I have the following fractional differential equations.
$D^{0.8}y_{1}=2y_{1}(1-\dfrac{y_{1}}{2y_{2}}-\dfrac{y_{1}}{2})$, $y_{1}(0)=0.5$
Using the following algorithm I want to try to solve it
(1) $y(t_{j})=\dfrac{h^{\alpha}}{\Gamma(\alpha+2)}((j-1)^{\alpha +1}-(j-\alpha -1)j^{\alpha})f(t_{0},y(t_{0}))+y(0)+\dfrac{h^{\alpha}}{\Gamma(\alpha +2)}\displaystyle\sum_{i=1}^{j-1}((j-i+1)^{\alpha +1}-2(j-i)^{\alpha +1}+(j-i-1)^{\alpha +1})f(t_{i},y(t_{i}))+\dfrac{h^{\alpha}}{\Gamma(\alpha +2)}f(t_{j},y(t_{j-1})+\dfrac{h^{\alpha}}{\Gamma(\alpha +1)}f(t_{j-1},y(t_{j-1})))$
Well, I honestly do not know how to program it,The only thing I did was do everything in manual calculation.
If $f(t,y_{1}(t),y_{2}(t))=2y_{1}(1-\dfrac{y_{1}}{2y_{2}}-\dfrac{y_{1}}{2})$
for $j=0$ then $y_{1}(t_{0})=0.5$
for $j=1$ then $y_{1}(t_{1})=0.498952082$
for $j=2$ then $y_{1}(t_{2})=0.507047263469$
for $j=3$ then $y_{1}(t_{3})=0.517592091433$
But I don't know how to continue this. Because it takes me a long time doing one by one
How to program (1) In any programming language.
Thank you.