Explanation Given: The initial temperature is f(x) and the ends x = 0 and x = L which are insulated. Procedure used: Heat equation is written as, k ∂ 2 u ∂ x 2 = ∂ u ∂ t , (1) Where, 0 < x < L , and t > 0 u ( 0 , t ) = 0 , u ( L , t ) = 0 , t > 0 (2) u ( x , 0 ) = f ( x ) , 0 < x < L (3) Where L is length of rod with the initial temperature f(x) throughout, whose ends are held at zero for all time t > 0 . Fourier cosine series, f ( x ) = a 0 2 + ∑ n = 1 ∞ a n cos n π p x And a n = 2 p ∫ 0 p f ( x ) cos n π p x d x Where, f defined on interval ( − p , p ) Calculation: First use u ( x , t ) = X ( t ) T ( t ) to separate variables in equation (1). Let − λ be the separation constant k X ″ ( t ) T ( t ) = X ( t ) T ' ( t ) X ″ ( t ) X ( t ) = T ' ( t ) k T ( t ) = − λ (4) Lead to the two ordinary differential equations, X ″ ( t ) + λ X ( t ) = 0 (5) And, T ' ( t ) + k λ T ( t ) = 0 (6) Use boundary conditions (2) in u ( x , t ) = X ( t ) T ( t ) are, u ( 0 , t ) = X ( 0 ) T ( t ) =0 And u ( L , t ) = X ( L ) T ( t ) =0 Since T ( t ) ≠ 0 for all t , so only condition which holds above when X ' ( 0 ) =0 and X ' ( L ) =0 . The corresponding solutions of the differential equations, given by X ( x ) = c 1 + c 2 x , λ = 0 (8) X ( x ) = c 1 cosh α x + c 2 sinh α x , λ = − α 2 < 0 (9) X ( x ) = c 1 cos α x + c 2 sin α x , λ = α 2 > 0 (10) Non trivial solution only gets from equation (10), So, put initial conditions X ' ( 0 ) =0 and X ' ( L ) =0 in equation (10), When put X ' ( 0 ) =0 in equation (10), get c 2 = 0

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

See similar textbooks

Concept explainers

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 differentia…

Videos

Question

Chapter 13.3, Problem 3E

To determine

To calculate: The temperature u(x,t) in a rod of length L, with the initial temperature f(x) and if the ends x=0 and x=L are insulated.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13.3, Problem 2E

Chapter 13.3, Problem 4E

Students have asked these similar questions

Detailed report without CHATGPT, accept if you can give with code and plots, previous reported . Do not waste my question.

Please do not give inappropriate solutions, previous question reported, i need correct report solution for this, NO CHATGPT

Need detailed report without CHATGPT, accept if you can give with code and plots, previous reported Plots are required.

Chapter 13 Solutions

Advanced Engineering Mathematics

Ch. 13.1 - Prob. 1E Ch. 13.1 - Prob. 2E Ch. 13.1 - Prob. 3E Ch. 13.1 - Prob. 4E Ch. 13.1 - Prob. 5E Ch. 13.1 - Prob. 6E Ch. 13.1 - Prob. 7E Ch. 13.1 - Prob. 8E Ch. 13.1 - Prob. 9E Ch. 13.1 - Prob. 10E

Knowledge Booster

$Advanced Math$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

The temperature u ( x , t ) in a rod of length L, with the initial temperature f(x) and if the ends x = 0 and x = L are insulated.

Solution Summary: The author calculates the temperature u(x,t) in a rod of length L, with the initial temperature f (x) and if the ends are insulated.

Concept explainers

Videos

To calculate: The temperature u(x,t) in a rod of length L, with the initial temperature f(x) and if the ends x=0 and x=L are insulated.

Want to see the full answer?

Chapter 13 Solutions