In this work, we present a new type of fractional derivatives (FD) involving exponential cotangent function in their kernels called Riemann–Liouville $D^{\sigma ,\gamma }$ and Caputo cotangent fractional derivatives ${}^C{D}^{\sigma ,\gamma }$, respectively, and their corresponding integral $I^{\sigma ,\gamma }$. The advantage of the new fractional derivatives is that they achieve a semi-group property, and we have special cases; if $\gamma =1$ 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 $D^{\sigma ,\gamma }$, ${}^C{D}^{\sigma ,\gamma }$ and $I^{\sigma ,\gamma }$. 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.

We present a new numerical approach to solving the fractional differential Riccati equations numerically. The approach—called the Mittag-Leffler–Galerkin method—comprises the finite Mittag-Leffler function and the Galerkin method. The error analysis of the method was studied. As a result, we present two theorems by which the error can be bounded. In addition to error analysis, the residual correction method, which allows us to estimate the error and obtain new approximate solutions, is also presented. To show how the method is applied, and the efficiency of the proposed method, some test examples were considered. When the numerical results obtained were examined, it was found that while the method achieves better results than some of the known methods in the literature, it also achieves results that are similar to those of others of the known methods.