5. Consider the heat problem that is described by:0.9urr = Ut,0 < x < 5, t> 0и (0, t) — 0, и(5, t) — 0, t>0и(х, 0) — 5х — 1?0 < x < 5(a) Identify the heat diffusivity constant a2, the length of the bar L, and the initial temperaturedistribution f (x).(b) Plug a² and L into u(x, t) to describe the solutions to this problem.-Σ-a²n²n?t/L²Cneu(x, t) =sinn=1(c) Plug L and f(x) into the following equation to describe the coefficients Cn-f(x)Cn sinn=1Note: You are not actually calculating anything in this problem. You are just setting up the solutionand the Fourier sine series.

Consider the general heat equation in one dimension, ut=α2uxx where α is the heat diffusivity…

5. Consider the heat problem that is described by: 0.9urr = Ut; 0 < x < 5, t > 0 и (0, t) — 0, и(5, t) — 0, t>0 u(x, 0) = 5x – x² 0 < x < 5 (a) Identify the heat diffusivity constant a2, the length of the bar L, and the initial temperature distribution f(x). (b) Plug a2 and L into u(x, t) to describe the solutions to this problem. u(x, t) = £ -a²n²n² Cne sin n=1

5. Consider the heat problem that is described by: 0.9urr = Ut; 0 < x < 5, t > 0 и (0, t) — 0, и(5, t) — 0, t>0 u(x, 0) = 5x – x² 0 < x < 5 (a) Identify the heat diffusivity constant a2, the length of the bar L, and the initial temperature distribution f(x). (b) Plug a2 and L into u(x, t) to describe the solutions to this problem. u(x, t) = £ -a²n²n² Cne sin 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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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

5. Consider the heat problem that is described by:
0.9urr = Ut,
0 < x < 5, t> 0
и (0, t) — 0, и(5, t) — 0, t>0
и(х, 0) — 5х — 1?
0 < x < 5
(a) Identify the heat diffusivity constant a2, the length of the bar L, and the initial temperature
distribution f (x).
(b) Plug a² and L into u(x, t) to describe the solutions to this problem.
-Σ
-a²n²n?t/L²
Cne
u(x, t) =
sin
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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