Terms Of Fractional Derivatives Research Articles

Lattice Boltzmann method for tempered time-fractional diffusion equation

Tempered fractional calculus, as an extension of fractional calculus, has been successfully applied in numerous scientific and engineering fields. Although several traditional numerical methods have been improved for solving a variety of tempered fractional partial differential equations, solving these equations by the lattice Boltzmann (LB) method is an unresolved issue. This paper is dedicated to presenting a novel LB method for the tempered time-fractional diffusion equation. The tempered time-fractional diffusion equation is first transformed into an integer-order partial differential equation by approximating the tempered fractional derivative term. Then the LB method is proposed to solve the transformed objective equation. The Chapman-Enskog procedure is conducted to confirm that the present LB method can accurately recover the objective equation. Some numerical examples with an analytical solution are employed to validate the present LB method, and a strong consistency is observed between the numerical and analytical solutions. The numerical simulations indicate that the LB method is a second-order accurate scheme. The proposed LB method presents a new approach to solving the tempered time-fractional diffusion equation, which is beneficial for the widespread application of the tempered time-fractional diffusion equation in addressing complex transport problems.

Combined effect of track-bridge interaction on the dynamic response of a simply supported railway bridge

In this work, the objective is to understand the effect of the nonlinear behaviour of the ballasted track, or the mechanism of vertical and horizontal track-bridge interaction behaviour, on the dynamic response of a simply supported railway bridge, carrying a single ballasted track characterized by a fractional order viscoelastic nonlinear Winkler foundation type, and acted by a moving train load with constant speed. The main feature of the model is that the horizontal and vertical interactions at the interface between the bridge deck and the ballasted track, composed by rails, sleepers and ballast, is introduced through a nonlinear cubic force-displacement relation, schematizing the nonlinear stiffness, plus fractional derivative term for simulating the rail bed’s damping behaviour and a system of horizontally orientated spring-damper elements. By applying the two numerical methods, fourth-order Runge-Kutta method and Finite Difference Method, the validation of the dynamic behaviour of the rail-bridge system is performed, and numerical simulations demonstrate the accuracy of the proposed theoretical model for predicting the real behaviour of the studied bridge taking into account the track-bridge interaction effect. The influence of the fractional-order derivative and the distance between sleepers on the critical velocities, associated with resonance effects, is also addressed.

Numerical approximation of a generalized time fractional partial integro-differential equation of Volterra type based on a meshless method

The moving least squares (MLS) approximation is used in this study to solve time-fractional partial integro-differential equations (TFPIDE). In our approach, we approximate the time Caputo fractional derivative term and the integral term by using a finite difference scheme and trapezoidal method, respectively, which yields an order of accuracy of O(τ+τ2−α), where 0<α<1. We thoroughly investigate the applicability and validity of the proposed method and provide an error estimate for its performance. Additionally, we solve various numerical problems to demonstrate the accuracy and computational efficiency of our approach, confirming the theoretical findings.

A discrete spectral method for time fractional fourth-order 2D diffusion-wave equation involving [formula omitted]-Caputo fractional derivative

In this work, the ψ-Caputo fractional derivative, as a generalization of the classical Caputo fractional derivative in which the fractional derivative of a sufficiently differentiable function is defined with respect to another strictly increasing function, ψ, is utilized to define the time fractional fourth-order 2D diffusion-wave equation. A Chebyshev–Gauss–Lobatto scheme is developed to solve this equation. In this way, a new operational matrix for the ψ-Riemann–Liouville fractional integration of the Chebyshev polynomials is derived. The solution of the equation is obtained by determining the solution of the algebraic system extracted from approximation the ψ-Caputo fractional derivative term via a finite discrete Chebyshev series and employing the expressed operational matrix. The validity of the established approach is examined by solving two examples.

On a localization-in-frequency approach for a class of elliptic problems with singular boundary data

We consider a class of nonhomogeneous elliptic equations in the half-space with critical singular boundary potentials and nonlinear fractional derivative terms. The forcing terms are considered on the boundary and can be taken as singular measure. Employing a functional setting and approach based on localization-in-frequency and Littlewood–Paley decomposition, we obtain results on solvability, regularity, and symmetry of solutions.

Iterative solutions for nonlinear equations via fractional derivatives: adaptations and advances

In recent years, the utilization of fractional calculus has witnessed a notable surge across various scientific and engineering domains. This manuscript delves into the exploration of adapted iterative techniques tailored for solving nonlinear equations, capitalizing on the diverse range of derivatives available for addressing different problem contexts. We scrutinize previously developed iterative methods, enhancing their efficacy by introducing an auxiliary parameter to the root search process for nonlinear equations (NLE), alongside a fixed order of fractional derivatives. The selection of the auxiliary parameter is confined to the interval (0,1] for convenience. A thorough convergence analysis is conducted, employing a fractional power series expansion of f(x) in terms of fractional derivatives. Subsequently, a series of NLEs is solved to showcase and contrast the efficiency of our proposed methods with established iterative techniques. This refined abstract aims to succinctly elucidate the objectives and contributions of our study, providing readers with a clearer understanding of the manuscript's scope and significance.

The Finite Volume Element Method for Time Fractional Generalized Burgers’ Equation

In this paper, we use the finite volume element method (FVEM) to approximate a one-dimensional, time fractional generalized Burgers’ equation. We construct the fully discrete finite volume element scheme for this equation by approximating the time fractional derivative term by the L1 formula and approximating the spatial terms using FVEM. The convergence of the scheme is proven. Finally, numerical examples are provided to confirm the scheme’s validity.

Steady-state solutions of the Whittaker–Hill equation of fractional order

We investigate the steady-state solutions of the Whittaker–Hill equation, including a fractional derivative term. Using appropriate Fourier series, we characterize the behavior of the eigenvalue surfaces as a function of the differential equation parameters and describe the corresponding eigensolutions. We also examine the situation where the fractional derivative is zero to serve as a comparison for the fractional solutions. For some special combinations of the parameters, the fractional derivative term leads to the appearance of degenerate complex eigenvalues.

The existence of solutions of Hadamard fractional differential equations with integral and discrete boundary conditions on infinite interval

&lt;abstract&gt;&lt;p&gt;In this article, the properties of solutions of Hadamard fractional differential equations are investigated on an infinite interval. The equations are subject to integral and discrete boundary conditions. A new proper compactness criterion is introduced in a unique space. By applying the monotone iterative technique, we have obtained two positive solutions. And, an error estimate is also shown at the end. This study innovatively uses a monotonic iterative approach to study Hadamard fractional boundary-value problems containing multiple fractional derivative terms on infinite intervals, and it enriches some of the existing conclusions. Meanwhile, it is potentially of practical significance in the research field of computational fluid dynamics related to blood flow problems and in the direction of the development of viscoelastic fluids.&lt;/p&gt;&lt;/abstract&gt;

Significance of Cu-Fe3O4 on fractional Maxwell fluid flow over a cone with Newtonian heating

The goal of this research is to investigate fractional Maxwell hybrid nanofluids utilizing partial differential equations in terms of Caputo time fractional derivatives. Specifically, the effect of Newtonian heating on the thermal performance of a fractional Maxwell hybrid nanofluid moving over a permeable cone in the presence of thermal radiation and heat generation is considered. The Crank–Nicolson method and L1 algorithmt of Caputo derivative are used to find numerical solutions to the considered nonlinear problem. The effects of significant flow factors on fluid properties are examined and illustrated in various graphs. According to the results, the thermal performance of the fluid raised by 0.4%, 6.1%, and 3.1% on adding 4% volume fraction of Fe3O4, Cu, and Cu−Fe3O4, respectively, in the base fluid.

Temporal second-order fully discrete two-grid methods for nonlinear time-fractional variable coefficient diffusion-wave equations

This paper proposes a temporal second-order fully discrete two-grid finite element method (FEM) for solving nonlinear time-fractional variable coefficient diffusion-wave equations. The method involves introducing an auxiliary variable and utilizing a reducing order technique for the fractional derivative. This reduces the original fractional wave equations to a coupled system consisting of two equations with lower order derivative in time. The time-fractional derivative is discretized using the L2-1σ formula, while a second-order scheme is used for discretizing the first-order time derivative. The spatial direction is discretized using an efficient two-grid Galerkin FEM. The paper demonstrates that the optimal error estimates in L2-norm and H1-norm of the two-grid methods can achieve optimal convergence order when the mesh size satisfies H=h12 and H=hr2r+2, respectively. To handle the initial singularity of the solution and the historical memory of the fractional derivative term, the paper presents the nonuniform and fast two-grid algorithms. The numerical results validate the theoretical analysis and demonstrate the effectiveness of the proposed two-grid methods.

A unified Maxwell model with time-varying viscosity via [formula omitted]-Caputo fractional derivative coined

In order to describe the mechanical behaviors of viscoelastic materials that couple memory effects and time-varying viscosity properties toggled between thixotropy and rheopexy, this paper explores and establishes the constitutive equation of a unified Maxwell model with a variable kernel in terms of the ψ-Caputo fractional derivative. By virtue of a Volterra integral equation of the second kind and the ψ-Laplace transform technique, the closed-form expressions of creep and relaxation responses for the proposed model are derived and compared in detail. Furthermore, to enhance practicality, the exponential and power-law time-varying viscosity candidates are embedded into the proposed model, along with exhibiting the corresponding creep and relaxation behaviors in light of illustration and reasoning, respectively. The results may provide a fresh perspective for detecting the relationship between the viscoelastic system with nonlinear time-varying viscosity and generalized fractional calculus.

General fractional integrals and derivatives and their applications

In this survey paper, some recent results obtained for the general fractional integrals and derivatives and for the regularized general fractional derivatives are discussed. We start with a short historical survey of the general Fractional Calculus operators with the Sonin kernels and continue with a presentation of the recent developments regarding the general fractional derivatives and their applications. In particular, we introduce the first and the second fundamental theorems of Fractional Calculus for the general fractional derivatives of arbitrary order, for the regularized general fractional derivatives of arbitrary order, and for the sequential general fractional derivatives. As an application of these results, the generalized convolution Taylor formulas and the generalized convolution Taylor series with the coefficients and remainders in terms of the general fractional derivatives and the regularized general fractional derivatives are presented. Finally, we consider some Cauchy problems for the fractional differential equations with the general fractional derivatives.

Numerical study of entropy generation in magneto-convective flow of nanofluid in porous enclosure using fractional order non-Darcian model

The present numerical work examines the effect of fractional order parameter on heat transfer and entropy generation for a thermo-magnetic convective flow of nanofluid (Cu-water) in a square porous enclosure that contains semi-circular bottom wall. The Darcy–Brinkmann–Forchheimer model is utilized to evaluate the momentum transfer in porous media, and the Caputo-time fractional derivative term is introduced in momentum as well as in the energy equation. Further, non-dimensional governing equations are simulated through the penalty finite element method, and the Caputo time derivative is approximated by L1-scheme. The study is carried out for various parameters, including Rayleigh number (Ra), Darcy number (Da), radius of the semicircle (r), fractional order (α), and Hartmann number (Ha). The comprehensive results are presented by the contour variation of isotherms, streamlines, and total entropy generation at the selected range of parameters. In addition, thermal transport and irreversibilities due to heat transfer, fluid friction, and magnetic field have been accounted through the numerical variation of mean Nusselt number (Num) and Bejan number due to heat transfer (Beht), fluid friction (Beff), and magnetic field (Bemf), respectively. The key findings of the present study reveal that during the initial evolution period, the Num value increases as α→1. Additionally, time taken to achieve the steady state condition varies and depends on fractional order α. Furthermore, in the absence of Ha, the heat transfer and entropy generation intensifies with augmentation of Ra and Da for all α, while, the increasing value of Ha shows an adverse impact on the heat transfer rate.

Time-fractional Caputo derivative versus other integrodifferential operators in generalized Fokker-Planck and generalized Langevin equations.

Fractional diffusion and Fokker-Planck equationsare widely used tools to describe anomalous diffusion in a large variety of complex systems. The equivalent formulations in terms of Caputo or Riemann-Liouville fractional derivatives can be derived as continuum limits of continuous-time random walks and are associated with the Mittag-Leffler relaxation of Fourier modes, interpolating between a short-time stretched exponential and a long-time inverse power-law scaling. More recently, a number of other integrodifferential operators have been proposed, including the Caputo-Fabrizio and Atangana-Baleanu forms. Moreover, the conformable derivative has been introduced. We study here the dynamics of the associated generalized Fokker-Planck equationsfrom the perspective of the moments, the time-averaged mean-squared displacements, and the autocovariance functions. We also study generalized Langevin equationsbased on these generalized operators. The differences between the Fokker-Planck and Langevin equations with different integrodifferential operators are discussed and compared with the dynamic behavior of established models of scaled Brownian motion and fractional Brownian motion. We demonstrate that the integrodifferential operators with exponential and Mittag-Leffler kernels are not suitable to be introduced to Fokker-Planck and Langevin equationsfor the physically relevant diffusion scenarios discussed in our paper. The conformable and Caputo Langevin equationsare unveiled to share similar properties with scaled and fractional Brownian motion, respectively.

Formation of multi-peak gap solitons and stable excitations for double-Lévy-index and mixed fractional NLS equations with optical lattice potentials

In this paper, we investigate the new two-Lévy-index fractional nonlinear Schrödinger (FNLS) equations with optical lattices, where two fractional derivative terms are introduced to control the diffractions. Especially, we explore the linear Bloch bandgaps and existence of gap solitons for the two-Lévy-index mixed FNLS equations with optical lattices. We analyse the effects of different coefficients of normal and abnormal diffraction terms on bandgap structures. Families of single-, double-, triple-, as well as quadruple-peak solitons are found in the first gap with the defocusing nonlinearity while such solitons can be found in the semi-infinite gap when it turns to the focusing regime. Meanwhile, we find that the above-obtained gap solitons can be roughly divided into two types, one of which branches out from the band edge, and another one cannot branch out from the band edge. Moreover, the stability of gap solitons is discussed via the linear stability analysis, and then their dynamical behaviours are verified by means of direct simulations. And stable multi-peak solitons can be obtained via analysing the phase structure. Meanwhile, we find the stable excitations of gap solitons via excitations of system parameters. The stable gap solitons are also verified to appear in the two-Lévy-index FNLS equation. These results may pave the way for the study of linear and nonlinear phenomena of two-Lévy-index and mixed FNLS equations or other mixed fractional physical models in optical lattices and the related physical experimental designs.

Extended statistical linearization approach for estimating non-stationary response statistics of systems comprising fractional derivative elements

An efficient technique to analyze non-stationary nonlinear random vibrations of dynamic systems endowed with a fractional derivative term is presented. The technique itself represents an extension of the concept of statistical linearization to this kind of systems, and it is applicable for both analytic and hysteretic system nonlinearities. The technique first resorts to harmonic balancing in deriving response-amplitude dependent equivalent damping and stiffness. This enables representation of the fractional derivative term as a linear combination of the system response displacement and velocity with amplitude dependent coefficients. Then, the expected values of these parameters are considered in proceeding to formulate a statistical linearization solution scheme. In this context, the solution procedure is completed by integrating in time the covariance Lyapunov equation associated with the derived equivalent linear system. The reliability of the proposed technique is tested by a series of germane Monte Carlo studies. This juxtaposition is also used to elucidate salient features of the technique, by varying the order of the fractional derivative term, and of the degree of the nonlinearity in the system. It also points out the versatility of the technique in determining the non-stationary values of auto-correlation and cross-correlations response parameters involving even the fractional derivative term.

Well‐posedness and Mittag–Leffler stability for a nonlinear fractional telegraph problem

AbstractThe ordinary diffusion models in continuous media are derived from basic principles: Conservation of energy and momentum. The classical telegraph equation is one of the earliest and most commonly used equation to describe the transmission of electromagnetic waves. In fractal and disordered media, to capture the complex diffusion feature, we consider rather fractional models. Indeed, the mean square displacement in anomalous media is no longer proportional to the time but rather proportional to a power of time whose exponent may be between zero and one or between one and two. This is the case for the present fractional telegraph equation in the presence of non‐negligible voltage wave. Here, the well‐posedness and stability of a telegraph problem with two time‐fractional derivatives is discussed. In addition to being a noninteger problem, the equation in the model is subject to a nonlinear source as well as a nonlinear lower order fractional derivative term. First, the well‐posedness in an appropriate underlying space is established by the use of resolvent operators. Then, it is proved that, despite the presence of these nonlinearities, solutions are Mittag–Leffler stable. This confirms (and extends) the fact that the lower order term plays the role of a damper in the fractional case. To this end, we combine the energy method with some properties of the Caputo fractional derivative.

Stochastic bifurcations induced by Lévy noise in a fractional trirhythmic van der Pol system

Trirhythmical nature has aroused considerable interest in describing dynamic behaviors of a fractional self-sustained system. In this paper, the fractional trirhythmic system is considered and a bifurcation analysis in a stochastic fractional trirhythmic self sustained system subjected to Lévy noise perturbation is presented. The fractional electronic circuit has been used to model the system and the oscillations are described by a nonlinear fractional differential equation to show a new bifurcation parameter. Then, based on the minimum mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of the damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Using the predictor–corrector simulation method, it is shown a good agreement between the fractional and the equivalent integer order system. The detailed parameter space study reveals that the multi-rhythmic properties of the oscillation of this system can be efficiently controlled by fractional order parameter and Lévy noise parameters. Finally, the analytical solutions are validated by numerical results of the Monte Carlo simulation of the original fractional modified van der Pol oscillator. The trirhythmic region can be detected in the parameter space (α,β,γ,δ) by choosing an approximate fractional order. The stability index and the noise intensity can in any case govern the dynamics of the self sustained system, but regulating the skewness parameter cannot bring about transitions between unimodal, bimodal and trimodal in this multi-rhythmic system. More bifurcations appear by adjusting the fractional order in this system. These results may be conducive to further exploring bifurcations in the real-world applications. This study sheds new insight lights on the understanding nontrivial effects of fractional derivative on a trirhythmic self sustained system.

A chaos control strategy for the fractional 3D Lotka–Volterra like attractor

In this paper, we have considered a three-dimensional Lotka–Volterra attractor in the frame of the Caputo fractional derivative to examine its dynamics. The theoretical concepts like existence and uniqueness and boundedness of the solution are analyzed. To regulate the chaos in this fractional-order system, we have developed a sliding mode controller and conditions for global stability of the controlled system with and without uncertainties and outside disruptions are derived. The ability of the designed controller is examined in terms of both commensurate and non-commensurate fractional order derivatives for all the aspects. The Lyapunov exponent is the novelty of this paper which is used to illustrate the behavior of the chaos and demonstrate the dissipativeness of the considered chaotic system. We have examined the effect of fractional order derivatives in this system. With the help of numerical simulations, the theoretical claims regarding the impact of the controller on the system are established.