(a) Explanation Given: The given boundary value problem is a 2 ∂ 2 u ∂ x 2 = ∂ 2 u ∂ t 2 + F 0 δ ( t − x v 0 ) , t > 0 , x > 0 and given boundary conditions are u ( 0 , t ) = 0 , t > 0 , lim x → ∞ u ( x , 0 ) = 0 , u ( x , 0 ) = 0 and ∂ u ∂ t | t = 0 = 0 , x > 0 . Calculation: When v 0 ≠ a The given boundary value problem is, a 2 ∂ 2 u ∂ x 2 = ∂ 2 u ∂ t 2 + F 0 δ ( t − x v 0 ) . (1) Take Laplace transform of both sides of the above equation and solve as follows. a 2 L { ∂ 2 u ∂ x 2 } = L { ∂ 2 u ∂ t 2 } + L { F 0 δ ( t − x v 0 ) } a 2 d 2 U d x 2 = s 2 U ( x , s ) − s u ( x , 0 ) − u t ( x , 0 ) + L { F 0 δ ( t − x v 0 ) } The particular solution of the above equation is, U ( x , s ) = c 1 e − s a x + c 2 e s a x + v 0 2 F 0 ( a 2 − v 0 2 ) s 2 e − s v 0 x At boundary conditions lim x → ∞ u ( x , 0 ) = 0 , lim x → ∞ U ( x , s ) = 0 c 2 = 0 Substitute the value of c 2 in equation (2) as follows. U ( x , s ) = c 1 e − s a x + v 0 2 F 0 ( a 2 − v 0 2 ) s 2 e − s v 0 x (b) To determine To solve: The partial differential equation when v 0 = a .

Bundle: Differential Equations with Boundary-Value Problems, Loose-leaf Version, 9th + WebAssign Printed Access Card for Zill's Differential Equations ... Problems, 9th Edition, Single-Term

9th Edition

ISBN: 9781337604901

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 14.2, Problem 28E

(a)

To determine

To solve: The partial differential equation when v0≠a.

(b)

To determine

To solve: The partial differential equation when v0=a.

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Chapter 14.2, Problem 27E

Chapter 14.2, Problem 29E

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Chapter 14 Solutions

Bundle: Differential Equations with Boundary-Value Problems, Loose-leaf Version, 9th + WebAssign Printed Access Card for Zill's Differential Equations ... Problems, 9th Edition, Single-Term

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 partial differential equation when v 0 ≠ a .

Solution Summary: The author explains how to solve the partial differential equation when v_0ne a.

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To solve: The partial differential equation when v0≠a.

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Chapter 14 Solutions