how to evaluate the integral of the expression involving bessel functions.
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I have the following expression which i need to find.

then how i can find the integration? here this k is vector which contain k0,k1,k2, and same d and e is also a vector which contain d1,d2,d3 and e1, e2 ,e3 . and
is also given. Then how i can evaluate this integration.

2 Commenti
Torsten
il 18 Lug 2025
Modificato: Torsten
il 18 Lug 2025
Why is psi a function of r ?
Which index p is meant when you write nu_D(r)*abs(nu_D(r))*besselj(chi_p^l * r) r as the integrand ?
Do you have functions to compute the derivatives of besselj and besseli ?
Why does your vector k has only 3 elements instead of 4 for k0,k1,k2 and k3 ?
Risposte (2)
Shishir Reddy
il 18 Lug 2025
Modificato: Shishir Reddy
il 18 Lug 2025
Hi @Javeria
Kindly refer the following steps to numerically evaluate the given integral using MATLAB.
1. Define the Parameters:
% Given values
RI = 0.21;
k = [0.281, 0.2, 0.1];
d = [0.1, 0.3, 0.4]; % d0, d1, d2, d3 → d(1), d(2), d(3), d(4)
e = [0.3, 0.09, 0.7];
chi = [0.01, 0.03, 0.02];
2. Define ν_D(r):
nu_D = @(r) ...
d(1) * besselj(0, k(1)*r) / besselj(0, k(1)*RI) + ...
d(2) * besseli(1, k(2)*r) / besseli(1, k(2)*RI) + ...
d(3) * besseli(1, k(3)*r) / besseli(1, k(3)*RI) + ...
e(1) * besselj(1, chi(1)*r) / besselj(1, chi(1)*RI) + ...
e(2) * besselj(1, chi(2)*r) / besselj(1, chi(2)*RI) + ...
e(3) * besselj(1, chi(3)*r) / besselj(1, chi(3)*RI);
3. Define the Integrand Function:
chi1 = chi(1);
integrand = @(r) abs(nu_D(r)).^2 .* besselj(0, chi1 * r);
4. Use 'integral' to compute the result:
psi = integral(integrand, 0, RI);
disp(['ψ = ', num2str(psi)]);
For more information regarding 'integral' function, kindly refer the following documentation -
I hope this helps.
Torsten
il 18 Lug 2025
Modificato: Torsten
il 18 Lug 2025
l = 0;
R_I = 2;
N = 3;
P = 3;
k = [0.28,0.2,0.1,0.05]; %[k0,k1,k2,k3]
e = [0.3,0.9,0.7]; %[e1,e2,e3]
d = [0.1,0.3,0.4,0.5]; %[d0,d1,d2,d3]
chi = [0.01,0.03,0.02]; %[chi1,chi2,chi3]
format long
psi = integral(@(r)fun(r,l,R_I,N,P,k,e,d,chi),0,R_I)
function values = fun(r,l,R_I,N,P,k,e,d,chi)
nu_D_values = nu_D(r,l,R_I,N,P,k,e,d,chi);
chip = chi(end); %-> which p-index from p has to be taken here ?
values = nu_D_values.*abs(nu_D_values).*besselj(l,chip*r).*r;
end
function nu_D_values = nu_D(r,l,R_I,N,P,k,e,d,chi)
nu_D_values = d(1)*besselj(l,k(1)*r)/dbesselj(l,k(1)*R_I);
for n = 1:N
nu_D_values = nu_D_values + d(n+1)*besseli(l,k(n+1)*r)/dbesseli(l,k(n+1)*R_I);
end
for p = 1:P
nu_D_values = nu_D_values + e(p)*chi(p)*besselj(l,chi(p)*r)/dbesselj(l,chi(p)*R_I);
end
end
function value = dbesseli(nu,z)
value = 0.5*(besseli(nu-1,z) + besseli(nu+1,z));
%value = nu./z.*besseli(nu,z) + besseli(nu+1,z);
end
function value = dbesselj(nu,z)
value = 0.5*(besselj(nu-1,z) - besselj(nu+1,z));
%value = nu./z.*besselj(nu,z) - besselj(nu+1,z);
end
5 Commenti
Torsten
il 21 Lug 2025
The first thing that I'd do is to output the matrices A, B, C, D E, F, W, G and H_I and inspect where the entries in the order of 1e13 - 1e29 stem from in your computations.
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