**Riemann-Liouville definition:**

$$aD\_t^a f(t) = \frac{1}{\Gamma(n-a)} \frac{d^n}{dt^n} \Big|\_{a}^{t} \frac{f(\tau)}{(t-\tau)^{a-n+1}} d\tau \tag{5}$$

where *n* ∈ þ. The condition for above equation is *n* � 1< *α*<*n*, *Γ*ð Þ*:* is called Gamma function, which its defination is given by:

$$I^\circ(X) = \bigcap\_{0}^{\infty} z^{X-1} e^{-z} dz \tag{6}$$

Laplace transform of differ-integral operator aD<sup>α</sup> t � � is given by expected form:

$$L\left[aD\_t^af(t)\right] = \int\_0^\infty e^{-st} aD\_t^af(t)dt\tag{7}$$

$$L\left[aD\_t^af(t)\right] = s^aF(s) - \sum\_{m=0}^{n-1} s(-1)^j 0D\_t^{a-m-1}f(t) \tag{8}$$

Where *F s*ðÞ¼ *L ft* f g ð Þ is the normal Laplace transformation and *n* is an integer number that satisfies *n* � 1< *α*≤*n* and *s* ¼ *jw* denotes the Laplace transform variable.
