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3 (a) Use the method of separation of variables to derive a series solution of the fol- lowing heat equation : ut (x, t) — kuxx (x, t) = 0, u(0, t) = u(L, t) = 0, u(x,0) = (x), 0

3 (a) Use the method of separation of variables to derive a series solution of the fol- lowing heat equation : ut (x, t) — kuxx (x, t) = 0, u(0, t) = u(L, t) = 0, u(x,0) = (x), 0

Advanced Engineering Mathematics
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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Ex.3.

3 (a) Use the method of separation of variables to derive a series solution of the fol- lowing heat equation : ut(x, t) - kuxx(x, t) = 0, u(0, t) = u(L, t) = 0, 0 < x < L, 0 < t, 0≤t, u(x,0) = (x), 0 ≤ x ≤ L, where k and I are given positive constants and is a given twice-differentiable function. (b) Write down a formula for the solution of the heat equation in part (a) when the function is given by the following formula: *(x) = 5 sin (3x) L (c) State the Maximum / Minimum Principle for the heat equation. (d) Show that the solution you found in part (a) satisfies the Maximum / Minimum Principle for the heat equation. (e) Use the Maximum / Minimum Principle for the heat equation to show that there is at most one solution of the heat equation in part (a).
Transcribed Image Text:3 (a) Use the method of separation of variables to derive a series solution of the fol- lowing heat equation : ut(x, t) - kuxx(x, t) = 0, u(0, t) = u(L, t) = 0, 0 < x < L, 0 < t, 0≤t, u(x,0) = (x), 0 ≤ x ≤ L, where k and I are given positive constants and is a given twice-differentiable function. (b) Write down a formula for the solution of the heat equation in part (a) when the function is given by the following formula: *(x) = 5 sin (3x) L (c) State the Maximum / Minimum Principle for the heat equation. (d) Show that the solution you found in part (a) satisfies the Maximum / Minimum Principle for the heat equation. (e) Use the Maximum / Minimum Principle for the heat equation to show that there is at most one solution of the heat equation in part (a).
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