This paper presents a restricted SIR mathematical model to analyze the evolution of a contagious infectious disease outbreak (COVID-19) using available data. The new model focuses on two main concepts: first, it can present multiple waves of the disease, and second, it analyzes how far an infection can be eradicated with the help of vaccination. The stability analysis of the equilibrium points for the suggested model is initially investigated by identifying the matching equilibrium points and examining their stability. The basic reproduction number is calculated, and the positivity of the solutions is established. Numerical simulations are performed to determine if it is multipeak and evaluate vaccination's effects. In addition, the proposed model is compared to the literature already published and the effectiveness of vaccination has been recorded.

The integral-order derivative is not suitable where infinite variances are expected, and the fractional derivative manages to consider effects with more precision; therefore, we considered time-fractional Emden–Fowler-type equations and solved them using the rational homotopy perturbation method (RHPM). The RHPM method is based on two power series in rational form. The existence and uniqueness of the equation are proved using the Banach fixed-point theorem. Furthermore, we approximate the term h(z) with a polynomial of a suitable degree and then solve the system using the proposed method and obtain an approximate symmetric solution. Two numerical examples are investigated using this proposed approach. The effectiveness of the proposed approach is checked by representing the graphs of exact and approximate solutions. The table of absolute error is also presented to understand the method’s accuracy.

In this article, we investigate the solution of the fractional multidimensional Navier–Stokes equation based on the Caputo fractional derivative operator. The behavior of the solution regarding the Navier–Stokes equation system using the Sumudu transform approach is discussed analytically and further discussed graphically.

This research work is dedicated to solving the n-generalized Korteweg–de Vries (KdV) equation in a fractional sense. The method is a combination of the Sumudu transform and the Adomian decomposition method. This method has significant advantages for solving differential equations that are both linear and nonlinear. It is easy to find the solutions to fractional-order PDEs with less computing labor.

The COVID-19 pandemic touched about 200 countries of the globe. A strategy is given in this paper by considering a seven-compartment mathematical model with the inclusion of asymptomatic and symptomatic populations with waning immunity under the piecewise derivative concept of singular and nonsingular kernels, respectively. We investigate the dynamics of COVID-19 with the new framework of piecewise fractional derivative in the sense of Caputo and Atangana–Baleanu–Caputo fractional operators. The said analysis includes at least one solution and unique solution analysis with piecewise derivative in two subintervals. The proposed model is carried out by the approximate solution of piecewise numerical iterative technique of Newton polynomial. Each equation is written separately for the algorithm of numerical technique. Graphical representation for the proposed piecewise derivable model has been simulated with the available data at various global orders lying between 0 and 1 for both the subintervals. Such type of analysis will be very good and helpful for all those global problems where sudden or abrupt variation occurs.