Fractional derivative to solve ODE in the sense of Caputo

I tried to write a matlab code for this Derivative

this to solve ODE :

I write this matlab code

but I dont know where is the error.

  % Numerical method
% Initial conditions and setup
h = 0.01;  % step size
x = 0:h:1;  % the range of x
a=0.5;
y = zeros(size(x));  % allocate the result y
y(0) = 1;  % the initial y value
n = numel(y);  % the number of y values
% The loop to solve the DE
for j=0:n 
    b=((j+1).^1-a)-j.^1-a;   
    
    for k=0:n
        f = (x(k)+2*(y(k))); 
    y(k+1)=f;
    end
    
     %here will be the implicit solution.
end
d=y(k+1).*b;
S=h-a; 
G=gamma(2-a);
F=(S./G)*d;
plot(x,F)

Answer 1

[Guru Mohanty]

Hi, I have gone through your code. I think you are getting Array indexing error. Its due to these following reasons.

1. In this statement

    y = zeros(size(x));  % allocate the result y
    y(0) = 1;  % the initial y value

The code trying to access Zeroth element, but as we know Array indexing in MATLAB is always positive, i.e. It will start from ‘1’.

To set initial value of y you can do

    y(1) = 1;  

2. Also, in the for loop you should start indexing from ‘1’ instead of ‘0’. i.e.

    for j=1:n
        % Expression
    end