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Fish farming, a potential activity for the production of protein with high nutritional value, has emerged as a substantial source of income in the modern world. The degradation of natural materials uses oxygen to reduce them below the life-support stage. A huge area with an excessive amount of water swamp for oxygen exchange makes for the ideal ecosphere for fish farming, yet the mussel population may grow swiftly in a really modest water swapping farm producing a lot of feed. In this study, we examine the dynamics and chaotic behavior of a fish farm model containing populations of nutrients and mussels with different kernels insight of fractal-fractional operator. We examined the positively invariant area as well as demonstrate positive, bounded solutions of the model. We also show the equilibrium states for the occurrence and extinction of species. We prove the existence and uniqueness of positive solutions through fixed-point theorems and verified the Ulam–Hyers stability for proposed model. To study the impact of the fractional operator through computational simulations, results are generated employing a two-step Lagrange polynomial in the generalized version for the various kernels, which include the power-law and Mittag–Leffler kernel. We develop some numerical simulations to test our theoretical conclusions with chaotic behavior at different fractional-order values.

The aim of this study is to analyze and investigate the COVID-19 transmission with effect of symptomatic and asymptomatic in the community. Mathematical model is converted into fractional order with the help of fractal fractional definition. The proposed fractional order system is investigated qualitatively as well as quantitatively to identify its stable position. Local stability of the COVID-19 system is verified and test the system is tested with flip bifurcation. Also the system is investigated for global stability using Lyapunov first and second derivative functions to see its rate of spread after recovery. The existence, boundedness and positivity of the COVID-19 are checked which are the key properties for such of type of epidemic problem to identify reliable findings. Effect of global derivative is demonstrated to verify its rate of effects according to their sub compartments to identify in which rate the symptomatic and asymptomatic transmission occurs. Solutions for fractional order system are derived with the help of advanced tool fractal fractional operator with generalized mittag-leffler kernel for different fractional values. Simulations are carried out to see symptomatic as well as asymptomatic effects of COVID-19 in the worldwide using MATLAB Coding. They show the actual behavior of COVID-19 especially for asymptomatic measures which will be helpful in early detection, also which will be helpful to understand the outbreak of COVID-19 as well as for future prediction and better control strategies.

Fractional calculus (FC) is a valuable tool in the modeling of many phenomena, and it has become a topic of great interest in science and engineering. This mathematical tool has proved its efficiency in modeling the intermediate anomalous behaviors observed in different physical phenomena. The Meissner–Ochsenfeld effect describes the levitation of superconductors in a nonuniform magnetic field if they are cooled below critical temperature. This paper presents analytical solutions of the fractional London equation that describes the Meissner–Ochsenfeld effect considering the Liouville–Caputo, Caputo–Fabrizio–Caputo, Atangana–Baleanu–Caputo, fractional conformable derivative in Liouville–Caputo sense and Atangana–Koca–Caputo fractional-order derivatives. Numerical simulations were obtained for different values of the fractional-order.

In this study, we consider a mathematical model for the disease dynamics in both prey and predator by considering the Susceptible–Infected–Recovered–Susceptible (SIRS) model with the prey–predator Lotka–Volterra differential equations. Carrying capacity, predation, migration, and immunity loss are also taken into account for both species. Using the law of mass action, the physical model is transformed into a nonlinear coupled system of ODEs. The classical/integer order ODE system is then generalized through the fractal-fractional differential operators of power law and Mittag-Leffler kernels. For the model under consideration, we additionally check for positivity, boundedness, the basic reproduction number, and equilibrium points. Graphical results are obtained by use of a numerical approach, and the existence and uniqueness of the solution are also established theoretically. The graphical solutions has been displayed via 2D and 3D phase plots. It has been shown that when the fractional order reduces, the amplitude of the chaotic attractor dynamics shrinks, along with the range of limit cycles and periodic trajectories while a drop in the fractal dimension parameter causes an increase in the time period of the chaotic attractor dynamics. This study not only improves the understanding of epidemic breakouts in predator–prey systems, but also highlights the efficiency of fractal-fractional calculus in ecological modeling.

In this paper, by using two local fractional integral operators with Mittag-Leffler kernel, we construct some new Hermite–Hadamard type and Hermite–Hadamard–Fejér type local fractional integral inequalities for generalized preinvex functions on Yang’s fractal sets. Finally, three examples are proposed to illustrate the main results. Meanwhile, we also proposed an identity and a bounded estimate for the moments of random variables to illustrate the applications of the results.

Dealing with nonsingular kernels is not an easy task due to their restrictions at origin. In this short paper, we suggest an extension of the fractional operator involving the Mittag-Leffler kernel which admits integrable singular kernel at the origin. New solutions of the related differential equations were reported together with some perspectives from the modelling viewpoint.

In this study, we model the fractal-fractional system of the Computer virus problem using the Atangana–Baleanu operator. Moreover, we have presented the existence and the uniqueness of the results under applying the Schauder fixed point and Banach fixed theorems. We have used the Atangana–Toufik technique to obtain the approximate solutions by choosing various values of orders. Different values of fractal-fractional orders along with different amounts of initial conditions are selected to examine the performance of the suggested numerical method on the new fractal-fractional system. Also, graphs in different dimensions are presented to exhibit the solutions, clearly.

This paper will introduce novel techniques for a fractional-order model of the human liver involving the Atangana–Baleanu, Atangana–Toufik, and the Fractal fractional method with the nonsingular kernel. These techniques give more accurate and appropriate results. Existence and uniqueness have been developed with the help of fixed-point theory results. Numerical simulations are done from the developed algorithm of the model to elaborate the effect of fractional values and justify the theoretical results. Such kind of analysis will be useful for further investigation of epidemic diseases, and also provide a better understanding of disease dynamics to overcome the effect of disease in a community.

The purpose of this work is to investigate the dynamical behavior of a measles disease model adopting fractional and fractal–fractional operators with Mittag-Leffler kernel using two distinct numerical algorithms. First, we discuss the measles model in a fractional framework with Atangana–Baleanu–Caputo derivative and examine some fundamental mathematical assumptions of the considered model. We implement fixed-point theory to explore the existence and uniqueness of model solutions. Next, we apply the novel fractal–fractional concept with Atangana–Baleanu derivative to the measles model and reveal that the model has unique solution. We present the approximate results for the proposed models with graphical illustrations. The results are presented with various choices of fractal and fractional orders. The system behavior to various biological parameters is also investigated. In addition, we compare the considered operators using novel numerical schemes that take into account different $\varrho$ values.

The solutions of systems of fractional differential equations depend on the type of the fractional derivative used in the system. In this paper, we present in closed forms the solutions of linear systems involving the modified Atangana–Baleanu derivative that has been introduced recently. For the nonlinear systems, we implement a numerical scheme based on the collocation method to obtain approximate solutions. The applicability of the results is tested through several examples. We emphasize here that certain systems with the Atangana–Baleanu derivative admit no solutions which is not the case with the modified derivative.

A random arbitrary-order mathematical system is investigated via the global and non-singular kernel of Atangana–Baleanu in the sense of Caputo $({𝒜}{ℬ}{𝒞})$ derivative in this study where the proposed problem is divided into four general compartments for the explanation. To show the existing result, the Krasnosilkii’s theorem from the theory of fixed points is used, whereas the well-known Banach theorem is utilized in order to show that the solution is unique to the proposed problem. Furthermore, by using the idea of Hyers–Ulam (UH) stability, the generalized problem is perturbed little for the purpose of checking its stability. The numerical solution is evaluated by applying the Adams–Bashforth iterative techniques. The numerical examples derived are tested in order to illustrate the established outcomes along with the numerical simulation to demonstrate the verification of the results obtained. The dynamics of every compartment is examined on different non-integer order ${𝒷}$ and by choosing arbitrary time $t$ by the taken approximate solution employing the AB numerical technique. Ultimately, the total continuous spectrum on the dynamics of each quantity in any arbitrary order lying between any of the two natural values, namely 0 and 1, has been achieved based on the investigated analyses.

A fast higher-order scheme is established for solving inhomogeneous time-fractional generalized Burgers’ equation. The time-fractional operator is taken as the modified operator with the Mittag-Leffler kernel. Through stability analysis, it has been demonstrated that the proposed numerical approach is unconditionally stable. The convergence of the numerical method is analyzed theoretically using von Neumann’s method. It has been proved that the proposed numerical method is fourth-order convergent in space and second-order convergent in time in the $L_2$-norm. The scheme’s proficiency and effectiveness are examined through two numerical experiments to validate the theoretical estimates. The tabular and graphical representations of numerical results confirm the high accuracy and versatility of the scheme.

A new definition of fractional differentiation of nonlocal and non-singular kernels has recently been developed to overcome the shortcomings of the traditional Riemann–Liouville and Caputo fractional derivatives. In this study, the dynamic behaviors of the fractional financial chaotic model have been investigated. Singular and non-singular kernel fractional derivatives are used to examine the proposed model. To solve the financial chaotic model with nonlocal operators, the fractional Adams–Bashforth method (ABM) is applied based on Lagrange polynomial interpolation (LPI). The existence and uniqueness of the solution of the model can be demonstrated using fixed point theory and nonlinear analysis. Further, the error analysis of the present method and Ulam–Hyers stability of the considered model have also been included. Obtained numerical simulations reveal that the model based on three different fractional derivatives shows various chaotic behaviors that may be useful in a practical sense which may not be observed in the integer case.

In an emergency, fear can spread among crowds through one-on-one encounters, with negative societal consequences. The purpose of this research is to create a novel theoretical model of fear (panic) spread in the context of epidemiology during an emergency using the fractal fractional operator. For quantitative analysis, the system’s boundedness and positivity are checked. According to the Arzela Ascoli theorem, the model is completely continuous. As a result of the discovery of Schauder’s fixed point, it has at least one solution. The existence and uniqueness of the concerned solution have been examined using the fixed point theory technique. Numerical simulations are used to demonstrate the accuracy of the proposed techniques using a generalized form of Mittag-Leffler kernel with a fractal fractional operator. Finally, simulations are utilized to represent the spread of group emotional contagion (spontaneous spread of emotions and related behaviors) dynamically.

In this research, the goal is to formulate a mathematical model that predicts the transference mechanisms of Ebola virus (EBOV) and Nipah virus (NIV) infections. They utilize fractional-order derivatives to describe the behavior of the viruses, particularly focusing on the difference in disease manifestation between humans and fruit bats, the putative natural reservoirs. Additionally, they employ fixed-point theory to analyze the qualitative aspects of their model. Also, we investigate the stability of the model using Ulam-Hyers-type results. Furthermore, we utilize the fractional Atangana–Baleanu integral and the Adams–Milton numerical method using MATLAB to provide graphical representations and insights into the behavior of the viruses under various conditions of transference.

This work provides a new fuzzy variable fractional COVID-19 model and uses a variable fractional operator, namely, the fuzzy variable Atangana–Baleanu fractional derivatives in the Caputo sense. Next, we explore the proposed fuzzy variable fractional COVID-19 model using the fixed point theory approach and determine the solution’s existence and uniqueness conditions. We choose an appropriate mapping and with the help of the upper/lower solutions method. We prove the existence of a positive solution for the proposed fuzzy variable fractional COVID-19 model and also obtain the result on the existence of a unique positive solution. Moreover, we discuss the generalized Hyers–Ulam stability and generalized Hyers–Ulam–Rassias stability. Further, we investigate the results on maximum and minimum solutions for the fuzzy variable fractional COVID-19 model.