One of the control measures available that are believed to be the most reliable methods of curbing the spread of coronavirus at the moment if they were to be successfully applied is lockdown. In this paper a mathematical model of fractional order is constructed to study the significance of the lockdown in mitigating the virus spread. The model consists of a system of five nonlinear fractional-order differential equations in the Caputo sense. In addition, existence and uniqueness of solutions for the fractional-order coronavirus model under lockdown are examined via the well-known Schauder and Banach fixed theorems technique, and stability analysis in the context of Ulam–Hyers and generalized Ulam–Hyers criteria is discussed. The well-known and effective numerical scheme called fractional Euler method has been employed to analyze the approximate solution and dynamical behavior of the model under consideration. It is worth noting that, unlike many studies recently conducted, dimensional consistency has been taken into account during the fractionalization process of the classical model.

This paper presents a class of implicit pantograph fractional differential equation with more general Riemann-Liouville fractional integral condition. A certain class of generalized fractional derivative is used to set the problem. The existence and uniqueness of the problem is obtained using Schaefer’s and Banach fixed point theorems. In addition, the Ulam-Hyers and generalized Ulam-Hyers stability of the problem are established. Finally, some examples are given to illustrative the results.

In this paper, we study and investigate an interesting Caputo fractional derivative and Riemann–Liouville integral boundary value problem (BVP): ^c $D_{0^{+}}^{q}u\left(t\right)=f(t,u\left(t\right)),t\in [0,T]$ , ${\displaystyle u^{\left(k\right)}\left(0\right)={\xi }_k,u\left(T\right)=\sum _{i=1}^m{{\beta }_i}_{RL}{I}_{0^{+}}^{p_i}u\left({\eta }_i\right),}$ where $n-1<q<n,n\ge 2,m,n\in \mathbb{N}$ , ${\xi }_k,{\beta }_i\in \mathbb{R}$ , $k=0,1,\dots ,n-2$ , $i=1,2,\dots ,m$ , and ^c $D_{0^{+}}^q$ is the Caputo fractional derivatives, $f:[0,T]\times C\left(\right[0,T],E)\to E$ , where E is the Banach space. The space E is chosen as an arbitrary Banach space; it can also be $\mathbb{R}$ (with the absolute value) or $C\left(\right[0,T],\mathbb{R})$ with the supremum-norm. _RL $I_{0^{+}}^{p_i}$ is the Riemann–Liouville fractional integral of order $p_i>0$ , ${\eta }_i\in (0,T)$ , and ${\displaystyle \sum _{i=1}^m{\beta }_i{\eta }_i^{p_i+n-1}\frac{\Gamma \left(n\right)}{\Gamma (n+p_i)}\ne T^{n-1}}$ . Via the fixed point theorems of Krasnoselskii and Darbo, the authors study the existence of solutions to this problem. An example is included to illustrate the applicability of their results.

In recent years, the availability of advanced computational techniques has led to a growing emphasis on fractional-order derivatives. This development has enabled researchers to explore the intricate dynamics of various biological models by employing fractional-order derivatives instead of traditional integer-order derivatives. This paper proposes a Caputo-Fabrizio fractional-order cholera epidemic model. Fixed-point theorems are utilized to investigate the existence and uniqueness of solutions. A recent and effective numerical scheme is employed to demonstrate the model's complex behaviors and highlight the advantages of fractional-order derivatives. Additionally, a sensitivity analysis is conducted to identify the most influential parameters.

This study investigates the solutions of an impulsive fractional differential equation incorporated with a pantograph. This work extends and improves some results of the impulsive fractional differential equation. A differential equation of an impulsive fractional pantograph with a more general anti-periodic boundary condition is proposed. By employing the well-known fixed point theorems of Banach and Krasnoselskii, the existence and uniqueness of the solution of the proposed problem are established. Furthermore, two examples are presented to support our theoretical analysis.