Explanation Given: The given boundary value problem involves evaluating the steady state temperature profile of the sphere in spherical coordinates, The situation given here is, a sphere in its first octant which is a hollow sphere with a < r < b and the inner radius at r = a is kept at f ( θ ) and its outer radius r = b is kept at temperature is zero. Calculation: If u = R ( r ) Θ ( θ ) , then the partial differential equation can be framed as follows. r 2 R ″ + 2 r R ′ R = Θ ″ + cot θ Θ ′ Θ = λ Compute the individual equations by taking r 2 R ″ + 2 r R ′ R and λ as below: r 2 R ″ + 2 r R ′ − λ R = 0 ( 1 ) Compute the individual equations by taking Θ ″ + cot θ Θ ′ Θ and λ as below: sin θ Θ ″ + cos θ Θ ″ + λ sin θ Θ = 0 ( 2 ) Substitute x = cos θ , 0 ≤ θ ≤ π , then by differentiating with respect to θ the above equation obtained as below: d Θ d θ = − sin θ d Θ d x 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 y d x 2 ⋅ d x d θ − cos θ d y d x Since, d x d θ = − sin θ , the above equation can be further written as follows: d 2 Θ d θ 2 = sin 2 θ d 2 y d x 2 − cos θ d y d x Transforming the above equation in the form of Legendre’s equation gives the following equation. ( 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

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ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 14.3, Problem 5E

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The steady state temperature u(r,θ) within the hollow sphere.

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Chapter 14.3, Problem 4E

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Advanced Engineering Mathematics

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The steady state temperature u ( r , θ ) within the hollow sphere.

Solution Summary: The author explains that the boundary value problem involves evaluating the steady state temperature profile of the sphere in spherical coordinates.

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The steady state temperature u(r,θ) within the hollow sphere.

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Chapter 14 Solutions