Effect Of Fractional Parameter Research Articles

On the fractional quark–antiquark confinement and symplectic quantum mechanics

Using the formalism of generalized fractional derivatives, a two-dimensional nonrelativistic meson system is studied. The mesons are interacting by a Cornell potential. The system is formulated in the domain of the symplectic quantum mechanics by means of the generalized fractional Nikiforov–Uvarov method. The corresponding Wigner function and the energy eigenvalues are then derived. The effect of fractional parameters [Formula: see text] and [Formula: see text] with the ground state solution is analyzed through the Wigner function for the charm–anticharm, bottom–antibottom and [Formula: see text] mesons. One of the fundamental achievements of such Cornell model is the determination of heavy quarkonia mass spectra. We have computed these masses and the present results are in agreement with the experimental data, improving previous theoretical results.

Traveling Wave Solutions of the Conformable Fractional Klein–Gordon Equation With Power Law Nonlinearity

This article investigates the construction of new traveling wave solutions for the conformable fractional Klein–Gordon equation, which is a well‐known mathematical and physical model that can be used to explain spinless pion and de Broglie waves. In order to accomplish this task, a classic and effective analysis method, namely, the extended tanh–coth method, was utilized. By introducing appropriate transformations, the conformable fractional Klein–Gordon equations are reduced to ordinary differential equations, and then the solutions with hyperbolic function form are obtained. In addition, the effect of fractional parameters on waveform is analyzed by drawing two‐ and three‐dimensional graphics. These results contribute to a deeper understanding of the dynamics of the conformable fractional Klein–Gordon equation. The research in this article also indicates that the extended tanh–coth method is a straightforward and concise technique that has the potential to be applicable to many other conformable fractional partial differential equations that appear in mathematical physics.

Homotopy time-fractional enhancement soliton solution for six component complex cometary plasma with thermal heavy ions

A dusty plasma system composed of three inertial species and three Kappa-distributed inertial-less species is considered. The hydrodynamic fluid equation used to describe investigated plasma system. The plasma system under consideration is a very rich system, so in this paper the effective Debye length and effective plasma frequency are considered to normalize all physical quantities. The Kortweg de-Veris equation is derived using the reductive perturbation method. Two envelope solitons (bright and dark) are propagating, and their appearance depends mainly on the values thermal parameter of heavy ions (oxygen ions) as well as the mass ratio for dust plasma. The homotopy perturbed transform method (HPTM) is used to solve the reduced time-fractional Korteweg-de Vries (TFKdV) equation. The effects of fractional parameters on the solution solutions are examined. Our results show that the shape of the solitary pulse is highly controlled by the fractional order, as well as the plasma parameters in addition to that the HPTM is very useful method, where few terms approximately three terms are enough to get successful approximate solution for fractional type equations.

Homotopy perturbation method-based soliton solutions of the time-fractional (2+1)-dimensional Wu–Zhang system describing long dispersive gravity water waves in the ocean

Physical phenomena and natural disasters, such as tsunamis and floods, are caused due to dispersive water waves and shallow waves caused by earthquakes. In order to analyze and minimize damaging effects of such situations, mathematical models are presented by different researchers. The Wu–Zhang (WZ) system is one such model that describes long dispersive waves. In this regard, the current study focuses on a non-linear (2 + 1)-dimensional time-fractional Wu–Zhang (WZ) system due to its importance in capturing long dispersive gravity water waves in the ocean. A Caputo fractional derivative in the WZ system is considered in this study. For solution purposes, modification of the homotopy perturbation method (HPM) along with the Laplace transform is used to provide improved results in terms of accuracy. For validity and convergence, obtained results are compared with the fractional differential transform method (FDTM), modified variational iteration method (mVIM), and modified Adomian decomposition method (mADM). Analysis of results indicates the effectiveness of the proposed methodology. Furthermore, the effect of fractional parameters on the given model is analyzed numerically and graphically at both integral and fractional orders. Moreover, Caputo, Caputo–Fabrizio, and Atangana–Baleanu approaches of fractional derivatives are applied and compared graphically in the current study. Analysis affirms that the proposed algorithm is a reliable tool and can be used in higher dimensional fractional systems in science and engineering.

Diverse geometric shape solutions of the time-fractional nonlinear model used in communication engineering

The nonlinear perturbed Radhakrishnan-Kundu-Lakshmanan (RKL) equation is a significant model in the engineering field for analyzing data transmission through nonlinear optical fibers. The two-variable (ψ′/ψ,1/ψ)-expansion method is a reliable scheme to compute self-controlled solitary wave solutions. In this study, we have constructed a wide range of geometric shapes and further inclusive soliton solutions comprising rational, trigonometric, and hyperbolic functions and their integration to the stated equation, devising the acknowledged method. The physical details of the ensued solitary wave solutions and the effects of fractional parameters are explained by sketching three- and two-dimensional graphs. For the appropriate values of the parameters, the periodic, V-shaped, bell-shaped, compacton, singular periodic, flat bell-shaped, plane-shaped, and some other types of solitons are formulated. The effect of the fractional derivative has been demonstrated by plotting the two-dimensional graphs for different values of the fractional order. It is important to note that changes in physical and auxiliary parameters cause variations in the wave profile. The formerly described physical phenomena may be better understood using the standard and new types of solitons. The results obtained exhibit the effectiveness, compatibility, and reliability of the two-variable (ψ′/ψ,1/ψ)-expansion approach for extracting soliton solutions of fractional-order nonlinear evolution equations in science, technology, and engineering.

New solutions of time-space fractional coupled Schrödinger systems

<abstract><p>The current manuscript focuses on the solution and analysis of space and time fractional coupled Schrödinger system that belongs to a class of evolution equations. These systems encounter in different fields related to plasma waves, optics, and quantum physics. The fractional He-Laplace approach is proposed for the series form solutions of fractional systems. This approach contains hybrid of Laplace transform and homotopy perturbation along with Caputo fractional derivative. The current study provide new results on time and space fractional coupled Schrödinger systems which are not captured in existing literature. Reliability of proposed algorithm in both time and space fractional scenarios is observed through residual error concept throughout fractional domains. The effect of fractional parameters on wave profiles are analyzed numerically and graphically as 2D and 3D illustrations. Analysis reveals that proposed algorithm is suitable for non-linear time-space fractional systems encountering in different fields of sciences.</p></abstract>

Fractional analysis of unsteady squeezing flow of Casson fluid via homotopy perturbation method

The objective of this article is to model and analyze unsteady squeezing flow of fractional MHD Casson fluid through a porous channel. Casson fluid model is significant in understanding the properties of non-Newtonian fluids such as blood flows, printing inks, sauces and toothpaste etc. This study provides important results as unsteady flow of Casson fluid in fractional sense with aforementioned effects has not been captured in existing literature. After applying similarity transformations along with fractional calculus a highly non-linear fractional-order differential equation is obtained. Modeled equation is then solved along with no-slip boundary conditions through a hybrid of Laplace transform with homotopy perturbation algorithm. For validity purposes, solution and errors at various values in fractional domain are compared with existing results. LHPM results are better in terms of accuracy than other available results in literature. Effects of fractional parameter on the velocity profile, skin friction and behaviors of involved fluid parameters is the focal point of this study. Comprehensive, quantitative and graphical analysis is performed for investigating the effects of pertinent fluid parameters on the velocity profile and skin friction. Analysis revealed that fractional parameter depicts similar effect in case of positive and negative squeeze number. Also, skin friction decreases with an increasing fractional parameter. Moreover, in fractional environment Casson parameter has shown similar effect on the velocity profile in case of positive and negative squeeze number.

"General analytical solution of fractional Klein–Gordon equation in a spherical domain"

"Time-fractional Klein–Gordon equation in a sphere is considered for the case of central sym- metry under the time-variable Dirichlet condition. The time-fractional derivative with the power-law kernel is used. The Laplace transform and convenient transformations of the independent variable and unknown func- tion are used to determine the general analytical solution of the problem in the Laplace domain. In order to obtain the solution in the real domain, the inverse Laplace transforms of two functions of exponential type whose expressions are new in the literature have been determined. The similar solution for ordinary Klein– Gordon equation is a limiting case of general solution but a simpler form for this solution is provided. The convergence of general solution to the ordinary solution and the effects of fractional parameter on this solution are graphically underlined."

Start-up flow in a pipe of a double distributed-order Maxwell fluid

The distributed-order model is equivalent to weighting the fractional-order model on a given interval. In this paper, a double time-distributed order Maxwell model is developed and formulated to study the characteristics of a start-up pipe flow. The numerical solution of the distributed-order governing equation is obtained by the finite difference method. In order to test the convergence of numerical solutions, the truncation error of the proposed numerical method be calculated. The characteristics of the flow and the effects of fractional parameters and weight functions on the velocity distribution are discussed in detail. By comparing the velocity distribution diagrams, it can be inferred that the distributed-order model controlled by the weight function can describe the memory and elastic properties of fluid more flexibly.

A global report on the dynamics of COVID-19 with quarantine and hospitalization: A fractional order model with non-local kernel

In this paper, a compartmental mathematical model has been utilized to gain a better insight about the future dynamics of COVID-19. The total human population is divided into eight various compartments including susceptible, exposed, pre-asymptomatic, asymptomatic, symptomatic, quarantined, hospitalized and recovered or removed individuals. The problem was modeled in terms of highly nonlinear coupled system of classical order ordinary differential equations (ODEs) which was further generalized with the Atangana-Balaeanu (ABC) fractional derivative in Caputo sense with nonlocal kernel. Furthermore, some theoretical analyses have been done such as boundedness, positivity, existence and uniqueness of the considered. Disease-free and endemic equilibrium points were also assessed. The basic reproduction was calculated through next generation technique. Due to high risk of infection, in the present study, we have considered the reported cases from three continents namely Americas, Europe, and south-east Asia. The reported cases were considered between 1st May 2021 and 31st July 2021 and on the basis of this data, the spread of infection is predicted for the next 200 days. The graphical solution of the considered nonlinear fractional model was obtained via numerical scheme by implementing the MATLAB software. Based on the fitted values of parameters, the basic reproduction number ℜ0 for the case of America, Asia and Europe were calculated as ℜ0≈2.92819, ℜ0≈2.87970 and ℜ0≈2.23507 respectively. It is also observed that the spread of infection in America is comparatively high followed by Asia and Europe. Moreover, the effect of fractional parameter is shown on the dynamics of spread of infection among different classes. Additionally, the effect of quarantined and treatment of infected individuals is also shown graphically. From the present analysis it is observed that awareness of being quarantine and proper treatment can reduce the infection rate dramatically and a minimal variation in quarantine and treatment rates of infected individuals can lead us to decrease the rate of infection.

The memory time effects for unsteady seas and oceans water flow through the limestone porous medium in the presence of chemical reaction and Soret effects

AbstractIn this work, we study the effects of the thermal and diffusion memories in the seas and oceans water using the methodology of fractional calculus for modeling a viscous fluid. In our model, we derive a model with two fractional parameters representing the thermal and diffusion effects for this type of fluid. We formulate a problem of mixed convection heat and mass transfer flow of a viscous fluid through a Limestone porous medium in a vertical channel whose walls are kept at variable temperature and concentration. The influences of heat‐mass memories in the presence of chemical reaction, heat generation/absorption and, the Soret effect are discussed and represented graphically on all variables fields. Also, the influences of Darcy, Prandtl, Schmidt, Grashof parameters are compared with the effect of fractional parameters.

The Effects of Fractional Time Derivatives in Porothermoelastic Materials Using Finite Element Method

In this work, a new model for porothermoelastic waves under a fractional time derivative and two time delays is utilized to study temperature increments, stress and the displacement components of the solid and fluid phases in porothermoelastic media. The governing equations are presented under Lord–Shulman theory with thermal relaxation time. The finite element method has been adopted to solve these equations due to the complex formulations of this problem. The effects of fractional parameter and porosity in porothermoelastic media have been studied. The numerical outcomes for the temperatures, the stresses and the displacement of the fluid and the solid are presented graphically. These results will allow future studies to gain a detailed insight into non-simple porothermoelasticity with various phases.

Effect of magnetic field on electroosmotic flow of viscoelastic fluids in a microchannel.

Electroosmotic flow is an efficient transportation technology driven by applying an external electric field across the microchannel, which has a great potential for futureapplication. This work is presented to study the unsteady electroosmotic flow of viscoelastic fluids combined with a constant pressure gradient and a vertical magnetic field through a parallel plate microchannel. For the reason that the upper and bottom walls of the parallel plate microchannel in microfluidic devices can be made of different materials, this leads to different hydrophobic properties, asymmetric zeta wall potentials, and different slip boundary conditions. The Navier slip model with different slip coefficients at walls is considered. The generalized Maxwell fluid with fractional derivative is adopted for the constitutive equation of the fluid. The analytical and numerical solutions of velocity are derived by employing the integral transform method and finite difference method, respectively. Excellent agreement is found between the numerical solutions and analytical solutions. Finally, the effects of fractional parameter , relaxation time , slip coefficients and , the ratio of wall zeta potentials , Hartmann number , and electrical field strength parameter on velocity profiles are interpreted graphically in detail.

Fractional order thermo-viscoelastic theory of biological tissue with dual phase lag heat conduction model

Viscoelasticity is an inherent property of the biological tissue and is often used as a diagnosis parameter in characterizing cancer. Recently, it has been found that fractional calculus agrees well with experimental results when describing the viscoelastic materials. In this study, the fractional linear thermo-viscoelastic theory is developed with considering the fractional relaxation effect of dual phase lag heat conduction model and viscoelastic constitutive relation simultaneously. The transient coupling thermal and mechanical behavior of biological tissue are investigated. The governing equations are solved by Laplace transformation. The effects of fractional parameter on the distributions of temperature, displacement and stress are obtained and illustrated graphically.

Memory effects and of the killing rate on the tumor cells concentration for a one-dimensional cancer model

Abstract In this article, the fractional models of cancer tumor are studied using the Laplace transform and numerical inversion. The generalized model with Caputo time-fractional derivative is considered for two different cases of the killing rate of the cancer cells. The models based on the time-fractional derivatives give a better description of the tumor evolution because, in such models, the history of the tumor cells concentration influences the time evolution of the tumor. To solve the initial-boundary value problems, some adequate transforms of variables and functions have been considered, together with the Laplace transform and the Bessel equation. A numerical method, namely the Stehfest's algorithm is used to determine the inverse Laplace transforms. The effect of fractional parameter on the tumor cells concentration has been highlighted by numerical simulations and graphical illustrations. The time variation of tumor cell concentration and its dependence of the memory parameter could provide useful information in choosing an appropriate treatment.

Analytical solutions to the advection-diffusion equation with Atangana-Baleanu time-fractional derivative and a concentrated loading

In this communication we have studied well known physical process of two-dimensional advection–diffusion phenomena. The advection–diffusion equation is time-fractionalized by exploiting Atangana-Baleanu fractional derivative operator. This fractionalization is achieved in the generalized constitutive equation of the mass flux density vector. The fractionalized two-dimensional advection–diffusion equation turns out to be a two-dimensional nonlinear fractional partial differential equation. This partial differential equation is considered under the hypothesis of an initial concentrated loading and Robin type boundary conditions. The analytical expression of the solution is determined for this boundary value problem by employing the integral transforms method, namely, the Laplace transform, sine-Fourier transform and finite sine–cosine Fourier transform. The effects of fractional parameter α on the concentration obtained from the analytical solution, for various parameters of interest, are illustrated graphically with the help of software Mathcad. The graphs illustrate that the memory effects are remarkable for small values of time and ordinary for large values of the time.

Convective flow of a Maxwell hybrid nanofluid due to pressure gradient in a channel

In this work, analytical solution of hybrid Maxwell nanofluid of the vertical channel due to pressure gradient is discussed. By introducing dimensionless variables the governing equations with all levied initial and boundary conditions is converted into dimensionless form. Fractional model for Maxwell fluid is developed by Caputo time fractional differential operator by using the constitutive relation. The dimensionless expression for temperature and velocity are found using Laplace transform. Draw graphs of temperature and velocity by Mathcad software and discuss the behavior of flow parameters and the effect of fractional parameters. As a result, we have found by increasing the volumetric fraction of copper and alumina temperature increases and velocity decreases. Also, fluid flow properties showed dual behavior for small and large time, respectively, by increasing fractional parameters values.

Heat transfer flow of Maxwell hybrid nanofluids due to pressure gradient into rectangular region

In this work, influence of hybrid nanofluids (Cu and mathrm{Al}_{2}mathrm{O}_{3}) on MHD Maxwell fluid due to pressure gradient are discussed. By introducing dimensionless variables the governing equations with all levied initial and boundary conditions are converted into dimensionless form. Fractional model for Maxwell fluid is established by Caputo time fractional differential operator. The dimensionless expression for concentration, temperature and velocity are found using Laplace transform. As a result, it is found that fluid properties show dual behavior for small and large time and by increasing volumetric fraction temperature increases and velocity decreases respectively. Further, we compared the Maxwell, Casson and Newtonian fluids and found that Newtonian fluid has greater velocity due to less viscosity. Draw the graphs of temperature and velocity by Mathcad software and discuss the behavior of flow parameters and the effect of fractional parameters.

Nonlinear vibration analysis of fractional viscoelastic cylindrical shells

Nonlinear vibrations of viscoelastic thin cylindrical shells are studied in this paper. The viscoelastic properties are modeled using the Kelvin–Voigt fractional-order constitutive relationship. Based on the nonlinear Love thin shell theory, the structural dynamics of the cylindrical shell is modeled by using the Newton’s second law, and the Galerkin method is used to discretize the nonlinear partial differential equations into the set of nonlinear ordinary differential equations. The method of multiple scales is used to solve the nonlinear ordinary differential equations, and the amplitude–frequency and phase–frequency equations are extracted. The obtained results are verified with available investigations, and the effects of fractional parameters, excitation, and nonlinearity on the amplitude–frequency and phase–frequency responses of the viscoelastic cylindrical shells are outlined.

Finite Volume Method for Fractional Maxwell Viscoelastic Fluid Over a Moving Plate With Convective Boundary Condition

Abstract A novel finite volume method about the boundary layer flow and heat transfer of fractional viscoelastic fluid over a moving plate with convective boundary condition is developed. The fractional Maxwell model and fractional Fourier's law are employed in the constitutive relations. Numerical solutions are obtained and validated by exact solutions of special case with source terms. The effects of fractional parameters on the flow and heat transfer characteristics are analyzed. Results show that the viscoelastic fluid performs shear-thickening property with the increase of fractional parameter. Moreover, the variations of the average Nusselt number demonstrate that the viscoelastic fluid characterized by fractional Fourier's law has short memory in heat conduction process.