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2. Find the steady-state temperature u(r,0) in a semicircular plate of radius r 2 if [10<0<...
Find the steady-state temperature u(r.0) in a circular plate of radius r = 1 if the temperature on the circumference is as given (show all work!): 0 0 T 1) u(1,0) = 0, T<02T
Find the steady-state temperature u(r.0) in a circular plate of radius r = 1 if the temperature on the circumference is as given (show all work!): 0 0 T 1) u(1,0) = 0, T
Estimate the steady state temperature on the plate a quarter
circle radius
r = 1 and f (θ) = θ2 - π shown in the
figure.
(a) Establish the equations to
be solved and their boundary conditions.
(b) Determine the coefficients of the series obtained.
(c) Give the solution to the problem.
VA u=f@) H=0 ON
3. A circular plate of unit radius, whose faces are insulated, has half of its boundary kept at constant temperature u, and the other half at temperature uz (see figure). Find the steady state temperature pf the plate. The steady state heat flow is written as oʻu 1du 1 8²u ar?'r or a dz = 0, with the boundary condition 0<¢ <T, luz, < < 21. u(1,0) = {u, After using the proper separation of variable method, we have u=dy...
1. Find vfor t> 0 if the circuit is in steady state at t 0 Ans: -4e2" (cos 2t +sin 2t) +4 Volts 2Ω 1 H -0 2Ω 1
3. (7 points) Let u(x, y) be the steady-state temperature u(r, y) in a rectangular plate whose vertical r0 and 2 are insulated. When no heat escapes, we have to solve the following the boundary value problem: a(z,0) = 0, u(z,2) = x, 0 < x < 2 (a) By setting u(x, g) -X(x)Y(u), separate the equation into two ODE 0 What ane the sewr homdany condiome hoald Xe) watiy (37)2. (c) Find x(r) for the case when λ-0 and...
3. (20 points) Denote u(ar, y) the steady-state temperature in a rectangle area 0 z 10, 0yS 1. Find the temperature in the rectangle if the temperature on the up side is kept at 0°, the lower side at 10° while the temperature on the left side is S0)= sin(y) and the right side is insulated. Answer the following questions. (a) (10 points) Write the Dirichlet problem including the Laplace's equation in two dimensions and the boundary conditions. (b) (10...
9. Find the steady-state temperature distribution in the plate shown below y= x и —D () insulated и 3D Т 1/2 1 и 3D 0
9. Find the steady-state temperature distribution in the plate shown below y= x и —D () insulated и 3D Т 1/2 1 и 3D 0
30] Find th e solution of the following boundary value problem. 1<r<2, u(r, θ = 0) = 0, u(r, θ = π) =0, 1,0-0, u(r-2,0)-sin(20), 0 < θ < π. u(r Please also draw the sketch associated with this problem. You may assume that An -n2, Hn(s)sin(ns), n 1,2,3,. are the eigenpairs for the eigenvalue problem H(0) 0, H(T)0.
30] Find th e solution of the following boundary value problem. 1
Find the steady-state temperature u(r,z) in a finite cylinder defined by 0< r < 1,0 < z < 1 in a finite cylinder defined by 0 <rs 1,0 < z <1 if the boundary conditions are as given: 0 z< 1 2) u(1, z) = Z, 0, az z 0 0r1
Find the steady-state temperature u(r,z) in a finite cylinder defined by 0
Find the steady-state temperature u(r,z) in a finite cylinder defined by 0< r < 1,0 < z < 1 in a finite cylinder defined by 0 <rs 1,0 < z <1 if the boundary conditions are as given: 0 z< 1 2) u(1, z) = Z, 0, az z 0 0r1
Find the steady-state temperature u(r,z) in a finite cylinder defined by 0