- #1
batoulsy
- 1
- 0
What is the transformation used
Is there any explanation for :
$$
\frac{\mathit{\lambda}}{\mathit{\Gamma}{\mathrm{(}}{q}{\mathrm{)}}}\mathop{\int}\limits_{t0}\limits^{t}{{\mathrm{(}}{t}\mathrm{{-}}{s}{\mathrm{)}}^{{q}\mathrm{{-}}{1}}}{x}_{0}\mathrm{(}s\mathrm{)}ds
$$
How did become like this
$$
\frac{{x}_{0}\hspace{0.33em}\mathit{\lambda}}{\mathit{\Gamma}{\mathrm{(}}{q}{\mathrm{)}}\mathit{\Gamma}{\mathrm{(}}{q}{\mathrm{)}}}\mathop{\int}\limits_{0}\limits^{1}{{\mathrm{(}}{t}\mathrm{{-}}{t}_{0}}{\mathrm{)}}^{{2}{q}\mathrm{{-}}{1}}{\mathrm{(}}{1}\mathrm{{-}}\mathit{\sigma}{\mathrm{)}}^{{q}\mathrm{{-}}{1}}{\mathit{\sigma}}^{{q}\mathrm{{-}}{1}}{d}\mathit{\sigma}
$$
Where:
$$
{x}_{0}{\mathrm{(}}{t}{\mathrm{)}}\mathrm{{=}}\frac{{x}_{0}{\mathrm{(}}{t}\mathrm{{-}}{t}_{0}{\mathrm{)}}^{{q}\mathrm{{-}}{1}}}{\mathit{\Gamma}{\mathrm{(}}{q}{\mathrm{)}}}
$$
Is there any explanation for :
$$
\frac{\mathit{\lambda}}{\mathit{\Gamma}{\mathrm{(}}{q}{\mathrm{)}}}\mathop{\int}\limits_{t0}\limits^{t}{{\mathrm{(}}{t}\mathrm{{-}}{s}{\mathrm{)}}^{{q}\mathrm{{-}}{1}}}{x}_{0}\mathrm{(}s\mathrm{)}ds
$$
How did become like this
$$
\frac{{x}_{0}\hspace{0.33em}\mathit{\lambda}}{\mathit{\Gamma}{\mathrm{(}}{q}{\mathrm{)}}\mathit{\Gamma}{\mathrm{(}}{q}{\mathrm{)}}}\mathop{\int}\limits_{0}\limits^{1}{{\mathrm{(}}{t}\mathrm{{-}}{t}_{0}}{\mathrm{)}}^{{2}{q}\mathrm{{-}}{1}}{\mathrm{(}}{1}\mathrm{{-}}\mathit{\sigma}{\mathrm{)}}^{{q}\mathrm{{-}}{1}}{\mathit{\sigma}}^{{q}\mathrm{{-}}{1}}{d}\mathit{\sigma}
$$
Where:
$$
{x}_{0}{\mathrm{(}}{t}{\mathrm{)}}\mathrm{{=}}\frac{{x}_{0}{\mathrm{(}}{t}\mathrm{{-}}{t}_{0}{\mathrm{)}}^{{q}\mathrm{{-}}{1}}}{\mathit{\Gamma}{\mathrm{(}}{q}{\mathrm{)}}}
$$