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du k Ət u with ди (0, t) = 0, du (L, t) = 0, and u(x,0) = f(x). Əx²

du k Ət u with ди (0, t) = 0, du (L, t) = 0, and u(x,0) = f(x). Əx²

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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Consider the following boundary value problem (if necessary, see Section 2.4.1): ди = k- Ət ди (0, t) = 0, (L, t) = 0, and u(x, 0) = f(x). with (a) Give a one-sentence physical interpretation of this problem. (b) Solve by the method of separation of variables. First show that there are no separated solutions which exponentially grow in time. [Hint: The answer is NTX u(x, t) = Ao + Ane-^nkt COS L n=1 What is An? (c) Show that the initial condition, u(x, 0) = f(x), is satisfied if 5 An NTX f (x) = Ao + ) An cos L n=1 (d) Using Exercise 2.3.6, solve for Ao and An (n > 1). (e) What happens to the temperature distribution as t → 0? Show that it ap- proaches the steady-state temperature distribution (see Section 1.4).
Transcribed Image Text:Consider the following boundary value problem (if necessary, see Section 2.4.1): ди = k- Ət ди (0, t) = 0, (L, t) = 0, and u(x, 0) = f(x). with (a) Give a one-sentence physical interpretation of this problem. (b) Solve by the method of separation of variables. First show that there are no separated solutions which exponentially grow in time. [Hint: The answer is NTX u(x, t) = Ao + Ane-^nkt COS L n=1 What is An? (c) Show that the initial condition, u(x, 0) = f(x), is satisfied if 5 An NTX f (x) = Ao + ) An cos L n=1 (d) Using Exercise 2.3.6, solve for Ao and An (n > 1). (e) What happens to the temperature distribution as t → 0? Show that it ap- proaches the steady-state temperature distribution (see Section 1.4).
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