Caputo Time-fractional Derivative Research Articles

Simulating time delays and space–time memory interactions: An analytical approach

This study introduces a novel analytical framework to explore the effects of Caputo spatial and temporal memory indices combined with a proportional time delay on (non)linear (1+2)-dimensional evolutionary models. The solution is expressed as a Cauchy product of an absolutely convergent series that effectively captures the dynamics of these parameters. By extending the differential transform method into higher-dimensional fractional space, we reformulate the evolution equation as a (non)linear higher-order recurrence relation, which enables the precise determination of fractional series coefficients. Our findings show that Caputo derivatives and time delay significantly impact the system’s behavior, with graphical analysis revealing a continuous transition from a stationary to an integer state solution. The study also identifies a quantitative analogy between the Caputo-time fractional derivative and proportional time delay that highlights the role of Caputo derivatives as memory indices. This method has proven highly effective in deriving solutions for fractional higher-dimensional extensions of evolutionary equations.

Navigating the Fractional Calcium Dynamics of Orai Mechanism in Polar Dimensions.

Calcium plays a crucial role as a second messenger in neuronal signal transduction pathways. The influx of calcium ions through various physicochemical gating channels activates neuronal calcium signaling. The Endoplasmic Reticulum (ER) is a significant intracellular structure that sequesters calcium and controls signaling through SERCA, IPR, and leak channel mechanisms. Disruption of calcium dynamics can trigger intrinsic dyshomeostasis, cell damage, and apoptosis. The present study articulates a Caputo fractional time derivative in the polar coordinate dimensions to investigate the role of nonlocal calcium-free ions in the neuron through the Orai channel, and ER fluxes, incorporating various physiological parameters. The solution was obtained through the hybrid integral transform technique for analytical form. The closed form was generated using Green's function in terms of Mainardi and Wright's functions. Our simulation uncovered the calcium concentration bandwidth of interaction with different neuronal parameters. Parameters and calcium ion synergy show normal and Alzheimer's disease-impacted interaction through different illustrations. Our simulation reveals that S100B and BAPTA have significant calcium-controlling behavior.

Axisymmetric stagnation-point flow of a fractional Oldroyd-B hybrid nanofluid over a vertical cylinder by improved Buongiorno's model

This paper investigated the unsteady axisymmetric stagnation-point flow, heat and mass transfer of Oldroyd-B hybrid nanofluid over a vertical cylinder by using Caputo time fractional derivatives for the first time, with the consideration of second-order velocity slip conditions. Improved Buongiorno's model was adopted to build the governing equations and finite difference combined with L1-algorithm was employed to solve the numerical solutions. Effects of key physical parameters on velocity, heat and mass transfer were presented graphically and discussed in detail. Outcomes show that the fractional derivatives are capable to describe the memory effect and thermoelasticity of viscoelastic nanofluids. Both Brownian motion and thermophoresis, especially Brownian motion, improve the heat transfer on the cylinder wall. In addition, the sensitivity of average Nusselt (or Sherwood) number to velocity fractional derivates α and β rises with increasing α and β, while the sensitivity to temperature fractional derivate γ declines with increasing γ. When γ increases from 0.2 to 0.3, the relative growth rate of average Nusselt number is about 9.26 %. This study provides a reference for exploring the stationary-point flow of fractional order nanofluids on vertical cylinders.

Numerical approximation of a generalized time fractional partial integro-differential equation of Volterra type based on a meshless method

The moving least squares (MLS) approximation is used in this study to solve time-fractional partial integro-differential equations (TFPIDE). In our approach, we approximate the time Caputo fractional derivative term and the integral term by using a finite difference scheme and trapezoidal method, respectively, which yields an order of accuracy of O(τ+τ2−α), where 0<α<1. We thoroughly investigate the applicability and validity of the proposed method and provide an error estimate for its performance. Additionally, we solve various numerical problems to demonstrate the accuracy and computational efficiency of our approach, confirming the theoretical findings.

Fully discrete Galerkin scheme for a semilinear subdiffusion equation with nonsmooth data and time-dependent coefficient

We couple the L1 discretization of the Caputo fractional derivative in time with the Galerkin scheme to devise a linear numerical method for the semilinear subdiffusion equation. Two important points that we make are nonsmooth initial data and time-dependent diffusion coefficient. We prove the stability and convergence of the method under weak assumptions concerning regularity of the diffusivity. We find optimal pointwise in space and global in time errors, which are verified with several numerical experiments.

Analysis of a fast element-free Galerkin method for the multi-term time-fractional diffusion equation

In this paper, a fast element-free Galerkin (EFG) method is proposed for solving the multi-term time-fractional diffusion equation (TFDE). Through the use of the multi-term L2−1σ formula to discrete the multi-term Caputo time-fractional derivative, a fast second-order scheme is presented for the temporal discretization, and then the EFG method is used to perform the spatial discretization. The stability of the temporal discretization scheme is discussed. By verifying the fractional Grönwall inequality, the global error of the proposed EFG method for the multi-term TFDE is analyzed in theory. Numerical results validate the theoretical results and the effectiveness of the method.

Polynomial decay of a linear system of PDEs via Caputo fractional‐time derivative

An in‐depth study and analysis of the stability of one‐dimensional Linear System of PDEs via Caputo time fractional derivative (LSCFD) was presented. We proved some stability results for the LSCFD in different Hilbert spaces. Indeed, by using Fourier analysis method and the properties of the Mittag–Leffler Function (MLF), some polynomial stability results for LSCFD have been established. Finally, as an application, we used finite difference methods well suited to integer and fractional order derivatives, and performed some numerical experiments to confirm the theoretical results.

Application of an Extended Cubic B-Spline to Find the Numerical Solution of the Generalized Nonlinear Time-Fractional Klein–Gordon Equation in Mathematical Physics

A B-spline function is a series of flexible elements that are managed by a set of control points to produce smooth curves. By using a variety of points, these functions make it possible to build and maintain complicated shapes. Any spline function of a certain degree can be expressed as a linear combination of the B-spline basis of that degree. The flexibility, symmetry and high-order accuracy of the B-spline functions make it possible to tackle the best solutions. In this study, extended cubic B-spline (ECBS) functions are utilized for the numerical solutions of the generalized nonlinear time-fractional Klein–Gordon Equation (TFKGE). Initially, the Caputo time-fractional derivative (CTFD) is approximated using standard finite difference techniques, and the space derivatives are discretized by utilizing ECBS functions. The stability and convergence analysis are discussed for the given numerical scheme. The presented technique is tested on a variety of problems, and the approximate results are compared with the existing computational schemes.

Entropy generation for thermo-magnetic fractional order convective flow in complex porous enclosures: a numerical study

PurposeThis study aims to analyze the impact of fractional derivatives on heat transfer and entropy generation during transient free convection inside various complex porous enclosures, such as triangle, L-shape and square-containing wavy surfaces. These porous enclosures are saturated with Cu-water nanofluid and subjected to the influence of a uniform magnetic field.Design/methodology/approachIn the present study, Darcy’s model is used for the momentum transport equation in the porous matrix. Additionally, the Caputo time fractional derivative is introduced in the energy equation to assess the heat transfer phenomenon. Furthermore, the total entropy generation has been computed by combining the entropy generation due to fluid friction (Sff), heat transfer (Sht) and magnetic field (Smf). The complete mathematical model is further simulated using the penalty finite element method, and the Caputo time derivative term is approximated using the L1 scheme. The study is conducted for various ranges of the Rayleigh number (102≤Ra≤104), Hartmann number (0≤Ha≤20) and fractional order parameter (0<α<1) with respect to time.FindingsIt has been observed that the fractional order parameter α governs the characteristics of entropy generation and heat transfer within the selected range of parameters. The Bejan number associated with heat transfer (Beht), fluid friction (Beff) and magnetic field (Bemf) further demonstrate the dominance of flow irreversibilities. It becomes evident that the initial evolution state of streamlines, isotherms and local entropy varies according to the choice of α. Additionally, increasing Ra values from 102 to 104 shows that the heat transfer rate increases by 123.8% for a square wavy enclosure, 7.4% for a triangle enclosure and 69.6% for an L-shape enclosure. Moreover, an increase in the value of Ha leads to a reduction in heat transfer rates and entropy generation. In this case, Bemf→1 shows the dominance of the magnetic field irreversibility in the total entropy generation.Practical implicationsRecently, fractional-order models have been widely used to express numerous physical phenomena, such as anomalous diffusion and dispersion in complex viscoelastic porous media. These models offer a more accurate representation of physical reality that classical models fail to capture; this is why they find a broad range of applications in science and engineering.Originality/valueThe fractional derivative model is used to illustrate the flow pattern, heat transfer and entropy-generating characteristics under the influence of a magnetic field. Furthermore, to the best of the author’s knowledge, a fractional-derivative-based mathematical model for the entropy generation phenomenon in complex porous enclosures has not been previously developed or studied.

The existence and asymptotic behavior of solutions to 3D viscous primitive equations with Caputo fractional time derivatives

On the one hand, the primitive three-dimensional viscous equations for large-scale ocean and atmosphere dynamics are commonly used in weather and climate predictions. On the other hand, ever since the middle of the last century, it has been widely recognized that the climate variability exhibits long-time memory. In this paper, we first prove the global existence of weak solutions to the primitive equations of large-scale ocean and atmosphere dynamics with Caputo fractional time derivatives. Then we establish the existence of an absorbing set, which is positively invariant. Finally, an attractor (strictly speaking, the minimal attracting set containing all the limiting dynamics) is constructed for the time fractional primitive equations, which means that the present state of a system may have long-time influences on the states in far future. However, there was no work on the long-time behavior of the time fractional primitive equations and we fill this gap in this paper.

Numerical recovery of a time-dependent potential in subdiffusion *

In this work we investigate an inverse problem of recovering a time-dependent potential in a semilinear subdiffusion model from an integral measurement of the solution over the domain. The model involves the Djrbashian–Caputo fractional derivative in time. Theoretically, we prove a novel conditional Lipschitz stability result, and numerically, we develop an easy-to-implement fixed point iteration for recovering the unknown coefficient. In addition, we establish rigorous error bounds on the discrete approximation. These results are obtained by crucially using smoothing properties of the solution operators and suitable choice of a weighted Lp(0,T) norm. The efficiency and accuracy of the scheme are showcased on several numerical experiments in one- and two-dimensions.

Significance of Cu-Fe3O4 on fractional Maxwell fluid flow over a cone with Newtonian heating

The goal of this research is to investigate fractional Maxwell hybrid nanofluids utilizing partial differential equations in terms of Caputo time fractional derivatives. Specifically, the effect of Newtonian heating on the thermal performance of a fractional Maxwell hybrid nanofluid moving over a permeable cone in the presence of thermal radiation and heat generation is considered. The Crank–Nicolson method and L1 algorithmt of Caputo derivative are used to find numerical solutions to the considered nonlinear problem. The effects of significant flow factors on fluid properties are examined and illustrated in various graphs. According to the results, the thermal performance of the fluid raised by 0.4%, 6.1%, and 3.1% on adding 4% volume fraction of Fe3O4, Cu, and Cu−Fe3O4, respectively, in the base fluid.

Problem of Determining the Density of Sources in a Multidimensional Heat Equation with the Caputo Time Fractional Derivative

In this paper, we propose a new formula for representing the solution of the third initial-boundary value problem for multidimensional fractional heat equation with the Caputo derivative. This formula is obtained by the continuation method used in the theory of partial differential equations with integer derivatives. The Green’s function of the problem is also constructed in terms of the Fox H- function. Involving the results of solving a direct problem and the overdetermination condition, a uniqueness theorem for the definition of the spatial part of the multidimensional source function is proved.

A CPC fractional model of the heat and mass transport mechanism in Carbon nanotubes with slip effects on velocity

A fractional technique is used to evaluate the temperature, mass, and velocity flow of single and double wall CNTs over a vertical plate. Slip boundary conditions and applied magnetic force are addressed. Human blood is used to examine how base fluid behaves. Applying the proper dimensionless variables results in the dimensionless formulation of initial and boundary conditions related to the governed dimensional concentration, momentum, and energy equations. The Laplace transform technique is used to resolve the dimensionless governing partial differential equations and get the solutions. The constant proportional Caputo (CPC) time-fractional derivative is a unique class of fractional model used in the simulation technique. The fundamental definitions are used to support the said model first. Using MWCNTs and SWCNTs in comparison to the flow characteristics, a thermal and mass study is given. The heat and mass transfer processes for single-walled carbon nanotubes (SWCNTs) have been shown to typically be progressive. The momentum profile decreases as the fractional variables rise. Multi-walled carbon nanotubes (MWCNTs) show more progressive velocity control as a result of the magnetic parameter. Graphs demonstrate the influence of embedding factors on the velocity, energy, and concentration profiles.

Regularized Perona & Malik model involving Caputo time-fractional derivative with application to image denoising

This work tackles a novel partial differential equation based on a time-fractional order derivative for image denoising and restoration. We ameliorate the classical Perona–Malik model by considering Caputo time-fractional order derivative with a regularized diffusion. We start by investigating the existence of the solution to the proposed model. As a first result, we use the Faedo–Galerkin approach to prove the existence and uniqueness of a weak solution to an auxiliary fractional problem. Secondly, we establish the existence and uniqueness of a weak solution to our model via Schauder fixed point method. To validate the efficiency of our model, we present some numerical experiments on images with different features and various noise levels. We begin by introducing an adaptive discrete scheme to the proposed model. We illustrate the sensitivity of the time-fractional order derivative with respect to some well-known criteria. Finally, the obtained numerical results show a great efficiency and robustness against noise compared to the competitive models.

Local discontinuous Galerkin method for a nonlocal viscous water wave model

In this article, the fully discrete local discontinuous Galerkin (LDG) method is presented for numerically solving a class of nonlocal viscous water wave model, where the BDF2 with the L1 formula of the time Caputo fractional derivative is applied to deal with the time direction, and the LDG method is developed to approximate the space direction. The stability of the fully discrete LDG scheme is proven, and the a priori error result with O(τ32+hk+12) in L2-norm is derived in detail. Finally, numerical examples supporting theoretical error result are provided, and the decay rates under different coefficients are also discussed.

A new high-accuracy difference method for nonhomogeneous time-fractional Schrödinger equation

The fractional Schrödinger equation is an important fractional nonlinear evolution equation, and the study of its numerical solution has profound scientific meaning and wide application prospects. This paper proposes a new high-accuracy difference method for nonhomogeneous time-fractional Schrödinger equation (TFSE). The Caputo time-fractional derivative is discretized by high-order formula and the fourth-order compact difference approximation is applied for spatial discretization. A new nonlinear compact difference scheme with temporal second-order and spatial fourth-order accuracy is constructed, which is solved by the efficient linearized iterative algorithm. The unconditional stability and convergence are analysed by the energy method. The unique existence and maximum-norm estimate of new compact difference scheme solution are obtained. Theoretical analysis shows that the convergence accuracy of new compact difference scheme is with the strong regularity assumption. Numerical experiments verify theoretical results and indicate that the proposed method is an efficient numerical method for solving TFSE.

Stable Computational Techniques for the Advection–Dispersion Variable Order Model

In this paper, a novel approximation to the Caputo time fractional derivative of variable order (VO) [Formula: see text] ([Formula: see text]) is established. Since time fractional derivatives are integral, with a weakly singular kernel the discretization on the uniform mesh may not lead to satisfactory accuracy. So, numerical approximation is constructed on nonuniform meshes based on Newton interpolation polynomial. As a result, the novel approximation can be viewed as the modification of the existing method [Shen, S., Liu, F., Chen, J., Turner, I. and Anh, V. [2012] “Numerical techniques for the variable order time fractional diffusion equation,” Appl. Math. Comput. 218(22), 10861–10870]. The truncation errors and some significant properties of coefficients are analyzed. Furthermore, we derived Scheme-I and Scheme-II with second-order accuracy in spatial direction to solve the mobile–immobile advection–dispersion model (MIMADM) based on the VO Caputo derivative. The designed scheme converts the considered problems into a linear system of equations. Moreover, the stability properties of Scheme-I are discussed in detail. Ultimately, the proposed scheme is examined on several text examples demonstrating the high precision and effectiveness of the scheme. Also, comparative study between the outcomes achieved by implementing the suggested scheme with other numerical scheme in the existing literature is provided and also the acquired numerical results of these applications indicate that the suggested technique performs extremely well in terms of reliability and capability.

Inverse problem for a time‐fractional differential equation with a time‐ and space‐integral conditions

We study the inverse problem for a differential equation of order with the time Caputo fractional derivative and a source with values in Schwartz‐type distributions. The generalized solution of the Cauchy problem for such an equation with initial values and a time‐dependent reaction coefficient are unknown. We find sufficient conditions for the unique solvability of the inverse problem under a time‐ and space‐integral overdetermination conditions.

Exponentially Convergent Numerical Method for Abstract Cauchy Problem with Fractional Derivative of Caputo Type

We present an exponentially convergent numerical method to approximate the solution of the Cauchy problem for the inhomogeneous fractional differential equation with an unbounded operator coefficient and Caputo fractional derivative in time. The numerical method is based on the newly obtained solution formula that consolidates the mild solution representations of sub-parabolic, parabolic and sub-hyperbolic equations with sectorial operator coefficient A and non-zero initial data. The involved integral operators are approximated using the sinc-quadrature formulas that are tailored to the spectral parameters of A, fractional order α and the smoothness of the first initial condition, as well as to the properties of the equation’s right-hand side f(t). The resulting method possesses exponential convergence for positive sectorial A, any finite t, including t=0 and the whole range α∈(0,2). It is suitable for a practically important case, when no knowledge of f(t) is available outside the considered interval t∈[0,T]. The algorithm of the method is capable of multi-level parallelism. We provide numerical examples that confirm the theoretical error estimates.