A Cotangent Fractional Derivative with the Application

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In this work, we present a new type of fractional derivatives (FD) involving exponential cotangent function in their kernels called Riemann–Liouville Dσ,γ and Caputo cotangent fractional derivatives CDσ,γ, respectively, and their corresponding integral Iσ,γ. The advantage of the new fractional derivatives is that they achieve a semi-group property, and we have special cases; if γ=1 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 Dσ,γ, CDσ,γ and Iσ,γ. 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.