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Consider a thin bar of length 14 with heat distribution T(z, f), where ar Ət (a) Suppose T satisfies homogeneous BCS and the IC Find T(,) by using a separation solution similar to the one in the course notes What are wys and A(n-1,2,...)? wys Apply the initial condition to determine T(x, f). Do not use the notations to and An in your solution. T(z,t) 2T 16- for 0 0. 02 T(0,1) 7(14,t)-0 for t> 0, T₂ (2,0) --sin() for 0

Consider a thin bar of length 14 with heat distribution T(z, f), where ar Ət (a) Suppose T satisfies homogeneous BCS and the IC Find T(,) by using a separation solution similar to the one in the course notes What are wys and A(n-1,2,...)? wys Apply the initial condition to determine T(x, f). Do not use the notations to and An in your solution. T(z,t) 2T 16- for 0 0. 02 T(0,1) 7(14,t)-0 for t> 0, T₂ (2,0) --sin() for 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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Consider a thin bar of length 14 with heat distribution 7(x, t), where ar 827 16 for 0 <z <14 and t> 0. 822 at (a) Suppose T satisfies homogeneous BCs T(0,t)-T(14,t)-0 for t> 0, and the IC T (2,0)=sin() for 0 <z<14. Find T(z, t) by using a separation solution similar to the one in the course notes. 1. What are wy and An (n=1,2,...)? An- Apply the initial condition to determine T(x, t). Do not use the notations ton and An in your solution. T(z,t) BD (b) Find the smallest time t when max(7) ≤ 0.03. Give your answer to two decimal places. t Number (c) Now suppose both ends of the bar are insulated so that T satisfies the BCs Tz (0,t) Tz (14,t)-0 for t> 0. Let the eigenvalues of the associated Sturm-Liouville problem be ko.k₁ kg.... where kol <|k₂| < |k₂|<... i. ko 1. ka i. Find T(x, t) as too if T(2,0) 13- -15 ()² +10 () ³. lim T(z,t) Number 1-x
Transcribed Image Text:Consider a thin bar of length 14 with heat distribution 7(x, t), where ar 827 16 for 0 <z <14 and t> 0. 822 at (a) Suppose T satisfies homogeneous BCs T(0,t)-T(14,t)-0 for t> 0, and the IC T (2,0)=sin() for 0 <z<14. Find T(z, t) by using a separation solution similar to the one in the course notes. 1. What are wy and An (n=1,2,...)? An- Apply the initial condition to determine T(x, t). Do not use the notations ton and An in your solution. T(z,t) BD (b) Find the smallest time t when max(7) ≤ 0.03. Give your answer to two decimal places. t Number (c) Now suppose both ends of the bar are insulated so that T satisfies the BCs Tz (0,t) Tz (14,t)-0 for t> 0. Let the eigenvalues of the associated Sturm-Liouville problem be ko.k₁ kg.... where kol <|k₂| < |k₂|<... i. ko 1. ka i. Find T(x, t) as too if T(2,0) 13- -15 ()² +10 () ³. lim T(z,t) Number 1-x
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