Time-fractional Derivative Of Order Research Articles

Rayleigh wave through half space semiconductor solid with temperature dependent properties

The article focuses on Rayleigh wave propagation in a homogeneous isotropic semi-conductor thermoelastic medium rotating with fixed angular frequency with temperature-dependent properties. The elastic constants depend upon the temperature function. The effects of temperature dependency parameter, time derivative, and fractional order are illustrated. By the theory of thermo-elasticity, waves result in the generation of thermal signals that propagate through the medium. A heat conduction model of three-phase lag (3PL) along with fractional order time derivative is used to analyze the thermal signals. The secular equations of Rayleigh waves are derived mathematically at the stress-free, carrier density and thermally insulated boundaries. Some specific properties like velocity, attenuation coefficient, specific heat loss and penetration depth for Rayleigh waves have been evaluated and presented graphically. The secular equations are computed numerically and depicted graphically using Matlab.

A novel refined Caputo kernel and constitutive concepts for semi-exact nonlinear dynamic and creep analyses of suddenly pressurized hollow fractional-order visco-hyperelastic cylinders

Some reports have confirmed that discretizing the mass, damping, and stiffness characteristics of a continuum can result in remarkable deviations from the experimental results; as all these properties are intertwined and coupled within a single material. The viscoelastic constitutive laws with fractional-order time derivatives can model this integrity much more accurately, due to their capability of using a set of an infinite number of orders. Here, an innovative constitutive model with a nonlinear hyperelastic memory element that treats simultaneously the fractional-order viscoelasticity and Mooney-Rivlin hyperelasticity of the continuum is proposed to track the suppression of the short-term dynamic stresses and displacements and the redistribution of the long-term creep stress and deformations of the suddenly pressurized thick cylinders. While the spatial variations of the quantities of the incompressible cylinder are treated by an exact analytical approach, the resulting time-dependent system of equations is converted by a novel numerical solution scheme to an integrodifferential system that contains integer-order partial derivatives and a time-growing number of integral terms. An enhancement to Caputo's singular-kernel integral definition of the fractional-order derivatives is proposed here to replace these derivatives with equivalent integrals whose integer derivatives are computed based on second-order Taylor's expansion. The integer-order and fractional-order time derivatives are then computed numerically using the second-order Runge-Kutta and accumulation-based trapezoidal techniques, respectively. Finally, comparisons are made among the dissipative pattern and time variations of the dynamic/vibration stresses and large-deformations of the hyperelastic and fractional-order visco-hyperelastic cylinders in addition to the assessment of the effects of the constitutive parameters of the fractional-order model. Results show that while the larger magnitudes of both the nonlinear viscosity coefficients and fractional-order time derivatives diminish the resulting deformations and stresses, they respectively lead to smaller and larger frequencies of oscillation.

Application of Homotopy Analysis Transform Method for Solving a Fractional Singular One-Dimensional Thermo-Elasticity Coupled System

This article extends the application of fractional-order time derivatives to replace their integer-order counterparts within a system comprising two singular one-dimensional coupled partial differential equations. The resulting model proves invaluable in representing radially symmetric deformation and temperature distribution within a unit disk. The incorporation of fractional-order derivatives in mathematical models is shown to significantly enhance their capacity for characterizing real-life phenomena in comparison to their integer-order counterparts. To address the studied system numerically, we employ the q-homotopy analysis transform method (q-HATM). We evaluate the efficiency of this method in solving the problem through a series of illustrative examples. The convergence of the derived scheme is assessed visually, and we compare the performance of the q-HATM with that of the Laplace decomposition method (LDM). While both methods excel in resolving the majority of the presented examples, a notable divergence arises in the final example: the numerical solutions obtained using q-HATM converge, whereas those derived from LDM exhibit divergence. This discrepancy underscores the remarkable efficiency of the q-HATM in addressing this specific problem.

A robust higher-order finite difference technique for a time-fractional singularly perturbed problem

A higher-order finite difference method is developed to solve the variable coefficients convection–diffusion singularly perturbed problems (SPPs) involving fractional-order time derivative with the order α∈(0,1). The solution to this problem class possesses a typical weak singularity at the initial time t=0 and an exponential boundary layer at the right lateral surface as the perturbation parameter ɛ→0. Alikhanov’s L2−1σ approximation is applied in the temporal direction on a suitable graded mesh, and the spatial variable is discretized on a piecewise uniform Shishkin mesh using a combination of midpoint upwind and central finite difference operators. Stability estimates and the convergence analysis of the fully discrete scheme are provided. It is shown that the fully discrete scheme is uniformly convergent with a rate of O(M−p+N−2(logN)2), where p=min{2,rα}, r is the graded mesh parameter and M,N are the number of mesh points in the time and space direction, respectively. Two numerical examples are taken in counter to confirm the sharpness of the theoretical estimates.

Block-Centered Finite-Difference Methods for Time-Fractional Fourth-Order Parabolic Equations

The block-centered finite-difference method has many advantages, and the time-fractional fourth-order equation is widely used in physics and engineering science. In this paper, we consider variable-coefficient fourth-order parabolic equations of fractional-order time derivatives with Neumann boundary conditions. The fractional-order time derivatives are approximated by L1 interpolation. We propose the block-centered finite-difference scheme for fourth-order parabolic equations with fractional-order time derivatives. We prove the stability of the block-centered finite-difference scheme and the second-order convergence of the discrete L2 norms of the approximate solution and its derivatives of every order. Numerical examples are provided to verify the effectiveness of the block-centered finite-difference scheme.

Timing of steering actions in locomotor interception of targets following curving trajectories.

Recent work on the visual guidance of locomotor interception of nonuniformly moving targets argued for an early reliance on first-order (velocity-based) changes in the target's bearing angle that was complemented approximately 1 second later with reliance on second-order (acceleration-based) changes. Here we provide further support for this hypothesis in a virtual driving task, in which 19 participants steered a vehicle to intercept targets moving along receding circular trajectories. Adopting a set of carefully designed target trajectories, we tested discriminating predictions with respect to the timing and direction of the first steering action. Analyses of temporal and directional characteristics of first steering events revealed a pattern of results that was fully compatible with our predictions. Moreover, application of the recently developed QuID method, focusing on the temporal co-evolution of steering behavior and the potential information sources driving it, confirmed the operative progression from early reliance on first-order changes to subsequent (after approximately 1 second) reliance on a combination of first- and second-order changes in the target's bearing angle over the course of action at the individual-trial level. The finding of an evolution over time toward higher-order informational variables, potentially captured by a fractional-order time derivative, may have consequences for other locomotor interception tasks such as running to catch a fly ball.

Characteristic analysis of waves through piezoelectric semiconductor medium with fractional order time derivative

The article aims to study the characteristics of the waves generated in a medium by reflection from the surface of the solid medium. The theory of non-local elasticity is adopted, in which stress acting at any point is a function of the strain of all points in the neighborhood. The selected medium is an n-type semiconductor and is piezoelectric in nature. Elastic deformation results in the propagation of thermal waves and the conduction process are encountered by the three-phase-lag heat conduction model. A secular equation is obtained for reflected waves generated when an injected wave hits the surface of the media. The analysis indicates four reflected waves with positive projection and one with vibrational mode having zero wave number. All the waves produced are coupled waves and quasi in nature. The obtained theoretical results are later represented graphically for a material with Piezoelectric material. The effect of the order of fractional derivative, non-local parameters, and carrier density on the amplitude ratio of waves is discussed.

Study on reflected waves through visco-elastic solid rotating with fixed angular frequency

The work is about the characteristics study of reflected waves through viscoelastic solid. The solid used has a nanostructure and is semiconductor in nature, with uniform angular frequency. The concept of nonlocal thermoelasticity is considered to study the disturbances propagating through the medium. The refined theory of the multiphase lag heat conduction model is used to encounter the heat disturbance that travels through the medium. We have also considered the concept of fractional order time derivative while analyzing the heat conduction model. Expressions for dispersion relation for the reflected medium are obtained. It is found that four coupled waves reflected into the medium and each wave is neither purely transverse nor dilatational. The analytical results obtained are computed graphically for a particular medium. Characteristics like velocity, attenuation coefficient, and specific heat loss are computed and represented graphically for the medium. In addition to semiconductor nanostructure devices, the findings might be useful in visco-elastic semiconductor nanostructures involved in geology and seismology studies.

Existence of Global and Local Mild Solution for the Fractional Navier–Stokes Equations

Navier–Stokes equations (NS-equations) are applied extensively for the study of various waves phenomena where the symmetries are involved. In this paper, we discuss the NS-equations with the time-fractional derivative of order β∈(0,1). In fractional media, these equations can be utilized to recreate anomalous diffusion equations which can be used to construct symmetries. We examine the initial value problem involving the symmetric Stokes operator and gravitational force utilizing the Caputo fractional derivative. Additionally, we demonstrate the global and local mild solutions in Hα,p. We also demonstrate the regularity of classical solutions in such circumstances. An example is presented to demonstrate the reliability of our findings.

The Kelvin-Voigt visco-elastic model involving a fractional-order time derivative for modelling torsional oscillations of a complex discrete biodynamical system.

Under external loads trees exhibit complex oscillatory behaviour: their canopies twist and band. The great complexity of this oscillatory behaviour consists to an important degree of torsional oscillations. Using a system of ordinary differential fractional-order equations, free and forced main eigen-modes of fractional-type torsional oscillations of a hybrid discrete biodynamical system of complex structures were done. The biodynamical system considered here corresponds to a tree trunk with branches and is in the form of a visco-elastic cantilever of complex structure. Visco-elasticity corresponds to different ages of a tree. We set up a new model of torsional oscillations of a complex discrete, biodynamical system, using the Kelvin-Voigt visco-elastic model involving a fractional-order time derivative. The analytical expressions describing the characteristic properties of its fractional-type oscillations are determined. Based on mathematical and qualitative analogies, this concept represents a new model of torsional oscillations of a light cantilever that takes into account visco-elastic, dissipative properties of the material. Rigid discs are attached to the cantilever. Expressions for kinetic energy, deformation work and a generalized function of the fractional-type energy dissipation of this biodynamical system are defined. Independent main eigen-modes of the fractional type for free and forced torsional oscillations were determined for a special class of such systems, using formulas for the transformation of independent generalized angle coordinates to the principal main eigen-coordinates of the system. The forms of their approximate analytical solutions are shown. In the general case for inhomogeneous biodynamical systems of fractional type, there are no independent main fractional-type eigen-modes of torsional oscillations. The system behaves as a nonlinear system. A new constitutive relation between coupling of torsion loading to a visco-elastic fractional-type cantilever with fractional-type dissipation of cantilever mechanical energy and angle of torsion deformation is determined using a fractional-order derivative. The main advantages of the proposed model are the possibility to analyse torsional oscillations of more complex structures and the possibility to analyse complex cantilevers with different cross-sectional diameters.

Effect of Fractional Order Time Derivative on the Out-Diffusion Profiles During Degassing a Thin Plate in Vacuum

Effect of Fractional Order Time Derivative on the Out-Diffusion Profiles During Degassing a Thin Plate in Vacuum

3-D Forward Modeling of Transient EM Field in Rough Media Using Implicit Time-Domain Finite-Element Method

In a heterogeneous medium (usually called a rough medium) with fractured formations, the propagation of an electromagnetic (EM) field is a type of subdiffusion. Current mainstream geophysical EM data processing methods cannot be applied to data acquired on heterogeneous Earth, as they are not governed by the classic diffusion theory. To evaluate the influence of roughness on the transient EM (TEM) signal for a complex model and contribute to data inversion, we proposed a novel three-dimensional (3-D) forward modeling scheme for TEM in rough media. First, we derived the governing equation with a fractional-order time derivative for the subdiffusion of EM waves in rough media. Then, we proposed a novel time discretization using an unequal step length for the Caputo operator, which significantly reduces the total number of time steps. Finally, an implicit time-domain finite-element method using unstructured tetrahedron discretization was adopted to solve the 3-D forward problem. Furthermore, an efficient time segmentation strategy combined with parallel RHS construction was proposed to accelerate modeling. The numerical results prove that the proposed method is accurate and efficient, and will be a powerful numerical method for analyzing TEM wave propagation and processing TEM data in areas with multiscale fractures or porosity.

Local and Global Mild Solution for Gravitational Effects of the Time Fractional Navier–Stokes Equations

The gravitational effect is a physical phenomenon that explains the motion of a conductive fluid flowing under the impact of an exterior gravitational force. In this paper, we work on the Navier–Stokes equations (NSES) of the fluid flowing under the impact of an exterior gravitational force inclined at an angle of 45∘ with A time-fractional derivative of order β∈(0,1). To encourage anomalous diffusion in fractal media, we apply these equations. In Hδ,r, we prove the existence and uniqueness of local and global mild solutions. Additionally, we provide moderate local solutions in Jr. Additionally, we establish the regularity and existence of classical solutions to these equations in Jr.

A fully discrete spectral scheme for time fractional Cahn-Hilliard equation with initial singularity

In this paper, we study the numerical approximation of the time fractional Cahn-Hilliard equation with initial singularity. A nonlinear fully discrete scheme is constructed using the L2- 1 σ formula of time fractional order derivative on graded meshes and the Fourier spectral method in space. We present a priori estimate and give the existence and uniqueness of the approximate solution. The stability and convergence of the proposed scheme are proved rigorously. A numerical example is carried out to illustrate the theoretical analysis.

Novel soliton solutions for the fractional three-wave resonant interaction equations

Abstract In this article, we obtained new infinite sets of exact soliton solutions for the nonlinear evolution system of three-wave resonant interaction equations. The solved system contains the non-zero second-order dispersion coefficients, the non-zero phase velocity mismatch, and the conformable fractional time derivative of order between zero and one. The solution method is a constructed ansatz that consists of linear combinations of the tan and cotan hyperbolic functions with complex coefficients. We stated clear systematic steps toward writing an exact soliton solution for the studied system. To show the efficiency of this method, we introduced some numerical examples on each obtained set of solutions. The computations showed that similar solutions can be obtained if one replaces the tan and cotan hyperbolic functions with the tan and cotan trigonometric functions. The new obtained fractional solutions could be useful in studying the broad applications of triad resonances in plasma physics and in nonlinear optics.

Study on velocity of waves through piezoelectric semiconductor rotating with fix angular velocity in context of fractional-order time derivative

Study on velocity of waves through piezoelectric semiconductor rotating with fix angular velocity in context of fractional-order time derivative

Local and Global Existence and Uniqueness of Solution for Time-Fractional Fuzzy Navier–Stokes Equations

Navier–Stokes (NS) equation, in fluid mechanics, is a partial differential equation that describes the flow of incompressible fluids. We study the fractional derivative by using fractional differential equation by using a mild solution. In this work, anomaly diffusion in fractal media is simulated using the Navier–Stokes equations (NSEs) with time-fractional derivatives of order β∈(0,1). In Hγ,℘, we prove the existence and uniqueness of local and global mild solutions by using fuzzy techniques. Meanwhile, we provide a local moderate solution in Banach space. We further show that classical solutions to such equations exist and are regular in Banach space.

EXACT SOLITON SOLUTIONS FOR CONFORMABLE FRACTIONAL SIX WAVE INTERACTION EQUATIONS BY THE ANSATZ METHOD

In this paper, a conformable fractional time derivative of order [Formula: see text] is considered in view of the Lax-pair of nonlinear operators to derive a fractional nonlinear evolution system of partial differential equations, called the Fractional-Six-Wave-Interaction-Equations, which is derived in terms of one temporal plus one and two spatial dimensions. Further, an ansatz consisting of linear combinations of hyperbolic functions with complex coefficients is utilized to obtain an infinite set of exact soliton solutions for this system. Certain numerical examples are introduced to show the effectiveness of the ansatz method in obtaining exact solutions for similar systems of nonlinear evolution equations.

Abstract differential equations and Caputo fractional derivative

In this work we consider the abstract Cauchy problem with Caputo fractional time derivative of order $$\alpha \in (0,1]$$ , and discuss the continuity of the respective solutions regarding the parameter $$\alpha $$ . We also present a study about the continuity of the Mittag-Leffler families of operators (for $$\alpha \in (0,1]$$ ), when they are induced by sectorial operators.

An inverse source problem of a semilinear time-fractional reaction–diffusion equation

In this paper, we investigate an inverse problem of determining the time-dependent source coefficient in a semilinear reaction–diffusion equation involving the Caputo fractional time derivative of order in the case of nonlocal boundary and integral overdetermination conditions. We prove the existence and uniqueness of the classical solution by Fourier analysis and the iteration method. Moreover, we show the continuous dependence of this solution upon the additional data of the inverse problem.