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(b) Calculate the 1.4.7 . For the following problems, determine an equilibrium temperature distribution (if one...
Determine an equilibrium temperature distribution (if one exists) for ди Әt д? и дх2 +x - В for 0 < x < L subject to the boundary conditions ди - (0,t) = 0, дх ди (L, t) = 0, дх and initial condition и(x, 0) = 1. For what values of B are there solutions?
сиргекыопог ше оar ueuaенстgу соmamea uue тоu. 2. Given the following initial boundary value problem Section 1.2 u u(r, 0 f(x) (L, t) B (0, t) 1 Әт дr a. determine an equilibrium temperature distribution, if one exists, and b. find the values of 3 for which there are such equilibrium solutions сиргекыопог ше оar ueuaенстgу соmamea uue тоu. 2. Given the following initial boundary value problem Section 1.2 u u(r, 0 f(x) (L, t) B (0, t) 1 Әт...
сиргекыопог ше оar ueuaенстgу соmamea uue тоu. 2. Given the following initial boundary value problem Section 1.2 u u(r, 0 f(x) (L, t) B (0, t) 1 Әт дr a. determine an equilibrium temperature distribution, if one exists, and b. find the values of 3 for which there are such equilibrium solutions
(a) Consider the one-dimensional heat equation for the temperature u(x, t), Ou,02u where c is the diffusivity (i) Show that a solution of the form u(x,t)-F )G(t) satisfies the heat equation, provided that 护F and where p is a real constant (ii) Show that u(x,t) has a solution of the form (,t)A cos(pr)+ Bsin(p)le -P2e2 where A and B are constants (b) Consider heat flow in a metal rod of length L = π. The ends of the rod, at...
Consider a uniform bar of length L having an initial temperature distribution given by f(x), 0 < x < L. Assume that the temperature at the end x=0 is held at 0°C, while the end x=L is thermally insulated. Heat is lost from the lateral surface of the bar into a surrounding medium. The temperature u(x, t) satisfies the following partial differential equation and boundary conditions aluxx – Bu = Ut, 0<x<l, t> 0 u(0,t) = 0, uz (L, t)...
Determine the transient temperature distribution in a one-dimensional (1-D) fluid with a thermal diffusivity a=2 and velocity u = 0.1, if the initial temperature in the fluid is 0.0° and if at all subsequent times, the temperature of the left side is held at 0° while the right side is held at To=2º. The distance between the two walls is L=1 meter. T=0 TT 0 The governing differential equation is the 1-D heat convection diffusion equation at at 22T +...
solve for An as well! Find the temperature function u(x,t) (where is the position along the rod in cm and t is the time) of a 6 cm rod with conducting constant 0.2 whose endpoint are insulated such that no heat is lost, and whose initial temperature distribution is given by: 4 if 1 x < 4 u (х, 0) — 0 otherwise To start, we have L =6 0.2 Because the rods are insulated, we will use the cosine...
Find the temperature distribution at equilibrium in a rectangular plate (0 ≤ x ≤ L, 0 ≤ y ≤ H) when the side at x = 0 is subject to the prescribed temperature f(y) = 1 + y, and the sides at x = L, y = 0 and y = H are insulated, by using the method of separation of variables.
Determinc the equilibrium temperature distribution for a one-dimensional rod with constant thermal properties with the following sources and bound- ary conditions: 1.4.1. * (a) Q=0, f) Ko 0
Let u be the solution to the initial boundary value problem for the Heat Equation u(t, x) 4ut, x) te (0, o0), т€ (0, 3)%; with initial condition 2. f(x) u(0, x) 3 0. and with boundary conditions ди(t, 0) — 0, и(t, 3) — 0. Find the solution u using the expansion u(t, a) "(2)"п (?)"а " п-1 with the normalization conditions Vn (0) 1, wn(0) = 1 a. (3/10) Find the functions wn. with index n > 1....