(a) Explanation Given: The given boundary value problem is a 2 ∂ 2 u ∂ x 2 = ∂ 2 u ∂ t 2 and the given boundary conditions are u ( 0 , t ) = 0 , t > 0 , u ( x , 0 ) = f ( x ) and ∂ u ∂ t | t = 0 = 0 , x > 0 . Calculation: The given boundary value problem is, a 2 ∂ 2 u ∂ x 2 = ∂ 2 u ∂ t 2 . (1) Take Fourier transform of both sides of the above equation, a 2 F { ∂ 2 u ∂ x 2 } = F { ∂ 2 u ∂ t 2 } − a 2 α 2 U ( α , t ) + a 2 α u ( 0 , t ) = d 2 U d t 2 − a 2 α 2 U ( α , t ) = d 2 U d t 2 Therefore, the equation is, d 2 U d t 2 + a 2 α 2 U ( α , t ) = 0 Apply Fourier sine transform then the particular solution of the above equation is, U ( α , t ) = c 1 cos α a t + c 2 sin α a t . (2) At the given boundary conditions, u ( x , 0 ) = f ( x ) Take Fourier transform of the above equation, F { u ( x , 0 ) } = F { f ( x ) } So, the equation is expressed as, U ( α , 0 ) = F ( α ) . (3) At the given boundary conditions, u t ( x , 0 ) = g ( x ) Take Fourier transform of the above equation, F { u t ( x , 0 ) } = F { g ( x ) } So, the equation is expressed as, U t ( α , 0 ) = G ( α ) (b) To determine To show: The solution of the given problem is u ( x , t ) = 1 2 [ f ( x + a t ) + f ( x − a t ) ] if g ( x ) = 0 .

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ISBN: 9781337604918

Author: Dennis G. Zill

Publisher: Cengage Learning

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Chapter 14.4, Problem 9E

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The u(x,t) under given boundary conditions.

(b)

To determine

To show: The solution of the given problem is u(x,t)=12[f(x+at)+f(x−at)] if g(x)=0.

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Chapter 14.4, Problem 8E

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Chapter 14 Solutions

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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 u ( x , t ) under given boundary conditions.

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The u(x,t) under given boundary conditions.

To show: The solution of the given problem is u(x,t)=12[f(x+at)+f(x−at)] if g(x)=0.

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