Explanation Given: The Function is defined as follows: f ( x ) = { 1 , 0 < x < 2 0 , 2 < x < 4 Boundary condition is as follows: J 0 ( 4 α ) = 0 Definition used: The Fourier Bessel series of a function f ( x ) on the interval 0 < x < b is defined as follows: f ( x ) = ∑ i = 1 ∞ C i J n ( α i x ) , where the coefficient C i is defined as follows: C i = 2 b 2 J 2 n + 1 ( α i b ) ∫ 0 b x J n ( α i x ) f ( x ) d x And α i are defined by J n ( α b ) = 0 . Calculation: For the given function, n = 0 and b = 4 . Therefore, substitute 0 for n and 4 for b in the C i formula and calculate as follows: C i = 2 4 2 J 0 + 1 2 ( α i 4 ) ∫ 0 4 x J 0 ( α i x ) f ( x ) d x = 2 16 J 1 2 ( 4 α i ) [ ∫ 0 2 x J 0 ( α i x ) ( 1 ) d x + ∫ 0 4 x J 0 ( α i x ) ( 0 ) d x ] = 1 8 J 1 2 ( 4 α i ) ∫ 0 2 α i ( t α i ) J 0 ( t ) ( d t α i ) ( Substitute α i x = t ) = 1 8 J 1 2 ( 4 α i ) 1 α i 2 ∫ 0 2 α i t J 0 ( t ) d t Further, solve the above equation as follows: C i = 1 8 J 1 2 ( 4 α i ) 1 α i 2 ∫ 0 2 α i

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 11, Problem 21RE

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The Fourier-Bessel series for the function f(x)={1, 0<x<20, 2<x<4 by the use of Bessel functions of order zero that satisfy the boundary condition J0(4α)=0.

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Chapter 11, Problem 20RE

Chapter 11, Problem 22RE

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Chapter 11 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. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - In problem 16 show that the given functions are...Ch. 11.1 - Prob. 6E Ch. 11.1 - Prob. 7E Ch. 11.1 - In Problems 712 show that the given set of...Ch. 11.1 - In Problems 712 show that the given set of...Ch. 11.1 - In Problems 712 show that the given set of...

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The Fourier-Bessel series for the function f ( x ) = { 1 , 0 < x < 2 0 , 2 < x < 4 by the use of Bessel functions of order zero that satisfy the boundary condition J 0 ( 4 α ) = 0 .

Solution Summary: The author explains the Fourier-Bessel series of a function f(x)=l1,

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The Fourier-Bessel series for the function f(x)={1, 0<x<20, 2<x<4 by the use of Bessel functions of order zero that satisfy the boundary condition J0(4α)=0.

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Chapter 11 Solutions

The Fourier-Bessel series for the function f ( x ) = { 1 , 0 &lt; x &lt; 2 0 , 2 &lt; x &lt; 4 by the use of Bessel functions of order zero that satisfy the boundary condition J 0 ( 4 α ) = 0 .

Solution Summary: The author explains the Fourier-Bessel series of a function f(x)=l1,

Concept explainers

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The Fourier-Bessel series for the function f(x)={1, 0<x<20, 2<x<4 by the use of Bessel functions of order zero that satisfy the boundary condition J0(4α)=0.

Want to see the full answer?

Chapter 11 Solutions

The Fourier-Bessel series for the function f ( x ) = { 1 , 0 < x < 2 0 , 2 < x < 4 by the use of Bessel functions of order zero that satisfy the boundary condition J 0 ( 4 α ) = 0 .