A Cotangent Fractional Derivative with the Application - Lakhlifa Sadek (Fractal and Fractional, 2023)
In this work, we present a new type of fractional derivatives (FD) involving exponential cotangent function in their kernels called Riemann–Liouville <semantics>Dσ,γ</semantics> and Caputo cotangent fractional derivatives <semantics>CDσ,γ</semantics>, respectively, and their corresponding integral <semantics>Iσ,γ</semantics>. The advantage of the new fractional derivatives is that they achieve a semi-group property, and we have special cases; if <semantics>γ=1</semantics> we obtain the Riemann–Liouville FD (RL-FD), Caputo FD (C-FD), and Riemann–Liouville fractional integral (RL-FI). We give some theorems and lemmas, and we give solutions to linear cotangent fractional differential equations using the Laplace transform of the <semantics>Dσ,γ</semantics>, <semantics>CDσ,γ</semantics> and <semantics>Iσ,γ</semantics>. Finally, we give the application of this new type on the SIR model. This new type of fractional calculus can help other researchers who still work on the actual subject.