Explanation Given: The given boundary conditions are u ( x , 0 ) = e − x and u ( 0 , y ) = e − y . Calculation: Consider the equation, u ( x , y ) = u 1 ( x , y ) + u 2 ( x , y ) . (1) The boundary value problem is, ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 = 0 . (2) Take Fourier sine transform with respect to y of both sides of equation (2), F { ∂ 2 u 1 ∂ x 2 } + F { ∂ 2 u 1 ∂ y 2 } = 0 d 2 U 1 d x 2 − α 2 U 1 ( x , α ) = 0 Apply Fourier sine transform then the particular solution of the above equation is, U 1 ( x , α ) = c 1 e − α x + c 2 e α x . (3) At the given boundary condition x → ∞ , c 2 = 0 Substitute the value of c 2 in equation (3), U 1 ( x , α ) = c 1 e − α x . (4) At boundary condition, u 1 ( 0 , y ) = e − y Take Fourier sine transform of the above equation, F { u 1 ( 0 , y ) } = F { e − y } = ∫ 0 ∞ e − y sin α y d y = α 1 + α 2 Therefore, the equation is expressed as, U 1 ( 0 , α ) = α 1 + α 2 . (5) Substitute x = 0 in equation (4), U 1 ( 0 , α ) = c 1 . (6) Equate the equations (5) and (6), c 1 = α 1 + α 2 Substitute the value of c 1 in equation (4), U 1 ( x , α ) = α 1 + α 2 e − α x Take inverse Fourier transform of the above equation and apply Fourier sine transform, F − 1 { U 1 ( x , α ) } = F − 1 { α 1 + α 2 e − α x } So, the equation is, u 1 ( x , y ) = 2 π ∫ 0 ∞ α e − α x 1 + α 2 sin α y d α . (7) Take Fourier sine transform with respect to x of both sides of equation (2), F { ∂ 2 u 2 ∂ x 2 } + F { ∂ 2 u 2 ∂ y 2 } = 0 d 2 U 2 d x 2 − α 2 U 2 ( α , y ) = 0 Apply Fourier sine transform then the particular solution of the above equation is, U 2 ( α , y ) = c 1 e − α y + c 2 e α y

Differential Equations with Boundary-Value Problems

9th Edition

ISBN: 9781337632515

Author: Dennis G. Zill

Publisher: Cengage Learning US

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Chapter 14.4, Problem 17E

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The steady state temperature in a plate under given boundary conditions.

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Chapter 14 Solutions

Differential Equations with Boundary-Value Problems

Ch. 14.1 - (a) Show that erf(t)=10ted. (b) Use part (a), the...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. 9E Ch. 14.1 - Use the third and fifth entries in Table 14.1.1 to...

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The steady state temperature in a plate under given boundary conditions.

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The steady state temperature in a plate under given boundary conditions.

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Chapter 14 Solutions