#### Project Title

Fractional derivative and Midpoint rules

#### Academic department under which the project should be listed

CSM - Mathematics

#### Research Mentor Name

Somayeh Mashayekhi

#### Abstract (300 words maximum)

The fractional derivative has a long history in mathematics dating back further than integer-order derivative, but it did not gain traction until recently. In recent years, fractional derivative has been shown to describe physical and biological phenomena better than integer-order derivatives. Fractional derivative generalize integer derivative to an arbitrary (non-integer) order. While the integer derivative is defined at a point, the fractional derivative is determined on an interval, and there is an integral involved in the definition of the fractional derivative. To develop the computation model in fractional calculus, we need to find an approximation of the integral. There are various definitions of fractional derivative. The widely used definition of a fractional derivative is the Caputo definition. In this research, we have used the Midpoint rules to find the approximation of the Caputo fractional derivative.

Fractional derivative and Midpoint rules

The fractional derivative has a long history in mathematics dating back further than integer-order derivative, but it did not gain traction until recently. In recent years, fractional derivative has been shown to describe physical and biological phenomena better than integer-order derivatives. Fractional derivative generalize integer derivative to an arbitrary (non-integer) order. While the integer derivative is defined at a point, the fractional derivative is determined on an interval, and there is an integral involved in the definition of the fractional derivative. To develop the computation model in fractional calculus, we need to find an approximation of the integral. There are various definitions of fractional derivative. The widely used definition of a fractional derivative is the Caputo definition. In this research, we have used the Midpoint rules to find the approximation of the Caputo fractional derivative.