Soft Numerical Computing in Uncertain Dynamic Systems

5.3.1: Analysis of the fuzzy modified Euler method

As with the fuzzy Euler method, in this section, consistency, convergence, and stability of the modified Euler method are discussed in detail.

5.3.1.1: Local truncation error and consistency

For all abovementioned cases of the fuzzy Euler method, we can define the local truncation error by using a residual value as follows:

$R_k=x\left(t_{k+1}\right){\ominus }_{\mathit{gH}}\left(x\left(t_k\right)\oplus h\odot \left(\frac{f\left(t_{k+1},x^{⁎}\left(t_{k+1}\right)\right)\oplus f\left(t_k,x\left(t_k\right)\right)}{2}\right)\right)$

$=\frac{h^3}{3!}\odot x_{i-\mathit{gH}}^{\prime \prime \prime }\left({\eta }_k\right)$

or:

$R_k=x\left(t_{k+1}\right){\ominus }_{\mathit{gH}}\left(x\left(t_k\right){\ominus }_H\left(-1\right)\left(\frac{f\left(t_{k+1},x^{⁎}\left(t_{k+1}\right)\right)\oplus f\left(t_k,x\left(t_k\right)\right)}{2}\right)\right)$

$={\ominus }_H\left(-1\right)\frac{h^3}{3!}\odot x_{\mathit{ii}-\mathit{gH}}^{\prime \prime \prime }\left({\eta }_k\right)$

and the mentioned error is defined as:

${\tau }_k=\frac{1}{}$