Explanation Given: The given Laplace equation is ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 = 0 , 0 < x < a , 0 < y < b and the given boundary conditions are u ( 0 , y ) = 0 , u ( a , y ) = 0 , u ( x , 0 ) = f ( x ) and u ( x , b ) = g ( x ) . Calculation: The given Laplace equation for rectangular plate is, ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 = 0 Consider the equation, u ( x , y ) = X ( x ) Y ( y ) (1) Separate the variables of the above equation, X ″ X = − λ and − Y ″ Y = − λ The equation of the variable X is, X ″ + λ X = 0 (2) The equation of the variable Y is, Y ″ − λ Y = 0 (3) When λ = α 2 > 0 , Substitute λ = α 2 in equation (2), X ″ + α 2 X = 0 The general solution of the above equation is, X ( x ) = c 1 cos α x + c 2 sin α x (4) At boundary conditions u ( 0 , y ) = 0 , u ( a , y ) = 0 , u ( x , 0 ) = f ( x ) and u ( x , b ) = g ( x ) , so the value of c 1 is 0 also written as c 1 = 0 . For non-trivial solution c 2 ≠ 0 , α = n π a Substitute c 1 = 0 and α = n π a in the equation (4), X ( x ) = c 2 sin n π a x (5) Substitute λ = α 2 in equation (3), Y ″ − α 2 Y = 0 The general solution of the above equation is, Y ( y ) = c 3 cosh α y + c 4 sinh α y (6) At boundary conditions u ( 0 , y ) = 0 , u ( a , y ) = 0 , u ( x , 0 ) = f ( x ) and u ( x , b ) = g ( x ) , Substitute α = n π a in the equation (6), Y ( y ) = c 3 cosh n π a y + c 4 sinh n π a y (7) Substitute the value of X ( x ) from equation (5) and the value of Y ( y ) from equation (7) in equation (1), u ( x , y ) = ( c 2 c 3 cosh n π a y + c 2 c 4 sinh n π a y ) sin n π a x Rewrite the term c 2 c 3 = A n and c 2 c 4 = B n in the above equation, u ( x , y ) = ∑ n = 1 ∞ (

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 12.5, Problem 13E

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Chapter 12.5, Problem 12E

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The Laplace equation for a rectangular plate under given boundary conditions.

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