How to perform correct numerical integration of equations containing Bessel functions ?

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ishan 2024-2-28

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回答： ishan 2024-4-30

采纳的回答： ishan

I am trying to numerically integrate ths improper integral using "integrate" but the output is un-converged:

What would be the right way to solve this equation in MATLAB ?

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Answer 1

ishan 2024-4-30

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I am very late to answer my own question but I have found a solution. The practical approach to this problem was that I used a finite upper limit of integration and made use of the "trapz" function. I noticed that the exponential term was decaying very rapidly to zero and there seemed no need to integrate it to infinity. My results matched with those from the reference.

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Answer 2

Torsten 2024-2-28

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编辑：Torsten 2024-2-28

I suspect you want exp(-zeta^2/(4*0.0277)) instead of exp(zeta^2/(4*0.0277)) in your integrand.

And why do you add the same Bessel function two times : J0(103.562*zeta)+J0(103.562*zeta) ? This gives 2*J0(103.562*zeta) :-)

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ishan 2024-2-28

在 MATLAB Online 中打开

Hello Torsten,

Yes, it should exp(-zeta^2/(4*0.0277)). It was a mistake while typing this question.

Those two values are two of the many values which I have. It will simplify to 2 J0 but it will not be everytime. I wrote them so that I can post the equation here wih example values.

I broke down the calculations into different parts and this is an isolated indefinite integral containing the Bessel function

fun = @(xi) besselj(0,106.4202*xi);
test = integral( fun,0,Inf);

This is the warning it throws up:

Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on
error is   3.3e+07. The integral may not exist, or it may be difficult to approximate
numerically to the requested accuracy. 
> In integralCalc/iterateScalarValued (line 372)
In integralCalc/vadapt (line 132)
In integralCalc (line 83)
In integral (line 87)
In test_program (line 55)

Torsten 2024-2-28

在 MATLAB Online 中打开

Seems to be difficult to integrate up to Inf:

syms x
vpa(int(besselj(0,106.4202*x),x,0,Inf))
ans = 0.0093967122783080660919954783140556
format long
fun = @(xi) besselj(0,106.4202*xi);
test = integral( fun,0,6)
test = 
   0.009399230000459
testInf = integral( fun,0,Inf)
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is   3.3e+07. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
testInf = 
    -3.106533748191589e+07