4. Consider the homogeneous heat-conduction problem wr =0, u(z,0)=f(x) (15) describing the temporal evolution of the...
Problem 1 (15 pts) Consider heat conduction on a slender homogeneous metal wire with constant crosssection as shown in Fig.1. L- 10cm Conductivity k = 100 w/m°C. Q(x)= 100.000W/㎡. At x = 0, q = 250 W/ m2. TL = 25 oC. Governing equation: _kdTeQ (0%L) Boundary condition: dT -k Figure 1 Heat conduction on a 1-D metal wire. a. Solve for T (x) with two linear elements (X1 = 0, x2-4cm, and X3 = 10cm) ; b. Compare with...
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)...
(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...
2. Consider a thin rod of length L = π (so that 0 x-7) with a general internal source of heat, Q(a,t) Ot (10) subject to insulated boundary conditions The initial temperature of the bar is zero a(x, 0) = 0 (12) (a) (3pts) What is k in (10)? (b) (10pts) Assume a separable solution to the homogeneous version of the PDE and boundary conditions (10)-(11) of the form u(r, t)- o(x)G(t). Write down or find the eigenvalues λη and...
1. Consider the heat flow problem on the real line, where u(x,t), t > 0 is the temperature at point x at time t: ди 1 a2u t>O (*) at 2 ar2 u(x,0) = sin(7x) = > (a) What is the thermal diffusitivity constant ß? (b) Find the intervals of x where the temparature will increase at t = 0. (c) Sketch the graph of the temperature at t = 0. (d) On the same axes as in (c), sketch...
3. Consider the following Neumann problem for the heat equation: 14(0,t)=14(L,t)=0, t>0 u(x,0)- f(x),0<x<L (a) Give a short physical interpretation of this problem. (b) Given the following initial condition, 2 *2 2 solve the initial boundary value problem for u(x,t. 3. Consider the following Neumann problem for the heat equation: 14(0,t)=14(L,t)=0, t>0 u(x,0)- f(x),0
please solve 17 for me thanks~~ :) ! temperature f(x) °C, where 5. f(x) = sin 0.1 x 6 f(x) = 4 - 08 |x - 5 7. fix) =x(10 - x) 8 Arbitrarytemperatures at ends. If the ends x = 0 and x= Lof the bar in the text are kept at constant 20. CAS PROJECT. Isotherms. Fim solutions (tempe rature s) in the squa with a 2 satisfying the followin tions. Graph isotherms. (a) u80 sin Tx on...
I need help with question 30d 16. y = 0 (that is, y(x) = 0 for all x, also written y(x) = 0) is a solution of (2) (not of (1) if (x) • o , called the trivial solution 17. The sum of a solution of (1) and a solution of (2) is a solution of (1). 18. The difference of two solutions of (1) is a solution of (2). 19. If yı is a solution of (1), what...