Time Fractional Operator Research Articles

A novel image denoising technique with Caputo type space–time fractional operators

A novel image denoising model, namely Full Fractional Total Variation (TVFF), based on the Rudin-Osher-Fatemi (ROF) and the fractional total variation models is presented. The leading advantage of TVFF model is that it uses fractional derivatives with length scale parameters instead of ordinary derivatives with respect to both time and spatial variables in the diffusion equation. The Riesz–Caputo fractional derivative operator is used to disperse nonlocal influence throughout all directions, whereas the Caputo fractional derivative concept is employed for time fractional derivatives. Therefore, the influence of neighboring pixels is given greater weight compared to those situated farther away and this reflects the consideration behind denoising process better. Moreover, the numerical approach is constructed, and its stability and convergence properties are thoroughly examined. To show the superiority of our model, the denoised images are subjected to visual and numerical comparisons using metrics such as the Signal-to-Noise Ratio (SNR), the Structural Similarity Index Measure (SSIM) and the Edge-Retention Ratio (ERR). The performance of the TVFF method is evaluated under various types of noise, including Poisson, Speckle, and Salt & Pepper, and the results are compared with those obtained using Gauss and Median Filters. Furthermore, the proposed method is applied to both blind and synthetic images, thereby showcasing its versatility and applicability across diverse datasets. The outcomes showcase the substantial potential of our enhanced model as a versatile and efficient tool for image denoising.

A meshless method based on the Laplace transform for multi-term time-space fractional diffusion equation

<abstract><p>Multi-term fractional diffusion equations can be regarded as a generalisation of fractional diffusion equations. In this paper, we develop an efficient meshless method for solving the multi-term time-space fractional diffusion equation. First, we use the Laplace transform method to deal with the multi-term time fractional operator, we transform the time into complex frequency domain by Laplace transform. The properties of the Laplace transform with respect to fractional-order operators are exploited to deal with multi-term time fractional-order operators, overcoming the dependence of fractional-order operators with respect to time and giving better results. Second, we proposed a meshless method to deal with space fractional operators on convex region based on quintic Hermite spline functions based on the theory of polynomial functions dense theorem. Meanwhile, the approximate solution of the equation is obtained through theory of the minimum residual approximate solution, and the error analysis are provided. Third, we obtain the numerical solution of the diffusion equation by inverse Laplace transform. Finally, we first experimented with a single space-time fractional-order diffusion equation to verify the validity of our method, and then experimented with a multi-term time equation with different parameters and regions and compared it with the previous method to illustrate the accuracy of our method.</p></abstract>

A HIGHER-ORDER APPROACH FOR TIME-FRACTIONAL GENERALIZED BURGERS’ EQUATION

A fast higher-order scheme is established for solving inhomogeneous time-fractional generalized Burgers’ equation. The time-fractional operator is taken as the modified operator with the Mittag-Leffler kernel. Through stability analysis, it has been demonstrated that the proposed numerical approach is unconditionally stable. The convergence of the numerical method is analyzed theoretically using von Neumann’s method. It has been proved that the proposed numerical method is fourth-order convergent in space and second-order convergent in time in the [Formula: see text]-norm. The scheme’s proficiency and effectiveness are examined through two numerical experiments to validate the theoretical estimates. The tabular and graphical representations of numerical results confirm the high accuracy and versatility of the scheme.

DOUBLE DIFFUSIVE MAGNETO-FREE CONVECTION FLOW OF A MAXWELL FLUID OVER A VERTICAL PLATE: SPECIAL FUNCTIONS BASED ANALYSIS USING LOCAL AND NONLOCAL KERNELS TO HEAT AND MASS FLUX SUBJECT TO EXPONENTIAL HEATING

The purpose of this research is to analyze the general equations of double diffusive magneto-free convection in a rate type fluid presented in non-dimensional form, and apply to a moving heated vertical plate as in the boundary layer flow up, with existence of externally magnetic fields which are either moving or fixed consistent with the plate. Thermal transport phenomenon is discussed in the presence of constant concentration coupled with first-order chemical reaction with exponential heating. An innovative definition in power law (CF) and Mittag-Leffler (ABC) kernels form time fractional operators are implemented to hypothecate the constitutive mass, heat and momentum equations. The results based on special functions are obtained by using the technique of Laplace transformation to tackle the non-dimensional equations for velocity, mass and energy. The contribution of mass, thermal and mechanical components on the dynamics of fluid is presented and discussed independently. An interesting property regarding the behavior of the fluid velocity is found when the movement is observed in the magnetic intensity along with the plate. In that situation, the fluid velocity is not to be zero when it is far and away from the plate. Moreover, the heat transfer aspects, flow dynamics and their credence on the parameters are drawn out by graphical illustrations along with some special cases for the movement of the plate which are also studied.

Stochastic solutions of generalized time-fractional evolution equations

We consider a general class of integro-differential evolution equations which includes the governing equation of the generalized grey Brownian motion and the time- and space-fractional heat equation. We present a general relation between the parameters of the equation and the distribution of the underlying stochastic processes, as well as discuss different classes of processes providing stochastic solutions of these equations. For a subclass of evolution equations, containing Marichev-Saigo-Maeda time-fractional operators, we determine the parameters of the corresponding processes explicitly. Moreover, we explain how self-similar stochastic solutions with stationary increments can be obtained via linear fractional Lévy motion for suitable pseudo-differential operators in space.

Fractional Modeling of Viscous Fluid over a Moveable Inclined Plate Subject to Exponential Heating with Singular and Non-Singular Kernels

In this paper, a new approach to investigating the unsteady natural convection flow of viscous fluid over a moveable inclined plate with exponential heating is carried out. The mathematical modeling is based on fractional treatment of the governing equation subject to the temperature, velocity and concentration field. Innovative definitions of time fractional operators with singular and non-singular kernels have been working on the developed constitutive mass, energy and momentum equations. The fractionalized analytical solutions based on special functions are obtained by using Laplace transform method to tackle the non-dimensional partial differential equations for velocity, mass and energy. Our results propose that by increasing the value of the Schimdth number and Prandtl number the concentration and temperature profiles decreased, respectively. The presence of a Prandtl number increases the thermal conductivity and reflects the control of thickness of momentum. The experimental results for flow features are shown in graphs over a limited period of time for various parameters. Furthermore, some special cases for the movement of the plate are also studied and results are demonstrated graphically via Mathcad-15 software.

Magnetic Field Effect on Heat and Momentum of Fractional Maxwell Nanofluid within a Channel by Power Law Kernel Using Finite Difference Method

The mathematical model of physical problems interprets physical phenomena closely. This research work is focused on numerical solution of a nonlinear mathematical model of fractional Maxwell nanofluid with the finite difference element method. Addition of nanoparticles in base fluids such as water, sodium alginate, kerosene oil, and engine oil is observed, and velocity profile and heat transfer energy profile of solutions are investigated. The finite difference method involving the discretization of time and distance parameters is applied for numerical results by using the Caputo time fractional operator. These results are plotted against different physical parameters under the effects of magnetic field. These results depicts that a slight decrease occurs for velocity for a high value of Reynolds number, while a small value of Re provides more dominant effects on velocity and temperature profile. It is observed that fractional parameters α and β show inverse behavior against u(y, t) and θ(y, t). An increase in volumetric fraction of nanoparticles in base fluids decreases the temperature profile of fractional Maxwell nanofluids. Using mathematical software of MAPLE, codes are developed and executed to obtain these results.

The time evolution of the large exponential and power population growth and their relation to the discrete linear birth-death process

<abstract><p>The Feller exponential population growth is the continuous analogues of the classical branching process with fixed number of individuals. In this paper, I begin by proving that the discrete birth-death process, $ M/M/1 $ queue, could be mathematically modelled by the same Feller exponential growth equation via the Kolmogorov forward equation. This equation mathematically formulates the classical Markov chain process. The non-classical linear birth-death growth equation is studied by extending the first-order time derivative by the Caputo time fractional operator, to study the effect of the memory on this stochastic process. The approximate solutions of the models are numerically studied by implementing the finite difference method and the fourth order compact finite difference method. The stability of the difference schemes are studied by using the Matrix method. The time evolution of these approximate solutions are compared for different values of the time fractional orders. The approximate solutions corresponding to different values of the birth and death rates are also compared.</p></abstract>

The Extension of the Physical and Stochastic Problems to Space-Time-Fractional Differential Equations

The fractional calculus gains wide applications nowadays in all fields. The implementation of the fractional differential operators on the partial differential equations make it more reality. The space-time-fractional differential equations mathematically model physical, biological, medical, etc., and their solutions explain the real life problems more than the classical partial differential equations. Some new published papers on this field made many treatments and approximations to the fractional differential operators making them loose their physical and mathematical meanings. In this paper, I answer the question: why do we need the fractional operators?. I give brief notes on some important fractional differential operators and their Grünwald-Letnikov schemes. I implement the Caputo time fractional operator and the Riesz-Feller operator on some physical and stochastic problems. I give some numerical results to some physical models to show the efficiency of the Grünwald-Letnikov scheme and its shifted formulae. MSC 2010: Primary 26A33, Secondary 45K05, 60J60, 44A10, 42A38, 60G50, 65N06, 47G30,80-99

Magnetic field diffusion in ferromagnetic materials: fractional calculus approaches

The paper addresses diffusion approximations of magnetic field penetration of ferromagnetic materials with emphasis on fractional calculus applications and relevant approximate solutions. Examples with applications of time-fractional semi-derivatives and singular kernel models (Caputo time fractional operator) in cases of field independent and field-dependent magnetic diffusivities have been developed: Dirichlet problems and time-dependent boundary condition (power-law ramp). Approximate solutions in all theses case have been developed by applications of the integral-balance method and assumed parabolic profile with unspecified exponents. Tow version of the integral method have been successfully implemented: SDIM (single integration applicable to time-fractional semi-derivative model) and DIM (double-integration model to fractionalized singular memory models). The fading memory approach in the sense of the causality concept and memory kernel effect on the model constructions have been discussed.

Numerical Solution of Fractional Order Anomalous Subdiffusion Problems Using Radial Kernels and Transform

By coupling of radial kernels and localized Laplace transform, a numerical scheme for the approximation of time fractional anomalous subdiffusion problems is presented. The fractional order operators are well suited to handle by Laplace transform and radial kernels are also built for high dimensions. The numerical computations of inverse Laplace transform are carried out by contour integration technique. The computation can be done in parallel and no time sensitivity is involved in approximating the time fractional operator as contrary to finite differences. The proposed numerical scheme is stable and accurate.

NEW EXISTENCE, UNIQUENESS RESULTS FOR MULTI-DIMENSIONAL MULTI-TERM CAPUTO TIME-FRACTIONAL MIXED SUB-DIFFUSION AND DIFFUSION-WAVE EQUATION ON CONVEX DOMAINS

In this paper, we investigate an efficient analytical method known as two step Adomian decomposition method (TSADM). This method does not require approximation/discretization, lengthy calculations and due to involvement of fractional operators and provides an exact solution. In this study, we generalize the multi-term time-fractional mixed sub-diffusion and diffusion-wave equation into multi dimensions with Caputo derivative for time fractional operators and obtain the exact solution. Furthermore, we establish the new results of existence and uniqueness of the solution using fixed point theory. To demonstrate the effectiveness of the proposed method, several generalized examples on the convex domain are considered.

Flow and heat transfer of generalized Maxwell fluid over a moving plate with distributed order time fractional constitutive models

The incompressible, steady and laminar fluid boundary flow and heat transfer through a moving plate subject to a kind of novel constitution relationships containing the relaxation time parameters and distributed order time fractional operators are originally introduced. Formulated distributed order time fractional equations governing the flow and heat consider the relaxation characteristic and a spectrum of memory and nonlocal characteristics. Solutions are obtained numerically that distributed order integrals are approximated by the summation of multi-fractional terms and the fractional derivatives are discretised by the L1 scheme and L2 scheme. Two source items are introduced and, consequently, the exact solutions are defined. Afterward, the comparison between the exact solutions and numerical solutions is given which verifies the correctness of the computed results. The repercussion of dynamic parameters on boundary layer flow and heat transfer is deliberated and shown by graphical illustrations. Results show that the velocity and temperature boundary layers become thicker with the increase of time parameter or with the decrease of relaxation time parameter. For a larger Prandtl number, the temperature boundary layer becomes thinner. Besides, the comparisons between the distributed order time fractional governing equation possessing the monotone decreasing gamma function distributed order and the uniformly distributed order with the classical time fractional governing equation are discussed and analysed.

Simulation of Nonlinear Pyroelectric Response of Ferroelectrics near Phase Transition: Fractional Differential Approach

The paper is devoted to mathematical modeling pyroelectric current of ferroelectric single crystal under the conditions of intensive light heating in view of fractal behavior of these materials. The proposed approach is based on numerical simulation of thermal distribution in a ferroelectric sample using time fractional operator as well as computation of pyroelectric response. The simulation results for typical TGS ferroelectric crystal were described in one-dimensional case of the model in comparison with experimental data. Pyroelectric signals depending on temperature pyroelectric coefficient and thermal physical characteristics were also analyzed.

Finite difference scheme for simulating a generalized two-dimensional multi-term time fractional non-Newtonian fluid model

A finite difference scheme, based upon the Crank–Nicolson scheme, is applied to the numerical approximation of a two-dimensional time fractional non-Newtonian fluid model. This model not only possesses a multi-term time derivative, but also contains a special time fractional operator on the spatial derivative. And a very important lemma is proposed and also proved, which plays a vital role in the proof of the unconditional stability. The stability and convergence of the finite difference scheme are discussed and theoretically proved by the energy method. Numerical experiments are given to validate the accuracy and efficiency of the scheme, and the results indicate that this Crank–Nicolson difference scheme is very effective for simulating the generalized non-Newtonian fluid diffusion model.

Novel Numerical Analysis of Multi-Term Time Fractional Viscoelastic Non-Newtonian Fluid Models for Simulating Unsteady Mhd Couette Flow of a Generalized Oldroyd-B Fluid

Abstract In this paper, we consider the application of the finite difference method for a class of novel multi-term time fractional viscoelastic non-Newtonian fluid models. An important contribution of the work is that the new model not only has a multi-term time derivative, of which the fractional order indices range from 0 to 2, but also possesses a special time fractional operator on the spatial derivative that is challenging to approximate. There appears to be no literature reported on the numerical solution of this type of equation. We derive two new different finite difference schemes to approximate the model. Then we establish the stability and convergence analysis of these schemes based on the discrete H 1 norm and prove that their accuracy is of O(τ + h 2) and O(τ min{3–γs ,2–αq ,2–β}+h 2), respectively. Finally, we verify our methods using two numerical examples and apply the schemes to simulate an unsteady magnetohydrodynamic (MHD) Couette flow of a generalized Oldroyd-B fluid model. Our methods are effective and can be extended to solve other non-Newtonian fluid models such as the generalized Maxwell fluid model, the generalized second grade fluid model and the generalized Burgers fluid model.

Finite Element Method for Two-Dimensional Unsteady MHD Couette Flow of a Generalized Oldroyd-B Fluid

In recent years, non-Newtonian fluids have been widely applied in engineering and industry. Due to the complexity of these fluids, there are several constitutive equations of non- Newtonian fluids. One particular subclass of non-Newtonian fluids is the class of a generalized Oldroyd-B fluids with fractional derivative constitutive equations, which has been found to approximate the response of many dilute polymeric liquids. Although the analytic solution of these equations can be derived, it contains a multi-level sum of an infinite series and Fox H-functions, which is extremely complex and difficult to evaluate. Therefore, seeking the numerical solution of the equation is of great importance. Different from the general time fractional diffusion equation, the constitutive equation is a novel multi-term time fractional diffusion equation and has a special time fractional operator on the spatial derivative, which is challenging to approximate. In this paper, we will consider the finite element method for the two-dimensional multi-term time fractional diffusion equation. Firstly, we utilise the L1 and L2 schemes to approximate the two different time fractional derivatives, respectively. Secondly, we apply the finite element method to the equation and establish the fully variational formulation. Then we adopt linear polynomials on triangular elements to derive the matrix form of the numerical scheme. Furthermore, we present the stability and convergence of the scheme. Finally, two numerical examples are investigated to show the effectiveness of our method, in which a two-dimensional unsteady MHD Couette flow of a generalized Oldroyd-B fluid is considered.

Application of the modern trend of fractional differentiation to the MHD flow of a generalized Casson fluid in a microchannel: Modelling and solution⋆

The motivation for writing this article is based on the contributions of Atangana and Baleanu to the fractal and fractional differentiation. Recently, Atangana and Baleanu launched a new fractional operator, namely, the Atangana-Baleanu fractional operator with the Mettang-Leffler function as the kernel of integration. This new operator is an efficient tool to model complex and real-world problems. This article deals with modeling and solution of the generalized magnetohydrodynamic (MHD) flow of a Casson fluid in a microchannel. This microchannel is taken of infinite length in the vertical direction and of finite width in the horizontal direction. The flow is modeled in terms of a set of partial differential equations involving the Atnagana-Baleanu time fractional operator along with physical initial and boundary conditions. The partial differential equations are transformed to ordinary differential equations via the fractional Laplace transformation and are solved for an exact solution using the transformed conditions. To explore the physical significance of various pertinent parameters affecting the flow, the numerical techniques of Zakian and Tzou are utilized for inversion of the Laplace transformation. The results obtained here may have useful industrial and engineering applications.

Parareal algorithms with local time-integrators for time fractional differential equations

It is challenge work to design parareal algorithms for time-fractional differential equations due to the historical effect of the fractional operator. A direct extension of the classical parareal method to such equations will lead to unbalance computational time in each process. In this work, we present an efficient parareal iteration scheme to overcome this issue, by adopting two recently developed local time-integrators for time fractional operators. In both approaches, one introduces auxiliary variables to localized the fractional operator. To this end, we propose a new strategy to perform the coarse grid correction so that the auxiliary variables and the solution variable are corrected separately in a mixed pattern. It is shown that the proposed parareal algorithm admits robust rate of convergence. Numerical examples are presented to support our conclusions.

Numerical methods and analysis for simulating the flow of a generalized Oldroyd-B fluid between two infinite parallel rigid plates

In recent years, non-Newtonian fluids have been widely applied in a number of engineering applications. One particular subclass of non-Newtonian fluids is the generalized Oldroyd-B fluids with fractional derivative constitutive equations, which can be used to approximate the response of many dilute polymeric liquids. Different to the general time fractional diffusion equation, the constitutive equation not only has a multi-term time derivative but also possesses a special time fractional operator on the spatial derivative, which is challenging to approximate. The literature reported on the numerical solution of this model is extremely sparse. In this paper, we will consider the finite difference method for its discretisation and propose a new scheme to approximate the time fractional derivative. Then we derive an implicit finite difference scheme and establish some new theoretical analysis of the stability and convergence. Furthermore, we present a numerical scheme to improve the time order. Finally, we present two numerical examples to show the effectiveness of our method and apply it to solve the generalized Oldroyd-B fluid model.