Narrow Results

The analysis of nonlinear models and their complex geometrical shapes with globalized piecewise fractional operators for past heredity characteristics serve as the novelty of and motivation for this research. We explore the dynamics of a financial model in the aforementioned framework, using the well known and extensively applied numerical technique of the fractional piecewise Adams–Bashforth method. The effect of the employed parameters on price index, interest rate, investment demand, variation, and expenditure per investment from the saving amount along with the elasticity of the commercial markets demand are investigated. Various fractional orders are simulated in the sense of the Caputo and Atangana–Baleanu Caputo derivatives in the first and second subintervals, respectively. Moreover, the piecewise existence and uniqueness of the solution for the proposed model are established. The concept of Ulam–Hyers stability is applied to the obtained solution. This study will be helpful for beginners to complex geometrical and crossover dynamical problems in various physical and biological sciences.

In this paper, the Homotopy perturbation Laplace method is implemented to investigate the solution of fractional-order Whitham–Broer–Kaup equations. The derivative of fractional-order is described in Caputo’s sense. To show the reliability of the suggested method, the solution of certain illustrative examples are presented. The results of the suggested method are shown and explained with the help of its graphical representation. The solutions of fractional-order problems as well as integer-order problems are determined by using the present technique. It has been observed that the obtained solutions are in significant agreement with the actual solutions to the targeted problems. Computationally, it has been analyzed that the solutions at different fractional-orders have a higher rate of convergence to the solution at integer-order of the derivative. Due to the analytical analysis of the problems, this study can further modify the solution of other fractional-order problems.

One of the most severe and troubling diseases these days is COVID-$19$ pandemic. The COVID-$19$ pandemic’s dangerous effects are extremely rapid, and infection normally results in death within a few weeks. As a consequence, it is important to delve deeper into the complexities of this elusive virus. In this study, we propose a Caputo-based model for increasing COVID-19 strains. The memory effect and hereditary properties of the fractional variant for the model enable us to fully comprehend the dynamics of the model’s features. The existence of unique solution using the fixed-point theorem and Arzelá–Ascoli principle as well as the stability analysis of the model by means of Ulam–Hyer stability (UHS) and generalized Ulam–Hyer stability (GUHS) have been discussed. Furthermore, the parameters of the model are estimated using $3$ months data points chosen from Nigeria using the nonlinear least-squares technique. The best-suited parameters and the optimized Caputo fractional-order parameter $\alpha$ are obtained by running simulations for both models. The proposed model is shown to comprehend the dynamical behavior of the virus better than the integer-order version. In addition, to shed more light on the model’s characteristics, various numerical simulations are performed using an efficient numerical scheme.

These days the whole world is facing a serious problem of infectious maladies and how to control the endemic of these diseases. Testing correctly is one of the most important procedures in preventing the spread of infectious diseases, as incorrect testing can turn a susceptible person into an infected person. In this paper, we study the dynamics of imperfect testing and diagnostics of infectious diseases model by replacing the integer order derivative in the sense of fractional order Atangana–Baleanu operator coupled with Caputo operator. This AB-fractional operator is the generalization of classical derivative and gives more data of the factors of the nonlocal dynamical frameworks. Fixed point theorems have been used for the verification of existence results and Picard’s stability technique utilized for the stability study of a fractional order imperfect testing infectious disease (ITID) model. Finally, numerical computations are implemented for the fractional order ITID model to illustrate the results graphically.

Many approaches have been proposed in recent decades to represent the behaviors of certain complicated global problems appearing in a variety of academic domains. One of these issues is the multi-step behavior that some situations exhibit. Abdon and Seda devised new operators known as “piecewise operators” to deal with such problems. This paper presents the dynamics of the tumor–immune–vitamins model in the sense of a piecewise derivative. The piecewise operator considered here is composed of classical and Caputo operators. The existence and uniqueness of the solution with a piecewise derivative are presented with the aid of fixed point results. With the help of the Newton polynomial, a numerical scheme is presented for the examined model. The attained results are visualized through simulations for different fractional orders.

The aim of this paper is to study the dynamics of Dengue disease model using a novel piecewise derivative approach in the sense of singular and non-singular kernels. The singular kernel operator is in the sense of Caputo, whereas the non-singular kernel operator is the Atangana–Baleanu Caputo operator. The existence and uniqueness of a solution with piecewise derivative are examined for the aforementioned problem. The suggested problem’s approximate solution was found using the piecewise numerical iterative Newton polynomial approach. In terms of singular and non-singular kernels, a numerical scheme for piecewise derivatives has been established. The numerical simulation for the piecewise derivable problem under consideration is drawn using data for various fractional orders. This work makes the idea of piecewise derivatives and the dynamics of the crossover problem clearer.

This paper investigates the mathematical result of the nonlinear fractional scheme of equations representing the singular and non-singular thermoelastic system. The suggested method is a graceful combination of Shehu transformation with homotopy perturbation method and fractional derivative described with the Caputo operator. To show and confirm the effectiveness of the future method, I examined three cases and evaluated in terms of fractional-order the projected model. Furthermore, the physical behavior of the fractional-order solution achieved has been shown in graphs, and the mathematical model is proved to confirm the accuracy. The solutions achieved clarify that the suggested methodology is simple to apply, highly organized, and accurate to study the attitude of nonlinear system differential equations of arbitrary order arising in the related fields of engineering and science.