Time-fractional Derivative Of Order Research Articles

A two-parameter Tikhonov regularization for a fractional sideways problem with two interior temperature measurements

This paper deals with a fractional sideways problem of determining the surface temperature of a heat body from two interior temperature measurements. Mathematically, it is formulated as a problem for the one-dimensional heat equation with Caputo fractional time derivative of order α∈(0,1], where the data are given at two interior points, namely x=x1 and x=x2, and the solution is determined for x∈(0,L),0<x1<x2≤L. The problem is challenging since it is severely ill-posed for x∉[x1,x2]. For the ill-posed problem, we apply the Tikhonov regularization method in Hilbert scales to construct stable approximation problems. Using the two-parameter Tikhonov regularization, we obtain the order optimal convergence estimates in Hilbert scales by using both a priori and a posteriori parameter choice strategies. Numerical experiments are presented to show the validity of the proposed method.

Error analysis of an L2-type method on graded meshes for semilinear subdiffusion equations

A semilinear initial–boundary value problem with a Caputo time derivative of fractional order α∈(0,1) is considered, solutions of which typically exhibit a singular behaviour at an initial time. For an L2-type discretization of order 3−α, we give sharp pointwise-in-time error bounds on graded temporal meshes with arbitrary degree of grading.

Numerical discretization of initial–boundary value problems for PDEs with integer and fractional order time derivatives

This paper is mainly concerned with introducing a numerical method for solving initial–boundary value problems with integer and fractional order time derivatives. The method is based on discretizing the considered problems with respect to spatial and temporal domains. With the help of finite difference methods, we transformed the studied problem into a set of fractional differential equations. Then, we implemented the fractional Adams method to solve this set in order to provide approximate solutions to the main problem. This combination results in an algorithm that can efficiently and accurately solve a general class of integer and fractional order initial–boundary value problems, such that it does not need to solve large systems of linear equations. In addition, we discussed the stability of the proposed scheme. Three illustrative examples are numerically solved to reveal the effectiveness and validity of the proposed technique.

A novel multi-frame image super-resolution model based on regularized nonlinear diffusion with Caputo time fractional derivative

In this work, we introduce an innovative fractional nonlinear parabolic model using a time-fractional order derivative, specifically employing the Caputo sense for fractional differentiation. This model aims to enhance traditional super-resolution models, particularly in the context of multi-frame image super-resolution. Additionally, we incorporate a regularized Perona–Malik diffusion mechanism to control the speed and direction of diffusion at each image location. We begin our study by exploring the theoretical solvability of our proposed model. Firstly, we employ the Faedo–Galerkin approach to establish the existence and uniqueness of a weak solution for an auxiliary fractional super-resolution model. Subsequently, we use the Schauder fixed point method to demonstrate the existence and uniqueness of a weak solution for our model. To validate the effectiveness of our model in the multi-frame super-resolution (SR) context, we conduct numerical experiments on images featuring diverse characteristics, including corners and edges, while applying various warping, decimation, and blurring matrices to the low-resolution (LR) images. We start the evaluation by introducing an adaptive discrete scheme tailored to the proposed model. To prove the robustness of our approach, we subject our images to varying levels of noise. Additionally, we perform simulations on real data (videos). The obtained high-resolution (HR) results demonstrate notable efficiency and robustness against noise, outperforming competitive models both visually and quantitatively.

Effects of Brownian motion on solitary wave structures for 1D stochastic Poisson–Nernst–Planck system in electrobiochemical

This study deals with the time fractional 1D stochastic Poisson–Nernst–Planck (TFSPNP) system under the effect of multiplicative time noise. The M-truncated derivative (MTD) takes into consideration the fractional order time derivative. This is a steady-state Poisson–Nernst–Planck (PNP) equations that have applications in bioelectric dressings and bandages. To obtain the soliton solutions of TFSPNP, we use the generalized exponential rational functional method. These findings are presented in the form of trigonometric, exponential, and hyperbolic functions. Moreover, to show the effect of multiplicative time noise and MTD, we construct the plot of some solutions in the form of three-dimensional, two-dimensional, and their corresponding contours. These plots clearly show the effect of randomness in the wave structures for the exact solitary wave solutions that are attained. In general, the solutions become more stable when a noise term disrupts their symmetry.

Analysis on reflected waves through semiconductor nanostructure medium with temperature dependent properties

The article is about the study of reflected waves through the surface of an elastic solid. The medium considered for the propagation of waves is homogeneous isotropic with semiconductor properties. The thermoelastic properties of the medium are a function of temperature. The governing equations are formulated by using non-local elastic theory. The conduction process of heat is studied by using the concept of three-phase lag along with fractional order time derivative. After reflecting through the surface one transverse and three longitudinal waves travel back into the media. The reflection coefficients and their energy ratios are computed analytically. Numerically simulated results are obtained for the intrinsic semiconductor material Silicon to depict the effect of temperature dependency parameters, nonlocal parameters, and time derivative fractional order on the different reflection coefficients. Some published results are also discussed as special cases.

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.

MATHEMATICAL MODELLING OF THE IMPACT OF CHEMOTHERAPY ON THE STATE OF A CANCEROUS TUMOR BASED ON FRACTIONAL CALCULUS

The article is dedicated to constructing difference approximations of fractal operators in a mathematical model of the impact of chemotherapy on the state of a cancerous tumor, based on fractional calculus using the Caputo derivative. A mathematical model of stem cells and chemotherapy is presented. Numerical algorithms for implementing fractional-order mathematical models have been developed using the Atangana-Toufik method. The UML diagram of the software application and its development process are described. The impact of fractal characteristics (long-term memory) of chemotherapy on the state of a cancerous tumor is analysed. The presence of a fractional-order time derivative as a parameter of the solutions provides important information for predicting the effects of chemotherapy on the tumor's state.

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.