 4u, y(0) = 0, for the following input: Solve: dt 0<t<T u(t) t>T Graph the solution (you may use Excel or Matlab) for T= 1sec, 0.1sec, 0.01sec, and 0.001sec. Do you see what is happening to the output? What is happening to the input?!
Find the solution to the heat equation on the infinite domain ∂u∂t=k∂2u∂x2,−∞<x<∞,t>0,u(x,0)={x,0,|x|<1|x|>1.∂u∂t=k∂2u∂x2,−∞<x<∞,t>0,u(x,0)={x,|x|<10,|x|>1. in terms of the error function. Q1 (10 points) Find the solution to the heat equation on the infinite domain azu ди at k -00<x<0, t>0, ar2 u(x,0) (X, 1x < 1 10, [] > 1. in terms of the error function. + Drag and drop your files or click to browse...
(1 point) Solve the nonhomogeneous heat problem u, = Uxx + 5 sin(5x), 0<x<1, u(0,t) = 0, u1,t) = 0 u(x,0) = 4 sin(4x) u(x, t) = Steady State Solution lim 700 u(x, t) =