how to use quadl function to integrate a function of combination Bessel function and a vector

Hello, I tried to use quadl command to build up a function of Bessel function multiply with a vector and integrate it from 0 to 1. My coding is below:

u=0.:0.01:1.0; nu=1.5; u=u.^1.5;
w=@(u) u.*besselj(nu,u);
quadl(w, 0, 1)

and it turns out as "identifier" expected, "(" found.

and then I modify my code using inline command and hope to create a function. the coding is below:

u=0.:0.01:1.0; nu=1.5; z=u.^1.5;
g=inline('besselj(nu,u).*z','u','nu','z');
quadl(g, 0, 1)

and it turns out as

Error using ==> inline/feval
Not enough inputs to inline function.

Could someone please to help me? Your prompt answer will be deeply appreciated. BR Donyau

[Donyau Chiang]

Thank you, Mr. Strider for your quick answer. Is it possible to use inline to create a function and integrate it because my MatLab version does not have quadv command yet? Otherwise, I have to update my version. Anyway, thank you for your instruction. BR Donyau

Answer 1

[John BG]

Donyau

1.- In MATLAB R2016 your initial code works just fine:

format long
u=0.:0.01:1.0; nu=1.5; u=u.^1.5;
w=@(u) u.*besselj(nu,u);
quadl(w, 0, 1)

ans =
   0.071277937291592

No such error message you mention in the question.

2.- There is not need to compess (chirp) the time reference with u=u.^1.5 because the result is the same:

u=0.:0.01:1.0; nu=1.5; % u=u.^1.5
w=@(u) u.*besselj(nu,u);
quadl(w, 0, 1)

ans =
   0.071277937291592

3.- Mathworks warns in http://uk.mathworks.com/help/matlab/ref/inline.html?s_tid=srchtitle

that command inline is going to be discontinued

4.- The function you want to integrate is the thicker line, Bessel 1st kind index 1.5 :

plot obtained with

X = 0:0.1:20;
J = zeros(9,201);
for i = 0:.5:9
    J(i*2+1,:) = besselj(i,X);
end
plot(X,J,'LineWidth',1.5)
axis([0 20 -.5 1])
grid on
L1=legend('J_0','J_{0.5}','J_1','J_{1.5}','J_2','J_{2.5}','J_3','J_{3.5}','J_4','Location','bestoutside')
% L1.Position=[12 .25 8 .65]
title('Bessel Functions of the First Kind for v = 0:0.5:4')
xlabel('X')
ylabel('J_v(X)')
hold on;
plot(X,J(4,:),'LineWidth',4,'Color',[.8 .8 0])

Since you only want to integrate within [0 1], and you are already multiplying besselj with the reference u*besselj(1.5,u) wouldn't you agree that it's quite a fair approximation to replace x*besselj(1) with a parabolic function?

5.- Perhaps what you considered replacing the integral with a Bessel function of higher order, from http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html

since m=1.5, the conditions z<<1 and z>>abs(m^2-.25) are not met, from the zoomed graph, the marker on besselj(1.5,u=1)=0.049 that is not quite the 0.07 from the integration.

6.- upgrade to R2016 and do not pick by line, do not use inline.

If you find this answer of any help solving your question,

please click on the thumbs-up vote link,

thanks in advance

John

[e-mail]

[Donyau Chiang]

Thank you, John, for your detailed explanation.

Answer 2

[Star Strider]

You’re integrating an array-valued funciton, so you need to use the quadv funciton:

u=0.:0.01:1.0; nu=1.5; u=u.^1.5;
w=@(u) u.*besselj(nu,u);
int_w = quadv(w, 0, 1);

If you have R2012a or later, a better choice is the integral function using the 'ArrayValued' argument:

int_w = integral(w, 0, 1, 'ArrayValued',true);

[Star Strider]

Donyau Chiang’s Comment is duplicated here:

Thank you, Mr. Strider for your quick answer. Is it possible to use inline to create a function and integrate it because my MatLab version does not have quadv command yet? Otherwise, I have to update my version. Anyway, thank you for your instruction. BR Donyau

[Star Strider]

My pleasure.

The quadv function was introduced prior to R2006a, so you should have it.

Using the anonymous function is preferable to using the inline function. Your anonymous function will work with both quadv and integral. I tested it with both.

[Donyau Chiang]

Thank you, Mr. Stride. I will ask my colleague to have updated version and try it because it is late at night now in Taiwan. Thank you again. BR Donyau

[Star Strider]

My pleasure.

What version are you using?

[Donyau Chiang]

I confirm it. You are a good teacher. My version is 6.1 released by year 2001. My version does not have commmand quadv. I think it is a good reason for me to ask the version updated. Thank you very much. BR Donyau

[Star Strider]

Thank you.

My pleasure.

I would have preferred that you Accept my Answer if it solved your problem.

[Donyau Chiang]

I definitely accept your answer and confirm to solve this problem. Thank you.

[Star Strider]

My pleasure.