Soft Numerical Computing in Uncertain Dynamic Systems

$x\left(t\right)=\left(0,1,2\right)\odot E_{\alpha }\left(-t^{\alpha }\right)≔c\odot E_{\alpha }\left(-t^{\alpha }\right)$

where:

$E_{\alpha }\left(-t^{\alpha }\right)=\sum _{i=0}^{\infty }\frac{{\left(-t^{\alpha }\right)}^i}{\mathrm{\Gamma }\left(\mathit{i\alpha }+1\right)}$

This function is ii − gH differentiable, because the length of x(t) is decreasing.

Therefore, the Euler method is in the form of Case 2:

$x\left(t_{k+1}\right)=x\left(t_k\right){\ominus }_H\left(-1\right)\frac{h^{\alpha }}{\mathrm{\Gamma }\left(\alpha +1\right)}\odot \left(0,1,2\right)\odot E_{\alpha }\left(-t_k^{\alpha }\right),k=0,1,\dots ,N-1$

For instance, consider the step size h = 0.1 and α = 0.5. The level-wise form of the Euler method is:

$x\left(t_{k+1},r\right)=x\left(t_k,r\right){\ominus }_H\left(-1\right)\frac{{0.5}^{\alpha }{E}_{0.5}\left(-t_k^{0.5}\right)}{\mathrm{\Gamma }\left(0.5+1\right)}\odot \left(r-1,2-r\right),$

for k = 0, 1, …, N − 1:

$\frac{0.687498}{\left(\sqrt{t_k}+1\right)\mathrm{\Gamma }\left(0.5+1\right)}=\frac{0.775758}{\left(\sqrt{t_k}+1\right)}$

Since the ...