Explanation Given: The boundary value problem is ∂ 2 u ∂ r 2 + 2 r ∂ u ∂ r = ∂ u ∂ t , 0 < r < 1 , t > 0 and the boundary conditions are u r ( 1 , t ) = − h [ u ( 1 , t ) − u 1 ] and u ( r , 0 ) = u 0 . Calculation: The given boundary value problem is, ∂ 2 u ∂ r 2 + 2 r ∂ u ∂ r = ∂ u ∂ t (1) Firstly verify the left hand side of the given boundary value problem is equals to given identity, 1 r ∂ 2 ∂ r 2 ( r u ) = 1 r ∂ ∂ r [ r ∂ u ∂ r + u ] = 1 r [ r ∂ 2 u ∂ r 2 + ∂ u ∂ r + ∂ u ∂ r ] = ∂ 2 u ∂ r 2 + 2 r ∂ u ∂ r Therefore, the above equation can be written as, 1 r ∂ 2 ∂ r 2 ( r u ) = ∂ u ∂ t Consider the equation, r u ( r , t ) = v ( r , t ) + ψ ( r ) (2) Partially differentiate both sides of the equation (2) with respect to r , r ∂ u ∂ r + u ( r , t ) = ∂ v ∂ r + ψ ′ ( r ) Partially differentiate both sides of the above equation with respect to r , r ∂ 2 u ∂ r 2 + 2 ∂ u ∂ r = ∂ 2 v ∂ r 2 + ψ ″ ( r ) Substitute r ∂ 2 u ∂ r 2 + 2 ∂ u ∂ r = ∂ 2 v ∂ r 2 + ψ ″ ( r ) and ∂ u ∂ t = ∂ v ∂ t in equation (1), ∂ 2 v ∂ r 2 + ψ ″ ( r ) = ∂ v ∂ t (3) Second derivative of ψ is zero in the equation (3) by homogeneous system, ψ ″ ( r ) = 0 Integrate the above equation, ψ ′ ( r ) = c 1 Integrate the above equation, ψ ( r ) = c 1 r + c 2 At boundary condition r = 0 , c 2 = 0 and substitute the value of c 2 in above equation, ψ ( r ) = c 1 r (4) At boundary condition u r ( 1 , t ) + h u ( 1 , t ) = h u 1 , u r ( 1 , t ) + h u ( 1 , t ) = v r ( 1 , t ) − v ( 1 , t ) + h [ v ( 1 , t ) + ψ ( 1 ) ] h u 1 = v r ( 1 , t ) + ( h − 1 ) v ( 1 , t ) + h c 1 h u 1 = h c 1 c 1 = u 1 Substitute the value of c 1 in the equation (4), ψ ( r ) = u 1 r (5) Take the equation (1) in the form of v , ∂ 2 v ∂ r 2 = ∂ v ∂ t In form of v the boundary conditions are v r ( 1 , t ) + ( h − 1 ) v ( 1 , t ) = 0 , lim r → 0 1 r v ( r , t ) < ∞ and v ( r , 0 ) = ( u 0 − u 1 ) r , Take separation of the variables of the above equation and separation constant is λ , R ″ R = − λ and T ′ T = − λ The equation of variable R is, R ″ + λ R = 0 (6) The equation of variable T is, T ′ + λ T = 0</

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 14.3, Problem 10E

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To solve: The boundary value problem for u(r,t).

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Chapter 14.3, Problem 9E

Chapter 14.3, Problem 11E

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Chapter 14 Solutions

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 boundary value problem for u ( r , t ) .

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To solve: The boundary value problem for u(r,t).

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Chapter 14 Solutions