Explanation Given: The function is f ( x ) = 1 − x 2 and the conditions are 0 < x < 1 , J 0 ( α ) = 0 . Concept used: Fourier Bessel series: “Fourier-Bessel series of a function f defined on the interval ( 0 , b ) is f ( x ) = ∑ i = 1 ∞ c i J n ( α i x ) where, c i = 2 b 2 J n + 1 2 ( α i b ) ∫ 0 b x f ( x ) J n ( α i x ) d x and α i is defined by J n ( α b ) = 0 .” Calculation: The function is given as follows: f ( x ) = 1 − x 2 , 0 < x < 1 The boundary conditions are given as follows: J 0 ( α ) = 0 The Fourier-Bessel series of a function f ( x ) = 1 − x 2 , J 0 ( α ) = 0 in the interval ( 0 , 1 ) is given a follows: f ( x ) = ∑ i = 1 ∞ c i J 0 ( α i x ) … … ( 1 ) The value of c i is calculated as follows: c i = 2 J 1 2 ( α i ) ∫ 0 1 x ( 1 − x 2 ) J 0 ( α i x ) d x = 2 J 1 2 ( α i ) ( ∫ 0 1 x J 0 ( α i x ) d x − ∫ 0 1 x 3 J 0 ( α i x ) d x ) Substitute t = x α i in the above integral

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 11.5, Problem 10E

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The Fourier-bessel series of order 0 for the function f(x)=1−x2 such that 0<x<1, J0(α)=0.

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Chapter 11.5, Problem 9E

Chapter 11.5, Problem 13E

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Differential Equations with Boundary-Value Problems (MindTap Course List)

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The Fourier-bessel series of order 0 for the function f ( x ) = 1 − x 2 such that 0 < x < 1 , J 0 ( α ) = 0 .

Solution Summary: The author explains the Fourier-bessel series of order 0 for the function f(x)=1-x

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The Fourier-bessel series of order 0 for the function f(x)=1−x2 such that 0<x<1, J0(α)=0.

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Chapter 11 Solutions

The Fourier-bessel series of order 0 for the function f ( x ) = 1 − x 2 such that 0 &lt; x &lt; 1 , J 0 ( α ) = 0 .

Solution Summary: The author explains the Fourier-bessel series of order 0 for the function f(x)=1-x

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The Fourier-bessel series of order 0 for the function f(x)=1−x2 such that 0<x<1, J0(α)=0.

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Chapter 11 Solutions

The Fourier-bessel series of order 0 for the function f ( x ) = 1 − x 2 such that 0 < x < 1 , J 0 ( α ) = 0 .