Explanation Given: The initial temperature is f ( x ) throughout and if the end x = 0 is kept at temperature zero and the end x = L is insulated. Calculation: Consider a function u as a function of ( x , t ) as follows: u ( x , t ) = X ( x ) T ( t ) The derivative of above expression with respect to x is given below. u x = X ′ ( x ) T ( t ) u x x = X ″ ( x ) T ( t ) The derivative of above expression with respect to t is given below. u t = X ( x ) T ′ ( t ) u t t = X ( x ) T ″ ( t ) Substitute X ( x ) T ′ ( t ) for u t and X ″ ( x ) T ( t ) for u x x in equation k ∂ 2 u ∂ x 2 = ∂ u ∂ t . k ( X ″ ( x ) T ( t ) ) = X ( x ) T ′ ( t ) k ( X ″ ( x ) X ( x ) ) = T ′ ( t ) T ( t ) X ″ ( x ) X ( x ) = 1 k ( T ′ ( t ) T ( t ) ) Consider X ″ ( x ) X ( x ) as − λ . where − λ is the separating constant. X ″ ( x ) X ( x ) = − λ X ″ ( x ) = − λ X ( x ) X ″ ( x ) + λ X ( x ) = 0 ( 1 ) Consider 1 k ( T ′ ( t ) T ( t ) ) as − λ . where − λ is the separating constant. 1 k ( T ′ ( t ) T ( t ) ) = − λ T ′ ( t ) = − λ k T ( t ) T ′ ( t ) + λ k T ( t ) = 0 ( 2 ) Substitute α 2 for λ in equation (1) and equation (2) as follows: X ″ ( x ) + ( α 2 ) X ( x ) = 0 T ′ ( t ) + ( α 2 ) k T ( t ) = 0 The solution of general equation of the above expression is X ( x ) = c 1 cos α x + c 2 sin α x and T ( t ) = c 3 e − k α 2 t . It is given that the boundary condition is u ( 0 , t ) = 0 . Substitute 0 for u ( 0 , t ) in equation X ( x ) = c 1 cos α x + c 2 sin α x to obtain X ( 0 ) = 0 . X ( 0 ) = c 1 cos α ( 0 ) + c 2 sin α ( 0 ) 0 = c 1 + 0 0 = c 1 Therefore, the value of c 1 is 0 . Substitute 0 for c 1 in equation X ( x ) = c 1 cos α x + c 2 sin α x . X ( x ) = ( 0 ) cos α x + c 2 sin α x = c 2 sin α x The partial derivative with respect to x of above expression is given below. ∂ X ∂ x = c 2 α cos α x ∂ X ∂ x | x = L = c 2 α cos α L ( x = L ) Now, substitute the boundary condition ∂ u ∂ x | x = L = 0 in equation ∂ X ∂ x | x = L = c 2 α cos α L

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 13.7, Problem 5E

To determine

The temperature u(x,t) in a rod of length L if the initial temperature is f(x) throughout and if the end x=0 is kept at temperature zero and the end x=L is insulated.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13.7, Problem 4E

Chapter 13.7, Problem 6E

Students have asked these similar questions

Between the function 3 (4)=x-x-1 Solve inside the interval [1,2]. then find the approximate Solution the root within using the bisection of the error = 10² method.

E10) Perform four iterations of the Jacobi method for solving the following system of equations. 2 -1 -0 -0 XI 2 0 0 -1 2 X3 0 0 2 X4 With x(0) (0.5, 0.5, 0.5, 0.5). Here x = (1, 1, 1, 1)". How good x (5) as an approximation to x?

by (2) Gauss saidel - - method find (2) و X2 for the sestem X1 + 2x2=-4 2x1 + 2x2 = 1 Such thef (0) x2=-2

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 if the initial temperature is f ( x ) throughout and if the end x = 0 is kept at temperature zero and the end x = L is insulated.

Solution Summary: The author calculates the temperature u(x,t) in a rod of length L.

Concept explainers

Videos

The temperature u(x,t) in a rod of length L if the initial temperature is f(x) throughout and if the end x=0 is kept at temperature zero and the end x=L is insulated.

Want to see the full answer?

Chapter 13 Solutions