Need help with complicated Legendre function to be integrated

I am trying to figure out how to integrate a complicated function that has legendre polynomials in it. Mu is the variable by which I am attempting to integrate, from 0 to Muo. I am not sure why the legendres are turning into arrays instead of scalars, which I believe is the main problem. Also, the dGam is an array as well, which might not be right. I'm thinking that I need to differentiate Phi first, then plug in the value of Mu, but I am not sure what the derivative would be, and wolframaplha wasn't much help. Any advice would be greatly appreciated. Thank you!

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m = 2; %choosing the index

muo = 5; %length of blade

nGam = 2; %order of gamma

B = quad(@Bfun,0.0001,muo,[],[],m,muo,nGam)

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function y = Bfun(mu,m,muo,nGam)

x = 2*mu/muo - 1; %change of variable (-1<x<1)

P1 = legendre(m+1,x); %Legendre polynomial

P1 = P1(m+2,:) %Selecting first order

P2 = legendre(m-1,x); %Legendre polynomial

P2 = P2(m,:) %Selecting first order

Phi2 = (P1 - P2)/(sqrt(2*(2*m-1))); %eqn for Phi2

dPhi2 = gradient(Phi2); %deriving Phi2

dPhi = 2/muo * dPhi2; %deriving Phi

dGam = [(nGam*(mu./sqrt(1 + mu.^2)).^nGam)./(nGam + mu.^3)]' %deriving gamma

y = -1* mu .* dPhi * dGam %sending final function

end

[Walter Roberson]

If you post the original formula, there is a possibility that I could figure it out in Maple.

Answer 1

[Mike Hosea]

I haven't looked at your example in any kind of detail, but I wonder if you're aware that QUAD requires your function to operate elementwise on an array of mu values? If this is inconvenient, you can meet the requirement in a trivial way with arrayfun. Try it this way (and then debug from there, if need be):

integrand = @(mu)Bfun(mu,m,muo,nGam); % scalar mu only
vintegrand = @(mu)arrayfun(integrand,mu); % vector mu supported
B = quad(vintegrand,0.0001,muo)

Notice that this also "binds" the values of m, muo, and nGam in the definition of the integrand function rather than passes them in as arguments to quad. If you do it that way, you can use quadgk if you want. -- Mike

Answer 2

[Andrew]

I was able to figure out a way to do it using symbolic variables. However when I tried to calculate the integral it said:

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Warning: Explicit integral could not be found.

B =

int(-(2*10^(1/2)*mu^3*(((2*mu)/5 - 1)^2 + ((2*mu)/5 - 1)*((4*mu)/5 - 2) - 1))/(25*(mu^2 + 1)*(mu^3 + 2)), mu = 0..5)

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I copied the above line into WolframAlpha and it was able to integrate fine. Is there anyway matlab would be able to do this integral? Thanks!

[Walter Roberson]

Maple does not have any difficulty integrating it in to a bunch of sqrt() and ln() and arctan()

Perhaps if you simplify() the expression before integrating it?

[Andrew]

Yeah, I tried the simplify, but it didn't work. Unfortunately I don't have maple, and I'm trying to use matlab. Do you know if there's any way to turn the symbolic equation into a numerical equation and then use quad( ) to evaluate it?

[Walter Roberson]

fun = matlabFunction(-(2*10^(1/2)*mu^3*(((2*mu)/5 - 1)^2 + ((2*mu)/5 - 1)*((4*mu)/5 - 2) - 1))/(25*(mu^2 + 1)*(mu^3 + 2)));

Answer 3

[Andrew]

I tried this:

quad(@(mu)Bfun(m,muo,nGam),0,5)

but it gave the following error message:

---------- ??? Undefined function or method 'isfinite' for input arguments of type 'sym'.

Error in ==> quad at 81 if ~isfinite(y(1)) ----------