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Variable-order derivatives are the natural extension of ordinary as well as of fractional-order differentiations and integration, respectively. Numerous suggestions for fractional variable-order operators have been made in the literature over time. Therefore, this is the moment to shine a light on the variable-order fractional calculus, due to the fact that it accurately describes the mathematical underpinnings and emphasizing the modeling utility via using contemporary numerical techniques. This study focuses on investigating a fractional variable-order model of lymphatic filariasis infection using with Atangana–Beleanue–Caputo derivative. Our investigations have led to the development of newly refined results, focusing on both qualitative and numerical aspects of analysis. To achieve our research objectives, we employ the fixed point theorems of Banach and Krasnoselskii. These theorems serve as powerful tools, allowing us to establish results regarding the existence of solutions to the model. Additionally, for precise numerical simulations, we employ the fractional Euler’s method, a sophisticated computational technique that allows us to effectively simulate and interpret the results both numerically and graphically. These graphs illustrate distinct variable-orders, providing a comprehensive understanding of the model’s behavior under different conditions. Here, it should be kept in mind that we have select various continuous functions for variable to present our graphical illustration.

A model describing the transmission dynamics of cholera is considered in this article. The concerned model is investigated under the Caputo-Fabrizio fractal fractional derivative. The objective of this article is to study theoretical and numerical results for the model under our consideration. Classical fixed point approach is used to obtain sufficient conditions for the existence of solution to the proposed model. Adam’s Bashforth numerical method is utilized for the numerical interpretation of the suggested model. The considered technique is a powerful mathematical tool, that provides a numerical solution for the concerned problem. To discuss the transmission dynamics of the considered model, several graphical presentations are given.

In this work, a numerical scheme based on shifted Jacobi polynomials (SJPs) is deduced for variable-order fractional differential equations (FDEs). We find numerical solution of consider problem of fractional order. The proposed numerical scheme is based on operational matrices of variable-order differentiation and integration. To create the mentioned operational matrices for variable-order integration and differentiation, SJPs are used. Using the aforementioned operational matrices, we change the problem under consideration into matrix equation. The resultant matrix equation is solved by using Matlab, which executes the Gauss elimination method to provide the necessary numerical solution. The technique is effective and produced reliable outcomes. To determine the effectiveness of the suggested method, the results are compared to the outcomes of some other numerical procedure. Additional examples are included in this article to further clarify the process. For various scale levels and fractional-order values, absolute errors are also recorded.

For the stochastic heat equation (SHE), a very accurate spectral method is considered. To solve the SHE, we suggest using a shifted Legendre Gauss–Lobatto collocation approach in combination with a shifted Legendre Gauss–Radau collocation technique. A comprehensive theoretical formulation is offered, together with numerical examples, to demonstrate the technique’s performance and competency. The scheme’s superiority in tackling the SHE is demonstrated.

The susceptible, exposed, infected, quarantined and vaccinated (SEIQV) population is accounted for in a mathematical model of COVID-19. This model covers the therapy for diseased people as well as therapeutic measures like immunization for susceptible people to enable understanding of the dynamics of the disease's propagation. Each of the equilibrium points, i.e., disease-free and endemic, has been proven to be globally asymptotically stable under the assumption that R0 is smaller or larger than unity, respectively. Although vaccination coverage is high, the basic reproduction number depends on the vaccine's effectiveness in preventing disease when R0 > 0. The Jacobian matrix and the Routh-Hurwitz theorem are used to derive the aforementioned analysis techniques. The results are further examined numerically by using the standard second-order Runge-Kutta (RK2) method. In order to visualize the global dynamics of the aforementioned model, the proposed model is expanded to examine some piecewise fractional order derivatives. We may comprehend the crossover behavior in the suggested model's illness dynamics by using the relevant derivative. To numerical present the results, we use RK2 method. [ABSTRACT FROM AUTHOR]

In this work, an epidemic model of a susceptible, exposed, infected and recovered SEIRtype is established for the distinctive dynamic compartments and epidemic characteristics of COVID-19 as it spreads across a population with a heterogeneous rate. The proposed model is investigated using a novel approach of fractional calculus known as piecewise derivatives. The existence theory is demonstrated through the establishment of sufficient conditions. In addition, result related to Hyers-Ulam stability is also derived for the considered model. A numerical method based on modified Euler procedure is also constructed to simulate the approximate solutions of the proposed model by employing various values of fractional orders. We testified the numerical results by using real available data of Japan. In addition, some results for the SEIR-type model are also presented graphically using the stochastic process, and the obtained results are discussed. [ABSTRACT FROM AUTHOR]

Recently, the area devoted to mathematical epidemiology has attracted much attention. Mathematical formulations have served as models for various infectious diseases. In this regard, mathematical models have also been used to study COVID-19, a threatening disease in present time. This research work is devoted to consider a SEIR (susceptible-exposed-infectious-removed) type mathematical model for investigating COVID-19 alongside a new scenario of fractional calculus. We consider piece-wise fractional order derivatives to investigate the proposed model for qualitative and computational analysis. The results related to the qualitative analysis are studied via using the tools of fixed point approach. In addition, the computational analysis is performed due to a significance of simulation to understand the transmission dynamics of COVID-19 infection in the community. In addition, a numerical scheme based on Newton's polynomials is established to simulate the approximate solutions of the proposed model by using various fractional orders. Additionally, some real data results are also shown in comparison to the numerical results. [ABSTRACT FROM AUTHOR]

Breast cancer ranks among the most prevalent malignancies affecting the female population and is a prominent contributor to cancer-related mortality. Mathematical modeling is a significant tool that can be employed to comprehend the dynamics of breast cancer progression and dissemination and to formulate novel therapeutic approaches. This paper introduces a mathematical model of breast cancer that utilizes the Caputo–Fabrizio fractal-fractional derivative. The aim is to elucidate and comprehend the intricate dynamics governing breast cancer cells and cytotoxic T lymphocytes in the context of the fractional derivative. The derivative presented herein offers a broader perspective than the conventional derivative, as it incorporates the intricate fractal characteristics inherent in the process of tumor proliferation. The significance of this study lies in its contribution to a novel mathematical model for breast cancer, which incorporates the fractal characteristics of tumor development. The present model possesses the capability to investigate the impacts of diverse treatment strategies on the proliferation of breast cancer, as well as to formulate novel treatment strategies that exhibit enhanced efficacy. [ABSTRACT FROM AUTHOR]

The flow of nanoparticles has many dynamic applications in solar systems, the thermal sciences, heating and cooling mechanisms, energy-producing sources, and many other disciplines. Following invaluable applications and inspiration, the current study is carried out by focusing on the thermal efficiency of spherical nanoparticles of A l 2 O 3 - C u in ethylene glycol through the non-Newtonian Maxwell fluid flow model. In the current analysis, the inclined stretching sheet equipped with suction effects is embedded in porous media, including the magnetohydrodynamics effects. The mathematical representation of the proposed problem is given a form in terms of partial differential equations. Then, this system is reduced to a system of ordinary differential equations by using appropriate similarity variable formulations. The obtained model is solved with bvp4c solver for the graphical and tabular aspects of the velocity field, the temperature field along with the skin friction coefficient, and the Nusselt number. The main outcomes of the results indicate that fluid velocity increases with increasing values for the angle of inclination, Maxwell fluid parameter, and suction parameter; however, the reverse process is seen for the porous medium parameter and magnetic field parameter. Moreover, the fluid temperature rises for augmenting values of the magnetic field parameter and porous medium parameter, whereas the opposite behavior is seen against the suction parameter. The present results are compared with the published ones and it is concluded that there is excellent agreement between them, which endorses the validity and accuracy of the current study. [ABSTRACT FROM AUTHOR]

Nanotechnology is well-known for its versatile and general thermal transport disciplines, which are used in semiconductors, spacecraft, bioengineering, functional electronics, and biosensors. As a result, process optimization has attracted the interest of scientists and technologists. The main aim of the current analysis is to explore the enhancement of energy/heat transfer via the dispersion of cylindrical-shaped nanoparticles of alumina and copper in ethylene glycol as a base fluid using a non-Newtonian Maxwell fluid model. In the current study, the effects of solar radiation, plate suction, and magnetohydrodynamics on a Maxwell hybrid nanofluid are encountered. The flow is induced by linearly stretching a sheet angled at ξ = π / 6 , embedded in a porous space. The proposed problem is converted into a mathematical structure in terms of partial differential equations and then reduced to ordinary differential equations by using appropriate similarity variables. In the similarity solution, all the curves for the velocity field and temperature distribution remain similar, which means that the symmetry between the graphs for the velocity and temperature remains the same. Therefore, there is a strong correlation between similarity variables and symmetry. The obtained model, in terms of ordinary differential equations, is solved using the built-in numerical solver bvp4c. It is concluded that more nanoparticles in a fluid can make it heat up faster, as they are typically better at conducting heat than the fluid itself. This means that heat is transferred more quickly, raising the temperature of the fluid. However, more nanoparticles can also slow the flow speed of the fluid to control the boundary layer thickness. The temperature field is enhanced by increasing the solar radiation parameter, the magnetic field parameter, and the porous medium parameter at an angle of   ξ = π / 6 , which serves the purpose of including radiation and the Lorentz force. The velocity field is decreased by increasing the values of the buoyancy parameter and the suction parameter effects at an angle of   ξ = π / 6 . The current study can be used in the improvement of the thermal efficiency of nanotechnological devices and in renewable energy sources to save energy in the energy sector. The present results are compared with the published ones, and it is concluded that there is excellent agreement between them, which endorses the validity and accuracy of the current study. [ABSTRACT FROM AUTHOR]

Fuzzy-interval valued functions (FIVFs) are the generalization of interval valued and real valued functions, which have a great contribution to resolve the problems arising in the theory of interval analysis. In this article, we elaborate the convexities and pre-invexities in aspects of FIVFs and investigate the existence of fuzzy fractional integral operators (FFIOs) having a generalized Bessel–Maitland function as their kernel. Using the class of convexities and pre-invexities FIVFs, we prove some Hermite–Hadamard (H-H) and trapezoid-type inequalities by the implementation of FFIOs. Additionally, we obtain other well known inequalities having significant behavior in the field of fuzzy interval analysis. [ABSTRACT FROM AUTHOR]

In this article, we discuss the existence and uniqueness results for mix derivative involving fractional operators of order β∈(1,2) and γ∈(0,1). We prove some important results by using integro-differential equation of pantograph type. We establish the existence and uniqueness of the solutions using fixed point theorem. Furthermore, one application is likewise given to represent our fundamental results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence-uniqueness, and Ulam Hyers stability (UHS) of solutions to a fractional-order pantograph differential equation (FOPDE) with two Caputo operators. Banach's fixed point (BFP) and Leray-alternative Schauder's are used to prove the existenceuniqueness of solutions. In addition, we discuss and demonstrate various types of Ulam-stability for our problem. Finally, an example is provided for clarity. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a generalization of the Kantrovich–Stancu-type Szasz operator asymmetry with hybrid families of special polynomials. Additionally, we construct certain positive linear operators together with the Sheffer–Appell polynomial sequences and then obtain the properties of convergence and the order of convergence, which is symmetric to these operators. For applications, we consider certain explicit examples including mixed-type special polynomials. [ABSTRACT FROM AUTHOR]

The present investigations are associated with designing Morlet wavelet neural network (MWNN) for solving a class of susceptible, infected, treatment and recovered (SITR) fractal systems of COVID-19 propagation and control. The structure of an error function is accessible using the SITR differential form and its initial conditions. The optimization is performed using the MWNN together with the global as well as local search heuristics of genetic algorithm (GA) and active-set algorithm (ASA), i.e., MWNN-GA-ASA. The detail of each class of the SITR nonlinear COVID-19 system is also discussed. The obtained outcomes of the SITR system are compared with the Runge-Kutta results to check the perfection of the designed method. The statistical analysis is performed using different measures for 30 independent runs as well as 15 variables to authenticate the consistency of the proposed method. The plots of the absolute error, convergence analysis, histogram, performance measures, and boxplots are also provided to find the exactness, dependability and stability of the MWNN-GA-ASA. [ABSTRACT FROM AUTHOR]