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.

This study investigates the chaotic behavior and stability of a nonlinear fractional-order financial system utilizing the Caputo fractional derivative. Initially, we formulate the nonlinear fractional-order financial model and establish the problem framework. Next, we prove the existence and uniqueness of solutions by applying the Banach and Schauder fixed-point theorems to the proposed system. Additionally, we analyze the Ulam–Hyers stability and discuss other significant findings related to the system’s stability. To simulate the proposed model, we develop numerical schemes based on fractional calculus, employing Lagrange polynomial interpolation. Finally, we present the numerical simulations to validate the theoretical results, highlighting the significant impact of fractional-order derivatives on the system’s behavior.

The primary objective of this study is to analyze the Hyers–Ulam stability of fractional derivatives for the Black–Scholes model, involving two underlying asset systems and utilizing Caputo fractional derivatives. We employ a fixed-point approach to examine the existence and uniqueness of solutions and to investigate the Hyers–Ulam stability of the given problem. Additionally, we analyze the graphical behavior of the obtained results, demonstrating that the analytical method is highly efficient and delivers precise results for determining approximate numerical solutions. The findings highlight the significant role of fractional approaches in studying nonlinear systems of scientific and physical importance. Furthermore, the graphical analysis, considering various fractional orders and parameter values, unveils new insights and intriguing phenomena associated with the Black–Scholes model.

In this paper, we consider the nonlinear cauchy problem

In this paper, we consider solving a system of fractional differential equations with nonlinear source term. The nonlocal source term is a convolution of an exponential with a kernel. The solved system is able to be seen as a generalization of a number of varied systems of equations with solutions that blows up in a finite time. Under initial conditions, the nontrivial global solutions blow-up and certain assumptions are established by the weak formulation of the problem with some estimation inequalities.

Wireless sensor networks have been extensively studied for their potential applications in both civil and military domains. However, due to their inherent resource constraints, sensor nodes are highly susceptible to cyber threats, particularly worm attacks, which pose significant security challenges. This study explores the dynamics of potential worm attacks within wireless sensor networks using a compartmental epidemic model. The model analyzes the temporal evolution of worm propagation while effectively capturing both spatial and temporal aspects. We determine the reproduction number and equilibrium of the system, with the local stability assessed using the Jacobian matrix. The linear growth and Lipschitz conditions are used to establish the existence and uniqueness of the solution. Furthermore, the Hyers–Ulam stability of the proposed model is also evaluated within the context of the fractal-fractional operator. Finally, a numerical method is developed to investigate the dynamic behavior of the wireless sensor network model under fractal-fractional orders, providing valuable insights into its robustness and security against potential worm attacks.

The study addresses the global impact of COVID-19 by developing a mathematical model that combines within-host and between-host factors to better understand the disease’s dynamics. It begins by describing SARS-CoV-2 dynamics within individual human hosts using fractional-order differential equations. The model is shown to be Ulam–Hyers stable, ensuring reliable predictions. The research then investigates virus transmission from infected to susceptible individuals using agent-based modeling (ABM). This approach allows us to capture the diversity and heterogeneity among individuals, including variations in internal state of individuals, immune response and responses to interventions, making the model more realistic compared to aggregate models. The agent-based model places individuals on a square lattice, assigns health states (susceptible, infectious, or recovered), and relies on infected individuals’ viral load for transmission. Parameter values are stochastically generated via Latin hypercube sampling. The study further explores the impact of viral mutation and control measures. Simulations demonstrate that vaccination substantially reduces transmission but may not eliminate it entirely. The strategy is more effective when vaccinated individuals are evenly distributed across the population, as opposed to concentrated on one side. The research further reveals that while reducing transmission probability decreases infections by implementing prevention protocols, it does not proportionally correlate with the reduction magnitude. This discrepancy is attributed to the intervention primarily addressing inter-host transmission dynamics without directly influencing intra-host viral dynamics.

This paper aims to contribute to and broaden the current understanding of fractional calculus and its uses in science. In general, determining the fractional derivative of a product of two functions is an obstacle. In this paper, we overcome this issue for Caputo and Riemann–Liouville fractional derivatives. Specifically, we provide explicit formulas for the derivatives of a product of two functions of order $\alpha \in (0,1)$ for these fractional derivatives. We also provided a few examples to help clarify and bolster our calculations.

In this work, a mathematical model of the fractional-order in non-Newtonian Powell–Eyring fluid flow (PEFF) and heat transfer is presented and solved numerically. The set of nonlinear differential equations in terms of velocity, temperature which describes our proposed problem is tackled through the spectral collocation method based on Chebyshev polynomials of the third kind. This method reduces the presented model to a nonlinear system of algebraic equations. This system is constructed as a constrained optimization problem and optimized to get the unknown coefficients of the series solution. The effects of the thermal radiation, Powell–Eyring parameters; suction parameter and Prandtl number on the PEFF are discussed. The numerical values of the dimensionless velocity and temperature are depicted graphically. The results show that the given procedure is an easy and efficient tool to investigate the solution of such models. Some of findings of this important work help to govern the velocity and the rate of heat transportation through the boundary layer. At the same time, this study highlights many applications in fields of engineering and industry, where the quality of the desired product depends on the rate of heat transfer, thermal radiation phenomenon, and the composition of the material used, and manufacturing processes.

In this paper, the fractional singular Lagrangian system is studied. The Hamilton–Jacobi treatment is developed to be applicable for fractional singular Lagrangian system. The equations of motion are obtained for the fractional singular systems and the Hamilton–Jacobi partial differential equations are obtained and solved to determine the action integral. Then the wave function for fractional singular system is obtained. Besides, to demonstrate this theory, the fractional Christ-Lee model is discussed and quantized using the WKB approximation.

In this paper, a convective fluid flow mixed with different nanoparticles (NPs) flowing on a vertical plate is studied. The impacts of slip boundary wall, Newtonian heating, and magnetic field are also pondered. Using Fourier’s rule of heat conductivity and Caputo-time (CT) fractional derivative (FD), the nondimensional governing equations for this mathematical model are obtained. The Laplace transformation (LT) technique is used to get a generalized solution to the equations of momentum and energy. The physical interpretation of velocity distribution in some particular cases is well recognized in the past research works and is also highlighted to scrutinize some further physical perception into the subject and to fortify the novelty of this research communication. Furthermore, graphical depictions are created using the mathematica program to assess the physical importance of the considered topic. As a result, we concluded that the blood-based aluminium oxide ($\mathrm{\text{A}}{\mathrm{\text{l}}}_2{\mathrm{\text{O}}}_3)$ NPs show more decaying behavior as compared to titanium dioxide ($\mathrm{\text{Ti}}{\mathrm{\text{O}}}_2)$ NPs for temperature and velocity profile. Moreover, changing the fractional parameter’s value for a long period of time improves thermal and velocity profiles.

The classical Hartmann flow problem is still interesting and novel due to its applications in MHD generators, plasma physics, power systems, etc. Owing to such importance in mind, this investigation explores the natural convection flow of viscous fluid following the Hartmann flow phenomenon due to a cylindrical tube. The heat transfer characteristics with diffusion phenomenon have been taken into consideration. The classical problem is further extended by countering the magnetic force impact. The fractional framework based on the Atangana–Baleanu (AB) and Caputo–Fabrizio (CF) is performed. The closed-form solutions are attained with Laplace as well as finite Hankel transforms. Further, the obtained results are stated as a combination of G-functions of Lorenzo and Hartley. The particular cases for the obtained simulations have been performed. The role of flow parameters governing the flow is graphically attributed.

The hybrid nanofluid is the modified class of nanomaterials with impressive thermal impact and preserves significant applications in extrusion systems, thermal management systems, engineering processes, cooling and heating materials and many more. The aim of this research is to communicate the thermal enhancement of water (H₂O) and sodium alginate (C₆H₉NaO₇) base fluids with interaction of hybrid nanofluids. The Casson fluid model is used to endorse the properties of such base liquids. The titanium dioxide and silver nanoparticles incorporate the thermal impact of hybrid nanofluid properties. The fractional simulations are performed by using the Caputo–Fabrizio (CF) time-fractional derivatives. The integration of problem is computed via Laplace technique. It is observed that the velocity profile enhanced the Grashof number. The increasing observations for Nusselt number and wall shear force due to fractional parameters are obtained.

The Liouville equation, first Bogoliubov hierarchy and Vlasov equations with derivatives of non-integer order are derived. Liouville equation with fractional derivatives is obtained from the conservation of probability in a fractional volume element. This equation is used to obtain Bogoliubov hierarchy and fractional kinetic equations with fractional derivatives. Statistical mechanics of fractional generalization of the Hamiltonian systems is discussed. Fractional kinetic equation for the system of charged particles are considered.

In this paper, an impulsive Kolmogorov-type system using the Caputo fractional-order derivative is developed. The fractional-order system displays many interesting dynamic behaviors and fractional integrals can be used to describe the fractal media. The existence and stability of integral manifolds for the impulsive fractional model are considered. The main results are proved by means of piecewise continuous Lyapunov functions and the new fractional comparison principle. The impulses are realized at variable impulsive moments of time and can be considered as a control. Finally, an example is given to illustrate our results.

In this research paper, we scrutinize the novel traveling wave solutions and other solutions with conformable, M-truncated and beta fractional derivatives for the nonlinear fractional Hirota–Maccari system. In order to acquire the analytical solutions, the Riccati–Bernoulli sub-ODE technique is implemented. Presented method is the very powerful technique to get the novel exact soliton and other solutions for nonlinear partial equations in sense of both integer and fractional-order derivatives. Mathematical properties of different kinds of fractional derivatives are given in this paper. A comparative approach is presented between the solutions with the fractional derivatives. For the validity of the solutions, the constraints conditions are determined. To illustrate the physical meaning of the presented equation, the 2D and 3D graphs of the acquired solutions are successfully charted by selecting appropriate values of parameters.

We study the dynamics of a mass, sliding on a dry surface and impacting against a rigid wall through a viscoelastic body, that we model as a straight rod of negligible mass. The problem comprises a constitutive model of the viscoelastic body with fractional derivatives of stress and strain, restrictions on the coefficients that follow from Clausius–Duhem inequality, and discontinuous inequality constraint conditions imposed by the Coulomb friction model. We show that the dynamics of the problem is governed by a single integro-differential inclusion. By use of the slack variable algorithm the problem was solved numerically. The predictions of the model concerning the duration of the impact, maximal values of the impacting force and deformation, as well as the restitution coefficient are determined for several values of system parameters. Depending on the dry friction coefficient three different impact scripts are identified: rebound after the impact, capture in the approaching phase, and capture in the rebound phase.

A distributed order fractional diffusion equation is considered. Distributed order derivatives are fractional derivatives that have been integrated over the order of the derivative within a given range. In this paper, sub-diffusive cases are considered. That is, the order of the time derivative ranges from zero to one. The equation is solved for Dirichlet, Neumann and Cauchy boundary conditions. The time dependence for each of the three cases is found to be a functional of the diffusion parameter. This functional is shown to have decay properties. Upper and lower bounds are computed for the functional. Examples are also worked out for comparative decay rates.

Different forms of diffusion equations on fractals proposed in the literature are reviewed and critically discussed. Variants of the known fractional diffusion equations are suggested here and worked out analytically. On the basis of these results we conclude that the quest: "what is the form of the diffusion equation on fractals," is still open, but we are possibly close to obtaining a satisfactory answer.

In this paper, we used the scaling concepts of Mandelbrot of fractals in variational problems of mechanical systems in order to re-write the action integral function as an integration over the fractional time. In addition, by applying the variational principle to this new fractional action, we obtained the modified Euler-Lagrange equations of motion in any fractional time of order 0 < α ≤ 1. Two examples are investigated in detail.