Alright! Through much investigation and work, I have determined the answer to this problem. Now, this is actually the first time that I'm answering a question on this site, so once more, please forgive any faux pax that I might commit.
First off, it appears that I was right about the Bessel-Fourier series. It is impossible to use just the Bessel series to get a function approximation. Through a lengthy process starting with the Bessel function and scaling the x and doing a whole host of tricks that include a Fourier transform, we get can determine that the Bessel-Fourier representation of a function is given in the form of
f(x) = A1*J_0(z1x) + A2*J_0(z2x) + ....
where z1 are the zeros of a First-Order Bessel Function, J_0 is the 0th First-Order Bessel Function and A1 are the coefficients of the Fourier-Bessel series. The zeros are easily acquired by just plotting the Bessel Function, but the coefficients are the tricky parts. They are given by the equation
An = (2/(J_1(zn)^2))*integral(x*f(x)*J_0(zn*x), 0, 1)
Needless to say, the difficult part of this is getting the integral of the Bessel functions. But by using this lovely list, the process can be made simple. Messy....but simple. I should also note that the the integral is from 0 to 1 because that was the nature of my assignment. If the question was to scale it from 0 to 2, then the equation would be
An = (2/(2^2)*(J_1(zn)^2))*integral(x*f(x)*J_0(zn*x), 0, 2)
But I digress. So, since my assignment also had me graph the first five values individually, and then add them together and compare the results with the actual function, that's what I did. And thus here is the code.
%%Plotting the Bessel Functions
a=[2.40483, 5.52008, 8.65373, 11.7915, 14.9309]; %a matrix with the first 5 zeros for a first order Bessle Function
A=zeros(5,1);
F=zeros(1, 100);
X=linspace(0,1);
for i=1:1:5
A(i,1)= (2*(besselj(1,a(i))*struve(0,a(i))-besselj(0,a(i))*struve(1,a(i))))/((a(i)^2)*(besselj(1,a(i)))^2)*(3.14/2);
%the equation to determine the coefficients of the Bessel-Fourier series
end
for i=1:1:5
figure(i);
plot(X, A(i,1)*besselj(0, (a(i)*X))); %plot the first 5 Bessel functions
end
for i=1:1:5
F=F+A(i,1)*besselj(0, (a(i)*X)); %adding all the Bessel functions together
end
figure(6);
plot(X, F); %plotting the Bessel functions and 1-x
hold on
plot(X, 1-X);
%%Struve Function
function f=struve(v,x,n)
% Calculates the Struve Function
%
% struve(v,x)
% struve(v,x,n)
%
% H_v(x) is the struve function and n is the length of
% the series calculation (n=100 if unspecified)
%
% from: Abramowitz and Stegun: Handbook of Mathematical Functions
% http://www.math.sfu.ca/~cbm/aands/page_496.htm
if nargin<3
n=100;
end
k=0:n;
x=x(:)';
k=k(:);
xx=repmat(x,length(k),1);
kk=repmat(k,1,length(x));
TOP=(-1).^kk;
BOT=gamma(kk+1.5).*gamma(kk+v+1.5);
RIGHT=(xx./2).^(2.*kk+v+1);
FULL=TOP./BOT.*RIGHT;
f=sum(FULL);
And there we go. The struve function code was from this place
I hope this helped, and if I made any egregious errors, please tell me, but to be honest, I can't explain the equations up there any further as I just got them from a textbook.
Best wishes!
