# Integration takes time too long

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Nurul Afrina on 8 Jun 2022
Commented: Nurul Afrina on 8 Jun 2022 Hi, i want to integrate this equation C(x,y,t) but it takes too long and I did not get the solution . What can I do ?
This is my code that I have try
x=-2:1:6;
y=0.4;
t=0.75;
v=1.5;
alpha=0.1;
gamma=1/(1-alpha);
syms r;
A=exp(v/2*(x+y)-alpha*gamma*t);
B=(r^3)/(r.^2+gamma+v^2/2)^2;
D=exp((alpha*gamma.^2*t)/(r.^2+gamma+v^2/2));
K=besselj(0,r*sqrt(x.^2+y.^2));
L=besselj(2,r*sqrt(x.^2+y.^2));
M=B*D.*(K+L);
C=(alpha*(gamma^2)*y*A).*int(M,r,0,inf);
plot(x,C)

Torsten on 8 Jun 2022
X=-2:0.1:6;
y=0.4;
t=0.75;
v=1.5;
alpha=0.1;
for i=1:numel(X)
x = X(i);
Fun = @(r)fun(r,x,y,t,v,alpha);
C(i) = integral(Fun,0,Inf);
end
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 5.9e-09. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 6.5e-09. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 6.7e-09. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 7.1e-09. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 7.4e-09. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 8.1e-09. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 8.3e-09. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 9.4e-09. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 9.7e-09. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 1.0e-08. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 1.1e-08. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 1.2e-08. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 1.2e-08. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 1.3e-08. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 1.4e-08. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 1.6e-08. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 1.6e-08. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 1.7e-08. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 1.9e-08. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 2.0e-08. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 2.2e-08. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 2.3e-08. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 2.4e-08. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 2.6e-08. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 2.8e-08. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 3.0e-08. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
Warning: Reached the limit on the maximum number of intervals in use. Approximate bound on error is 3.2e-08. The integral may not exist, or it may be difficult to approximate numerically to the requested accuracy.
plot(X,C) function value = fun(r,x,y,t,v,alpha)
gamma=1/(1-alpha);
A=exp(v/2*(x+y)-alpha*gamma*t);
B=(r.^3)./(r.^2+gamma+v^2/2).^2;
D=exp((alpha*gamma.^2*t)./(r.^2+gamma+v^2/2));
K=besselj(0,r.*sqrt(x.^2+y.^2));
L=besselj(2,r.*sqrt(x.^2+y.^2));
M=B.*D.*(K+L);
value = alpha*gamma^2*y*A*M;
end
Nurul Afrina on 8 Jun 2022
Thank you so much

SALAH ALRABEEI on 8 Jun 2022
Matlab is not good enough to symoblically ( analyitcally) integarate or solve such complex equations). If you want the the analytical integaration, it is better to simplify it yourslf by hand. However, you integarte (numerical approximation) over a truncated domain from [0, infinity) to [0, M], where M is a large number. This approached is already done by @Torsten here
Nurul Afrina on 8 Jun 2022
Alright noted. Thank you for your explanation sir .

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