Caputo Time-fractional Derivative Research Articles

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

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ L1-2 $\end{document}</tex-math></inline-formula> 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.</p>

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.

Non-uniform L1/discontinuous Galerkin approximation for the time-fractional convection equation with weak regular solution

In this paper, an efficient method seeking the numerical solution of a time-fractional convection equation whose solution is not smooth at the starting time is presented. The Caputo time-fractional derivative of order in (0,1) is discretized by the L1 finite difference method using non-uniform meshes; and, for the spatial derivative the discontinuous Galerkin (DG) finite element method is used. The stability and convergence of the method are analyzed for two-dimensional domains, using Cartesian and a particular class of unstructured grids. At last, several numerical examples are carried out which support the theoretical analysis.

Finite Difference Method on Non-Uniform Meshes for Time Fractional Diffusion Problem

Abstract 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 rate when the time discretization is performed on uniform meshes. Therefore, in order to improve the convergence order, the Caputo time fractional derivative term is discretized by the L ⁢ 2 - 1 σ {L2-1_{\sigma}} format on non-uniform meshes, with σ = 1 - α 2 {\sigma=1-\frac{\alpha}{2}} , while the spatial derivative term is approximated by the classical central difference scheme on uniform meshes. According to the summation formula of positive integer k power, and considering k = 3 , 4 , 5 {k=3,4,5} , we propose three non-uniform meshes for time discretization. Through theoretical analysis, different time convergence orders O ⁢ ( N - min ⁡ { k ⁢ α , 2 } ) {O(N^{-\min\{k\alpha,2\}})} can be obtained, where N denotes the number of time splits. Finally, the theoretical analysis is verified by several numerical examples.

Numerical Solution of a Quasilinear Parabolic Equation with a Fractional Time Derivative

A homogeneous Dirichlet initial-boundary value problem for a quasilinear parabolic equation with Caputo fractional time derivative is considered. The coefficients of the elliptic part of the equation depend on the derivatives of the solution and satisfy the conditions providing strong monotonicity and Lipschitz-continuity of the corresponding operator. The equation is approximated by two finite-difference schemes: implicit and fractional step scheme. The stability of these finite difference schemes is proved and accuracy estimates are obtained under the condition of sufficient smoothness of the input data and the solution of the differential problem. A number of iterative methods for implementing the constructed nonlinear mesh problems are analyzed. The convergence and convergence rate of the iterative methods are substantiated. The results of numerical experiments confirming the theoretical conclusions are presented.

Determination of Time-Dependent Sources and Parameters of Nonlocal Diffusion and Wave Equations from Final Data

Abstract Two inverse problems with final overdetermination for diffusion and wave equations containing the Caputo fractional time derivative and a fractional Laplacian of distributed order are considered. They are: 1) the problem to reconstruct a time-dependent source term; 2) the problem to recover simultaneously the source term, the order of the time derivative and the fractional Laplacian. Uniqueness of solutions of these problems is proved. Sufficient conditions for the uniqueness are stricter for the 2nd problem than for the 1st problem.

On the existence, uniqueness and stability results for time-fractional parabolic integrodifferential equations

We study the time-fractional parabolic integrodifferential equations with the Caputo fractional time derivative of order α∈(0,1) in a bounded convex polygonal domain in ℝd. We prove the existence and uniqueness of the solution using the eigenfunction expansion and establish a priori bounds for the solution under various regularity assumptions on the initial data and the source function. Our study includes the initial data in the spaces Ḣ2(Ω), H01(Ω) and L2(Ω) while the source function belongs to the class of Hölder continuous and bounded functions. It is shown that the solution of the corresponding homogeneous problem is infinitely differentiable with respect to time t when the initial function is an element of L2(Ω). Finally, we derive a general stability results for the solution of the homogeneous problem.

Finite difference/finite element method for two-dimensional time–space fractional Bloch–Torrey equations with variable coefficients on irregular convex domains

In magnetic resonance imaging of the human brain, the diffusion process of tissue water is considered in the complex tissue environment of cells, membranes and connective tissue. Models based on fractional order Bloch–Torrey equations are known to provide insights into tissue structures and the microenvironment.In this paper, we consider new two-dimensional multi-term time and space fractional Bloch–Torrey equations with variable coefficients on irregular convex domains, which involve the Caputo time fractional derivative and the Riemann–Liouville space fractional derivative. An unstructured-mesh Galerkin finite element method is used to discretize the spatial fractional derivative, while for each time fractional derivative we use the L1 scheme on a temporal graded mesh. The stability and convergence of the fully discrete scheme are proved. Numerical examples are given to verify the efficiency of our method.

Error analysis of the unstructured mesh finite element method for the two-dimensional time-space fractional Schrödinger equation with a time-independent potential

In this paper, error analysis of the unstructured mesh Galerkin finite element method for the two-dimensional time-space fractional Schrödinger equation with a time-independent potential defined on a finite domain is studied. The finite difference method is used to discretize the Caputo time fractional derivative, while the finite element method using unstructured mesh is used to deal with the Riesz fractional operators in space. Both the stability and convergence analysis of the numerical scheme are constructed. Numerical example is conducted to testify the validity of the proposed method. The conservation of the space fractional Schrödinger equation and the non-conservation of the time fractional Schrödinger equation in quantum mechanical system are achieved. This paper proposes an efficient numerical method as well as its theoretical analysis for the two-dimensional time-space fractional Schrödinger equation with time-independent potentials.

Thermal investigation for electrified convection flow of Newtonian fluid subjected to damped thermal flux on a permeable medium

The main purpose of this manuscript is to study an unsteady, MHD and free convection flow with the thermal transport for an electrical conducted, incompressible, and linearly viscous fluid flow close to a moving vertical plate in the existence of an exponential heating process. The flow area is considered permeable half-space as well as the utilization of an electromagnetic field perpendicular to fluid motion. The Caputo time-fractional derivative is utilized to present a heat flux constitutive-equation through a weak memory. The Laplace transformation technique is utilized to achieve the exact solutions of the dimensionless governing equations for the temperature field, Nusselt number, fluid velocity, and skin friction coefficient and stated in connection with Wright functions as well as a generalized G-Lorenzo-Hartley function. As a limiting case, a comparison with (Awan et al 2019 Chin. J. Phys. 60 98–106) equation (38) is done to verify the applied technique as well as for validity of our results. The influences of a variety of fractional as well as material factors are pictured with MathCad15. For low estimations of time, Fluid temperature and velocity are increased by decreasing fractional factor γ and both illustrate a contrary behavior for larger estimations of the time The fluid velocity is decreased through the large estimations of the magnetic factor