Explanation Given: The boundary value problem is ∂ 2 u ∂ r 2 + 1 r ∂ u ∂ r + 1 r 2 ∂ 2 u ∂ θ 2 = 0 , 0 < r < c , 0 < θ < 2 π and the boundary condition is u r ( c , θ ) = f ( θ ) . Calculation: The given boundary value problem is, ∂ 2 u ∂ r 2 + 1 r ∂ u ∂ r + 1 r 2 ∂ 2 u ∂ θ 2 = 0 Consider the equation, u ( r , θ ) = R ( r ) Θ ( θ ) (1) Take separation of the variables of the above equation and separation constant is λ , r 2 R ″ + r R ′ R = λ and − Θ ″ Θ = λ The equation of variable R is, r 2 R ″ + r R ′ − λ R = 0 (2) The equation of variable θ is, Θ ″ + λ Θ = 0 (3) When λ = 0 The general solution of the equation (3) is, Θ ( θ ) = c 1 + c 2 ln θ At boundary condition u θ ( r , 0 ) = 0 and u θ ( r , 2 π ) = 0 , Θ ′ ( 0 ) = 0 and Θ ′ ( 2 π ) = 0 , c 2 = 0 and substitute in above equation, Θ ( θ ) = c 1 (4) The general solution of the equation (2) is, R ( r ) = c 3 + c 4 ln r At boundary condition, c 4 = 0 and substitute in above equation, R ( r ) = c 3 (5) Substitute the value of Θ ( θ ) from equation (4) and the value of R ( r ) from equation (5) in equation (1), u ( r , θ ) = c 1 c 3 Rewrite the term c 1 c 3 = A 0 in the above equation and apply superposition principle in the above equation, u ( r , θ ) = A 0 (6) When λ = α 2 The general solution of the equation (3) is, Θ ( θ ) = c 1 cos α θ + c 2 sin α θ At given boundary conditions u θ ( r , 0 ) = 0 and u θ ( r , 2 π ) = 0 , Θ ′ ( 0 ) = 0 and Θ ′ ( 2 π ) = 0 , α = n and substitute in the above equation, Θ ( θ ) = c 1 cos n θ + c 2 sin n θ (7) The general solution of the equation (2) is, R ( r ) = c 3 r α + c 4 r − α At given boundary condition, c 4 = 0 and α = n

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 13.1, Problem 21E

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To find: The steady state temperature in circular plate.

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Chapter 13.1, Problem 20E

Chapter 13.1, Problem 22E

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The steady state temperature in circular plate.

Solution Summary: The author calculates the steady state temperature in a circular plate using the boundary value problem u(r,theta )=displaystyle

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To find: The steady state temperature in circular plate.

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