u(x,0)= Consider the following wave equation U, = U23 -00<x<00,t> 0 (0, -0<x<-1, _x+1, -1<< <0,...
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) =
Problem #9: Consider the below wave equation with the given conditions. clu olu 16 0<x< 5, t > 0, Ox2 u(0, 1) = u(5, 1) = 0, t > 0 700 u(x, 0) = 0, -0 = 7x(5– x) = Σ {1-(-1)"} sin(ntx/5), 0< x < 5. n=1 The solution to the above boundary-value problem is of the form U (x, t) = g(n, t) sin 97 * n=1 Find the function g(n, t). Problem #9: Enter your answer as...
Repeat the flat-plate momentum analysis by replacing the equation u(x, y) ~U ( ) 0<y>$(x) using a trigonometric profile approximation: 5 = sin()
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...
Solve the wave equation on the domain 0 < x < , t > 0 ? uxx Utt = with the boundary condition u (0, t) = 0 and the initial conditions u (x,0) = x2 u (x,0) = x
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...
b) Consider the wave equation azu azu at2 0 < x < 2, t>0, ar2 with boundary conditions u(0,t) = 0, u(2, t) = 0, t> 0, and initial conditions u(x,0) = x(2 – x), ut(x,0) = = 0, 0 < x < 2. Use the method of separation of variables to determine the general solution of this equation. (15 marks)
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
zone 1 Consider the following piecewise continuous, finite potential energy: ro; x < -a V(x)={-U, ; -a sxs a zone II U, > 0 (+ve) 10 ; x> a We consider zone III E>0: Unbound or scattering states (a) State the Time independent Schrödinger's Equation (TISE) and the expression of wave number k in each zone for the case of unbound state (b) Determine the expression of wave function u in each zone. (e) Determine the expression of probability Density...
2. Solve the linear homogeneous IVP U+ rtuz = 0, u.1,0) = sinr, -o0<< 0, t> 0.