Explanation Given: The radius of the sphere is 1 unit. The boundary value problem is as follows: ∂ 2 u ∂ r 2 + 2 r ∂ u ∂ r + 1 r 2 ∂ 2 u ∂ θ 2 + cot θ r 2 ∂ u ∂ θ = 0 u ( 1 , θ ) = { 100 0 < θ < π 2 − 100 π 2 < θ < π Calculation: The boundary value problem is as follows: ∂ 2 u ∂ r 2 + 2 r ∂ u ∂ r + 1 r 2 ∂ 2 u ∂ θ 2 + cot θ r 2 ∂ u ∂ θ = 0 Take separation of the variables of the above equation and separation constant is λ . r 2 R ″ + 2 r R ′ R = λ − Θ ″ + ( cot θ ) Θ ′ Θ = λ Consider the equation as follows: u ( r , θ ) = R ( r ) Θ ( θ ) Compute the individual equations by taking r 2 R ″ + 2 r R ′ R and λ as follows: r 2 R ″ + 2 r R ′ − λ R = 0 ...... ( 1 ) Compute the individual equations by taking − Θ ″ + ( cot θ ) Θ ′ Θ and λ as follows: sin θ Θ ″ + cos θ Θ ″ + λ sin θ Θ = 0 ...... ( 2 ) Consider x = cos θ and apply the chain rule to obtain the value of d Θ d θ as follows: d Θ d θ = d Θ d x d x d θ d Θ d θ = − sin θ d Θ d x ( d x d θ = − sin θ ) Further, differentiate the above equation with respect to θ with the help of chain rule. d 2 Θ d θ 2 = − sin θ d 2 Θ d x 2 ⋅ d x d θ − cos θ d Θ d x Since, d x d θ = − sin θ then the above equation is further written as follows: d 2 Θ d θ 2 = sin 2 θ d 2 Θ d x 2 − cos θ d Θ d x Substitute the value of d 2 Θ d θ 2 , d Θ d θ and x in equation ( 2 ) and transform the equation ( 2 ) in the form of Legendre’s equation as follows: ( 1 − x 2 ) d 2 Θ d x 2 − 2 x d Θ d x + λ Θ = 0 , − 1 ≤ x ≤ 1 The above equation is a form of Legendre’s equation and the solution of the equation ( 1 ) is as follows: Θ ( θ ) = P n ( cos θ ) … … ( 3 ) Where P n ( x ) are the Legendre polynomials for λ = n ( n + 1 ) , where n = 0 , 1 , 2 , 3 ⋅ ⋅ ⋅ ⋅ . The general solution to the equation ( 1 ) (Cauchy-Euler equation) is as follows: R ( r ) = c 1 r n + c 2 r − ( n + 1 ) … … ( 4 ) Consider that u ( r , θ ) is bounded at r = 0 for which the value of c 2 is necessarily 0 . Substitute 0 for c 2 in equation ( 4 ) . R ( r ) = c 1 r n Therefore, the product solution is written as follows: u n ( r , θ ) = c 1 r n P n ( cos θ ) The summation form of the above equation is written as follows: u ( r , θ ) = ∑ n = 1 ∞ A n r n P n ( cos θ ) … … ( 5 ) The given condition is follows: u ( 1 , θ ) = { 100 0 < θ < π 2 − 100 π 2 < θ < π As considered above x = cos θ so, the above condition is transformed as follows: g ( x ) = { 100 0 < x < 1 − 100 − 1 < x < 0 The coefficients A n in equation ( 5 ) are the coefficients of Fourier Legendre series which is calculated as follows: A n = ( 2 n + 1 2 ) ∫ 0 π u ( 1 , θ ) P n ( cos θ ) d θ As considered above x = cos θ so, the above equation is transformed as follows: A n = ( 2 n + 1 2 ) ∫ − 1 1 g ( x ) P n ( x ) d x … … ( 5 ) The Rodrigues’ formula to obtain the value of P n ( x ) is as follows: P n ( x ) = 1 2 n n ! d n ( x 2 − 1 ) n d x n … … ( 6 ) Substitute 0 for n in equation ( 6 ) and obtain the value of P 0 ( x ) as follows: P 0 ( x ) = x 2 − 1 Substitute 0 for n in the equation ( 5 ) to obtain the value of A 0

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 14, Problem 15CR

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The steady state temperature u(r,θ) in a sphere of unit radius provided that u(1,θ)={100 0<θ<π2−100 π2<θ<π.

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Advanced Engineering Mathematics

Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Prob. 5E Ch. 14.1 - Prob. 6E Ch. 14.1 - Prob. 7E Ch. 14.1 - Prob. 8E Ch. 14.1 - Prob. 10E Ch. 14.1 - Prob. 11E

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The steady state temperature u ( r , θ ) in a sphere of unit radius provided that u ( 1 , θ ) = { 100 0 < θ < π 2 − 100 π 2 < θ < π .

Solution Summary: The author explains the steady state temperature u(r,theta ) in the sphere. The boundary value problem is as follows:

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The steady state temperature u(r,θ) in a sphere of unit radius provided that u(1,θ)={100 0<θ<π2−100 π2<θ<π.

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Chapter 14 Solutions

The steady state temperature u ( r , θ ) in a sphere of unit radius provided that u ( 1 , θ ) = { 100 0 &lt; θ &lt; π 2 − 100 π 2 &lt; θ &lt; π .

Solution Summary: The author explains the steady state temperature u(r,theta ) in the sphere. The boundary value problem is as follows:

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The steady state temperature u(r,θ) in a sphere of unit radius provided that u(1,θ)={100 0<θ<π2−100 π2<θ<π.

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Chapter 14 Solutions

The steady state temperature u ( r , θ ) in a sphere of unit radius provided that u ( 1 , θ ) = { 100 0 < θ < π 2 − 100 π 2 < θ < π .