Fractional Derivatives and Integrals

Definitions

Fractional derivatives and integrals are generalizations of the classical derivatives and integrals to non-integer orders. They are widely used in various fields such as physics, engineering, and finance. The most common definitions of fractional derivatives and integrals are the Riemann-Liouville and Caputo definitions. We give a brief introduction to their definitions in this section.

Standard interval

For and , the left and right Riemann-Liouville fractional integrals are respectively defined as

Moreover, for real with , the Riemann-Liouville fractional derivative are defined as

Furthermore, the Caputo fractional derivatives of order are defined as

General interval

For general interval , the corresponding left-side and right-side Riemann-Liouville fractional integrals of order are defined as

for , and

for . The left-side and right-side Riemann-Liouville fractional derivatives of order are defined as

for , and

for .

Similarly, the left-side and right-side Caputo fractional derivatives of order are defined as

and

Properties

Relationship between Riemann-Liouville and Caputo types.

The Riemann-Liouville and Caputo fractional derivatives are linked by the following relations:

and

Inverse operator.

The inverse operator of the Riemann-Liouville fractional integral is the Riemann-Liouville fractional derivative, and vice versa, that is

and

Fractional Intergration by Parts.

The fractional integration by parts formula states that for and , we have

and

Superposition.

Assuming and , we have

for , and similarly for the right-side Riemann-Liouville fractional derivatives.