Theory And Numerical Approximations Of Fractional Integrals And Derivatives Direct

In conclusion, fractional integrals and derivatives have significant potential in modeling complex phenomena in various fields. The development of efficient numerical methods and applications in emerging fields are exciting areas of research.

The reverses the order of operations—it first differentiates integer-order, then integrates fractionally: \quad n-1 &lt

$$^C D^\alpha_a f(x) = I^n-\alpha_a \left[ \fracd^n fdx^n \right] (x) = \frac1\Gamma(n-\alpha) \int_a^x (x-t)^n-\alpha-1 f^(n)(t) , dt$$ which have limited physical interpretation.

Differentiate an integer-order derivative of a fractional integral. $$ a^RLD^\alpha t f(t) = \fracd^ndt^n \left( aI^n-\alpha t f(t) \right), \quad n-1 < \alpha \le n$$ Limitation: Requires fractional-order initial conditions, which have limited physical interpretation. \quad n-1 &lt