Caputo Time-fractional Derivative Research Articles

The Space-Time Coupled Fractional Cattaneo-Friedrich Maxwell Model with Caputo Derivatives.

In the current article, we have thoroughly investigated the collective impact of mixed convection with thermal radiation and chemical reaction on MHD flow of viscous and electrically conducting fluid (Cattaneo–Friedrich Maxwell-CFM model) over a permeable surface embedded in a porous medium. Here we have utilized the Caputo time-fractional derivatives and mechanical laws (generalized shear stress constitutive equation and generalized Fourier’s and Fick’s laws) are being used to fractionalize the presented model. The effects of radiative heat flux, Ohmic dissipation, and internal absorption are presented through generalized Fourier’s law while Fick’s law or mass transfer equation offers the effects of first order chemically reactive species. The finite element method and finite difference method are being utilized to numerically solve the nonlinear coupled differential equations. It is established, through compression of numerical and analytical solutions, that the presented model is convergent. Further, error analysis of the subject model is also carried out. Moreover, for better illustration of results, we have also offered a graphical and tabular presentation of impacts of the parameters of interest on velocity, temperature, concentration profile, local skin friction coefficient, and heat and mass transfer. It is evident from the obtained results that velocity near and away from the surface increases with the enhancement of fractional derivative parameter whereas an opposite trend is observed in the case of temperature. Furthermore, it is noticed that temperature shows a decreasing behavior for the value {Lambda }_{theta }<2 and {Lambda }_{phi }<2, on the other hand entirely opposite trend is witnessed for {Lambda }_{theta }ge 3 and {Lambda }_{phi }ge 3. From an engineering perspective, we have acquired comprehensive outcomes such that the heat transfer offers an increasing trend in the case of TR and thermal fractional parameter beta_{1} . Additionally, the chemical reaction parameter and Sc significantly contribute towards the mass transfer rate. Since, in literature, one cannot refer to such results with non-integer Caputo fractional derivatives thus the results obtained through the current assessment hold significance for future research avenues. Moreover, the numerical inferences of the subject study may contribute to an advanced thermal processing method in the food industry to swiftly increase the temperature for cooking or sterilization drives.

Analytical Solutions of the Fractional Mathematical Model for the Concentration of Tumor Cells for Constant Killing Rate

Two generalized mathematical models with memory for the concentration of tumor cells have been analytically studied using the cylindrical coordinate and the integral transform methods. The generalization consists of the formulating of two mathematical models with Caputo-time fractional derivative, models that are suitable to highlight the influence of the history of tumor evolution on the present behavior of the concentration of cancer cells. The time-oscillating concentration of cancer cells has been considered on the boundary of the domain. Analytical solutions of the fractional differential equations of the mathematical models have been determined using the Laplace transform with respect to the time variable and the finite Hankel transform with respect to the radial coordinate. The positive roots of the transcendental equation with Bessel function J0(r)=0, which are needed in our study, have been determined with the subroutine rn=root(J0(r),r,(2n−1)π/4,(2n+3)π/4),n=1,2,… of the Mathcad 15 software. It is found that the memory effects are stronger at small values of the time, t. This aspect is highlighted in the graphical illustrations that analyze the behavior of the concentration of tumor cells. Additionally, the concentration of cancer cells is symmetric with respect to radial angle, and its values tend to be zero for large values of the time, t.

Towards the exact solutions of Burger’s fluid flow through arteries with fractional time derivative magnetic field and thermal radiation effects

We consider the unsteady flow of Burger fluid within a circular cylindrical tube, driven by a time-dependent pressure gradient, a body acceleration and a magnetic field acting normal to the flow direction. The solutions of the fractional constitutive equations governing the unsteady Burger’s fluid flow through arterial walls were obtained via the Laplace transform and the finite Hankel transform. The effects of magnetic field on parameters such as blood temperature and velocity were studied by Caputo time-fractional derivatives. We note that the solutions of many particular models such as fractional Oldroyd-B fluid, fractional Maxwell fluid, fractional second grade fluid and fractional Newtonian fluid models can be recovered from the solutions of the fractional constitutive equations governing the unsteady Burger’s fluid flow by particularizing the material coefficients (i.e. special and limiting cases of the earlier Burger’s fluid model). The numerical computations have been carried out to analyze the effects of fractional parameter α, similarity parameter β, relaxation time λ1, retardation time λ3, radius of the circular cylinder R0 and material parameter λ2 on the blood velocity and temperature. Some interesting flow and temperature characteristics are presented graphically and discussed. The study reveals that blood velocity, temperature and fractional parameters are reduced in the presence of magnetic field. The importance of this study can be found in the application fields of magnetic field control of biotechnological processes, bio magnetic device technology, biomedical engineering, medicine, etc.

A fast difference scheme for the variable coefficient time-fractional diffusion wave equations

In this paper, we focus on the fast computation for solving the fourth-order time fractional diffusion wave equations with variable coefficient. A fast difference scheme is derived by applying the FL2−1σ formula to approximate the time Caputo fractional derivative. The resulting FL2−1σ scheme keeps the almost same accuracy with the traditional L2−1σ scheme, but it reduces the computational complexity significantly. The solvability, unconditional stability and convergence under the maximum norm of the scheme are proved strictly by using the discrete energy method. Numerical results are given to verify the performance of our scheme.

Analysis of a robust implicit scheme for space–time fractional stochastic nonlinear diffusion wave model

The current paper develops and analyzes a numerical scheme for the space–time fractional stochastic nonlinear diffusion wave equations. The implicit scheme is based on the matrix transform technique for discretizing the Riesz-space fractional derivative, -order approximation to the Caputo-fractional derivative in time and Taylor's series method to linearize the nonlinear source term, and has been successfully applied to solve a class of nonlinear fractional diffusion wave equation. We prove that the implicit scheme is convergent with β-order in space and order in time, respectively. The optimum error estimates in the temporal-spatial direction and unconditional stability of the implicit scheme have been theoretically investigated. Moreover, several specific numerical experiments confirm the consistency and high efficacy of the provided algorithms, which minimizes the computational costs.

A collocation method based on cubic trigonometric B-splines for the numerical simulation of the time-fractional diffusion equation

Fractional differential equations sufficiently depict the nature in view of the symmetry properties, which portray physical and biological models. In this paper, we present a proficient collocation method based on cubic trigonometric B-Splines (CuTBSs) for time-fractional diffusion equations (TFDEs). The methodology involves discretization of the Caputo time-fractional derivatives using the typical finite difference scheme with space derivatives approximated using CuTBSs. A stability analysis is performed to establish that the errors do not magnify. A convergence analysis is also performed The numerical solution is obtained as a piecewise sufficiently smooth continuous curve, so that the solution can be approximated at any point in the given domain. Numerical tests are efficiently performed to ensure the correctness and viability of the scheme, and the results contrast with those of some current numerical procedures. The comparison uncovers that the proposed scheme is very precise and successful.

Simultaneous Flow of n-Immiscible Fractional Maxwell Fluids with Generalized Thermal Flux and Robin Boundary Conditions

In a rectangular region, the multilayered laminar unsteady flow and temperature distribution of the immiscible Maxwell fractional fluids by two parallel moving walls are studied. The flow of the fluid occurs in the presence of Robin’s boundaries and linear fluid-fluid interface conditions due to the motion of the parallel walls on its planes and the time-dependent pressure gradient. The problem is defined as a mathematical model which focuses on the fluid memory, which is represented by a constituent equation with the Caputo time-fractional derivative. The integral transformations approach (the Laplace transform and the finite sine-Fourier transform) is used to determine analytical solutions for velocity, shear stress, and the temperature fields with fluid interface, initial, and boundary conditions. For semianalytical solutions, the algorithms of Talbot are used to calculate the Laplace inverse transformation. We used the Mathcad software for graphical illustration and numerical computation. It has been observed that the memory effect is significant on both fluid motion and temperature flow.

L1/LDG method for the generalized time-fractional Burgers equation

In this paper, we study the generalized time fractional Burgers equation, where the time fractional derivative is in the sense of Caputo with derivative order in (0,1). If its solution u(x,t) has strong regularity, for example u(⋅,t)∈C2[0,T] for a given time T, then we use the L1 scheme on uniform meshes to approximate the Caputo time-fractional derivative, and use the local discontinuous Galerkin (LDG) method to approach the space derivative. However, the solution u(x,t) likely behaves a certain regularity at the starting time, i.e., ∂u∂t and ∂2u∂2t can blow up as t→0+ albeit u(⋅,t)∈C[0,T] for a given time T. In this case, we use the L1 scheme on non-uniform meshes to approximate the Caputo time-fractional derivative, and use the LDG method to discretize the spatial derivative. The fully discrete schemes for both situations are established and analyzed. It is shown that the derived schemes are numerically stable and convergent. Finally, several numerical experiments are provided which support the theoretical analysis.

A fast high order method for time fractional diffusion equation with non-smooth data

&lt;p style='text-indent:20px;'&gt;In this paper, we consider the time fractional diffusion equation with Caputo fractional derivative. Due to the singularity of the solution at the initial moment, it is difficult to achieve an ideal convergence order on uniform meshes. Therefore, in order to improve the convergence order, we discrete the Caputo time fractional derivative by a new &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ L1-2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; format on graded meshes, while the spatial derivative term is approximated by the classical central difference scheme on uniform meshes. We analyze the approximation about the time fractional derivative, and obtain the time truncation error, but the stability analysis remains an open problem. On the other hand, considering that the computational cost is extremely large, we present a reduced-order finite difference extrapolation algorithm for the time-fraction diffusion equation by means of proper orthogonal decomposition (POD) technique, which effectively reduces the computational cost. Finally, several numerical examples are given to verify the convergence of the scheme and the effectiveness of the reduced order extrapolation algorithm.&lt;/p&gt;

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

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

A mathematical model for thermography on viscous fluid based on damped thermal flux

Abstract Thermography is a fully noninvasive technique that discerns the thermal profiles of highly viable rheological parameters in heat and mass transference. In this paper, the free convection flow of viscous fluid among two vertical and parallel plates in the existence of a transverse magnetic field is investigated. The Caputo time-fractional derivative is manipulated for introducing a thermal transport equation along with a weak memory. The analytical and closed-form fractional solution for the temperature and velocity profiles are obtained through Laplace paired in conjunction with the finite Sine-Fourier transforms technique. The solution to the classical model is concluded as a special case for the solutions to the fractional modeled problem when the memory factor (the order of fractional derivative) approaches 1. Also, the solutions are stated in connection with the Mittag–Leffler function. The influences of variations of fractional and material parameters are depicted through MathCad15.

A thermal optimization throughan innovative mechanism of free convection flow of Jeffrey fluid using non-local kernel

Optimality for operating temperature can improve the reliability of thermodynamics of convection flow of Jeffrey fluid through mathematical techniques; this is because thermal convection is invoked in several engineering problems for knowingan accurate assessment and stable or unstable heat transference characteristics. In this context, the free convection flow of Jeffrey fluid among two vertical plates is characterized by stable or unstable heat transfer. The heat transference mechanism is designed by employing a generalized and a fractional form of Fourier's law that delivers damping for thermal flux. In this process, we make use of the Caputo time-fractional derivative (CTFD) having a power-law singular kernel. The free convection is induced by the temperature gradient. The analytical solutions have been obtained through a method of Laplace coupled with finite sine-Fourier transform and have been embedded with regards to the Mittag-Leffler function. The behavior of velocity and temperature profiles is analyzed through numerical computations and graphical representations for different embedded parameters with Mathcad. Certainly, this article presents a comprehensive discussion as well as a graphical interpretation of the achieved results.

A high-order compact finite difference method on nonuniform time meshes for variable coefficient reaction–subdiffusion problems with a weak initial singularity

A high-order compact finite difference method on nonuniform time meshes is proposed for solving a class of variable coefficient reaction–subdiffusion problems. The solution of such a problem in general has a typical weak singularity at the initial time. Alikhanov’s high-order approximation on a uniform time mesh for the Caputo time fractional derivative is generalised to a class of nonuniform time meshes, and a fourth-order compact finite difference scheme is used for approximating the spatial variable coefficient differential operator. A full theoretical analysis of the stability and convergence of the method is given for the general case of the variable coefficients by developing an analysis technique different from the one for the constant coefficient problem. Taking the weak initial singularity of the solution into account, a sharp error estimate in the discrete $$L^{2}$$ -norm is obtained. It is shown that the proposed method attains the temporal optimal second-order convergence provided a proper mesh parameter is employed. Numerical results demonstrate the sharpness of the theoretical error analysis result.

Alikhanov Legendre—Galerkin Spectral Method for the Coupled Nonlinear Time-Space Fractional Ginzburg–Landau Complex System

A finite difference/Galerkin spectral discretization for the temporal and spatial fractional coupled Ginzburg–Landau system is proposed and analyzed. The Alikhanov L2-1σ difference formula is utilized to discretize the time Caputo fractional derivative, while the Legendre-Galerkin spectral approximation is used to approximate the Riesz spatial fractional operator. The scheme is shown efficiently applicable with spectral accuracy in space and second-order in time. A discrete form of the fractional Grönwall inequality is applied to establish the error estimates of the approximate solution based on the discrete energy estimates technique. The key aspects of the implementation of the numerical continuation are complemented with some numerical experiments to confirm the theoretical claims.

Optimal vaccination and treatment policies for regional approximate controllability of the time-fractional reaction–diffusion SIR epidemic systems

This paper addresses the optimal vaccination and treatment control problems for regional approximate controllability of a new spatiotemporal epidemic model that is developed by afterwards adding at the basic susceptible–infected–recovered (SIR) epidemic system with the Caputo time-fractional derivative of order α∈(0,1] and the diffusion term in each compartment. The obtained results can be used by policy-makers in any nation to optimally plan the limited vaccination and treatment resources ahead of ongoing outbreaks. Toward this aim the Sakawa-type controller is introduced, which is a finite-dimensional controller that prevents and controls the spreading of infection. Using the semigroup theory, we provide a framework to analyze the sufficient conditions under which the considered time-fractional reaction–diffusion SIR epidemic system is regionally approximately controllable with the proposed control actuation architecture. An approach on finding the optimal solution that minimizes the cost of corresponding vaccination and treatment control programs for the studied regional controllability problem is then presented. Finally, we finish with numerical results to illustrate our theoretical results.

Modified homotopy methods for generalized fractional perturbed Zakharov\u2013Kuznetsov equation in dusty plasma

We propose a new modification of homotopy perturbation method (HPM) called the δ-homotopy perturbation transform method (δ-HPTM). This modification consists of the Laplace transform method, HPM, and a control parameter δ. This control convergence parameter δ in this new modification helps in adjusting and controlling the convergence region of the series solution and overcome some limitations of HPM and HPTM. The δ-HPTM and q-homotopy analysis transform method (q-HATM) are considered to study the generalized time-fractional perturbed (3+1)-dimensional Zakharov–Kuznetsov equation with Caputo fractional time derivative. This equation describes nonlinear dust-ion-acoustic waves in the magnetized two-ion-temperature dusty plasmas. The selection of an appropriate value of δ in δ-HPTM and the auxiliary parameters n and ħ in q-HATM gives a guaranteed convergence of series solution, but the difference between the two techniques is that the embedding parameter p in δ-HPTM varies from zero to nonzero δ, whereas the embedding parameter q in q-HATM varies from zero to frac{1}{n}, ngeq{1}. We examine the effect of fractional order on the considered problem and present the error estimate when compared with exact solution. The outcomes reveal complete reliability and efficiency of the proposed algorithm for solving various types of physical models arising in sciences and engineering. Furthermore, we present the convergence and error analysis of the two methods.

Fractional optimal control problem for an age-structured model of COVID-19 transmission.

The aim of this study is to model the transmission of COVID-19 and investigate the impact of some control strategies on its spread. We propose an extension of the classical SEIR model, which takes into account the age structure and uses fractional-order derivatives to have a more realistic model. For each age group the population is divided into seven classes namely susceptible exposed infected with high risk infected with low risk hospitalized recovered with and without psychological complications and respectively. In our model, we incorporate three control variables which represent: awareness campaigns, diagnosis and psychological follow-up. The purpose of our control strategies is protecting susceptible individuals from being infected, minimizing the number of infected individuals with high and low risk within a given age group as well as reducing the number of recovered individuals with psychological complications. Pontryagin’s maximum principle is used to characterize the optimal controls and the optimality system is solved by an iterative method. Numerical simulations performed using Matlab, are provided to show the effectiveness of three control strategies and the effect of the order of fractional derivative on the efficiency of these control strategies. Using a cost-effectiveness analysis method, our results show that combining awareness with diagnosis is the most effective strategy. To the best of our knowledge, this work is the first that propose a framework on the control of COVID-19 transmission based on a multi-age model with Caputo time-fractional derivative.

A generalized kinetic model of the advection-dispersion process in a sorbing medium

A new time-fractional derivative with Mittag-Leffler memory kernel, called the generalized Atangana-Baleanu time-fractional derivative is defined along with the associated integral operator. Some properties of the new operators are proved. The new operator is suitable to generate by particularization the known Atangana-Baleanu, Caputo-Fabrizio and Caputo time-fractional derivatives. A generalized mathematical model of the advection-dispersion process with kinetic adsorption is formulated by considering the constitutive equation of the diffusive flux with the new generalized time-fractional derivative. Analytical solutions of the generalized advection-dispersion equation with kinetic adsorption are determined using the Laplace transform method. The solution corresponding to the ordinary model is compared with solutions corresponding to the four models with fractional derivatives.

Memory effects on the proliferative function in the cycle-specific of chemotherapy

A generalized mathematical model of the breast and ovarian cancer is developed by considering the fractional differential equations with Caputo time-fractional derivatives. The use of the fractional model shows that the time-evolution of the proliferating cell mass, the quiescent cell mass, and the proliferative function are significantly influenced by their history. Even if the classical model, based on the derivative of integer order has been studied in many papers, its analytical solutions are presented in order to make the comparison between the classical model and the fractional model. Using the finite difference method, numerical schemes to the Caputo derivative operator and Riemann-Liouville fractional integral operator are obtained. Numerical solutions to the fractional differential equations of the generalized mathematical model are determined for the chemotherapy scheme based on the function of “on-off” type. Numerical results, obtained with the Mathcad software, are discussed and presented in graphical illustrations. The presence of the fractional order of the time-derivative as a parameter of solutions gives important information regarding the proliferative function, therefore, could give the possible rules for more efficient chemotherapy.

Redefined Extended Cubic B-Spline Functions for Numerical Solution of Time-Fractional Telegraph Equation

This work is concerned with the application of a redefined set of extended uniform cubic B-spline (RECBS) functions for the numerical treatment of time-fractional Telegraph equation. The presented technique engages finite difference formulation for discretizing the Caputo time-fractional derivatives and RECBS functions to interpolate the solution curve along the spatial grid. Stability analysis of the scheme is provided to ensure that the errors do not amplify during the execution of the numerical procedure. The derivation of uniform convergence has also been presented. Some computational experiments are executed to verify the theoretical considerations. Numerical results are compared with the existing schemes and it is concluded that the present scheme returns superior outcomes on the topic.