Consider the heat problem that is described by: 0.9urr = Ut, 0 < x < 5, t > 0 u(0, t) = 0, u(5, t) = 0, t>0 u(x, 0) = 5x – a² 0 < x < 5 (a) Identify the heat diffusivity constant a?, the length of the bar L, and the initial temperature distribution f(x). (b) Plug a? and L into u(x, t) to describe the solutions to this problem. u(x, t) = Cne-a²n²x?t/L² sin ("T* ) n=1 (c) Plug L and f(x) into the following equation to describe the coefficients Cn. f(2) = Cn sin (T) n=1
Consider the heat problem that is described by: 0.9urr = Ut, 0 < x < 5, t > 0 u(0, t) = 0, u(5, t) = 0, t>0 u(x, 0) = 5x – a² 0 < x < 5 (a) Identify the heat diffusivity constant a?, the length of the bar L, and the initial temperature distribution f(x). (b) Plug a? and L into u(x, t) to describe the solutions to this problem. u(x, t) = Cne-a²n²x?t/L² sin ("T* ) n=1 (c) Plug L and f(x) into the following equation to describe the coefficients Cn. f(2) = Cn sin (T) n=1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
solve all. a, b and c.
Transcribed Image Text:Consider the heat problem that is described by:
0.9urr = Ut,
0 < x < 5, t > 0
u(0, t) = 0, u(5, t) = 0, t>0
u(x, 0) = 5x – a²
0 < x < 5
(a) Identify the heat diffusivity constant a?, the length of the bar L, and the initial temperature
distribution f(x).
(b) Plug a? and L into u(x, t) to describe the solutions to this problem.
u(x, t) =Cne-a²n²x?t/L² sin ( "T* )
n=1
(c) Plug L and f(x) into the following equation to describe the coefficients Cn.
f (x) = Cn sin
n=1
Note: You are not actually calculating anything in this problem. You are just setting up the solution
and the Fourier sine series.
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