Caputo Fractional Derivative Research Articles

Bifurcation analysis of a fractional-order Cohen–Grossberg neural network with three delays

This paper pays more attention to the dynamic bifurcations of a fractional-order Cohen–Grossberg neural network in the sense of the Caputo fractional derivative with three delays. In the beginning, the dynamical behaviors of the system in several different cases are discussed when time delay can be viewed as zero, and the corresponding Hopf bifurcation critical values are derived. Subsequently, the system’s dynamics with three delays are investigated for three cases where none are zero. The reduction method of transcendental terms is applied to solve the third transcendental term in the characteristic equation. The exact bifurcation conditions are obtained by choosing one of the delays as the bifurcation parameter, respectively. It is worth noting that previous studies on multi-delay systems only discuss the case where the multi-delay can be transformed into a single delay or two delays based on certain transformations. This paper explores how to study the problem of a system with three delays without transforming the delays. In addition, the effects of time delays on the critical points of the system are examined. Finally, to verify the correctness of the proposed results, numerical simulations are addressed.

Grey Time Power Model with Caputo Fractional Derivative

In this paper, the inverse cumulative grey time power model with a Caputo fractional derivative is established, and the solution to the whitening equation is given by the Laplace transform. To improve the prediction accuracy of the model, the linear-tangent-function transformation is used to improve the smoothness of the data sequence, and a grey time power model is obtained, which has higher accuracy than the sinusoidal-function transformation, negative-exponential-function transformation and logarithmic-function transformation. The form and application range of the model are generalized.

A Novel Approach for Solving Nonlinear Time Fractional Fisher Partial Differential Equations

This study focuses on solving non-linear time fractional Fisher partial differential equations using analytical series solutions. The authors consider the Caputo fractional derivative in their formulas, which adds accuracy to the results. They introduce a novel method called LRPS which proves to be a powerful tool for obtaining precise analytical and numerical solutions for these equations. The LRPS method emphasizes precision, effectiveness, and practical application, making it suitable for various fields such as physics, engineering, and finance. Due to the importance of accuracy, effectiveness and method of application in this approach, it is highlighted that accurate solutions can be obtained when there is a pattern in the parts of the series, while approximate estimates are provided otherwise. The LRPS method is presented as a powerful technique specifically designed for solving nonlinear fractional Fisher partial differential equations.

Complete infinitesimal prolongation of the Riemann–Liouville and Caputo derivatives

This paper presents the infinitesimal prolongation to Riemann–Liouville and Caputo fractional derivatives without the restrictive lower limit fixed in the integrals, when applicated to the transformation group. The properties are presented, and the examples are illustrated along with the symmetry to fractional derivative criteria.

Implicit fractional differential equations: Existence of a solution revisited

This paper focuses on revisiting and improving the results regarding the existence of a solution to the implicit fractional differential equations (FDEs) given in the following form: for all with the initial condition , where is the Caputo fractional derivative (CFD). Mainly, we aim to weaken the necessary conditions often found in the literature for guaranteeing the existence of solution.

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.

An efficient numerical simulation of chaos dynamical behaviors for fractional-order Rössler chaotic systems with Caputo fractional derivative

This paper introduces a numerical approach for the fractional-order Rössler chaotic systems and gives error analysis. The effectiveness of the present method is determined by comparing the numerical results of the high-precision difference scheme and predictor-correctors scheme. Some complex dynamical behaviors for the fractional-order Rössler chaotic systems are shown.

Solving Nonlinear Fractional PDEs with Applications to Physics and Engineering Using the Laplace Residual Power Series Method

The Laplace residual power series (LRPS) method uses the Caputo fractional derivative definition to solve nonlinear fractional partial differential equations. This technique has been applied successfully to solve equations such as the fractional Kuramoto–Sivashinsky equation (FKSE) and the fractional generalized regularized long wave equation (GRLWE). By transforming the equation into the Laplace domain and replacing fractional derivatives with integer derivatives, the LRPS method can solve the resulting equation using a power series expansion. The resulting solution is accurate and convergent, as demonstrated in this paper by comparing it with other analytical methods. The LRPS approach offers both computational efficiency and solution accuracy, making it an effective technique for solving nonlinear fractional partial differential equations (NFPDEs). The results are presented in the form of graphs for various values of the order of the fractional derivative and time, and the essential objective is to reduce computation effort.

Proposal for Use of the Fractional Derivative of Radial Functions in Interpolation Problems

This paper presents the construction of a family of radial functions aimed at emulating the behavior of the radial basis function known as thin plate spline (TPS). Additionally, a method is proposed for applying fractional derivatives, both partially and fully, to these functions for use in interpolation problems. Furthermore, a technique is employed to precondition the matrices generated in the presented problems through QR decomposition. Similarly, a method is introduced to define two different types of abelian groups for any fractional operator defined in the interval [0,1), among which the Riemann–Liouville fractional integral, Riemann–Liouville fractional derivative, and Caputo fractional derivative are worth mentioning. Finally, a form of radial interpolant is suggested for application in solving fractional differential equations using the asymmetric collocation method, and examples of its implementation in differential operators utilizing the aforementioned fractional operators are shown.

Ulam-Type Stability Results for Variable Order Ψ-Tempered Caputo Fractional Differential Equations

An initial value problem for nonlinear fractional differential equations with a tempered Caputo fractional derivative of variable order with respect to another function is studied. The absence of semigroup properties of the considered variable order fractional derivative leads to difficulties in the study of the existence of corresponding differential equations. In this paper, we introduce approximate piecewise constant approximation of the variable order of the considered fractional derivative and approximate solutions of the given initial value problem. Then, we investigate the existence and the Ulam-type stability of the approximate solution of the variable order Ψ-tempered Caputo fractional differential equation. As a partial case of our results, we obtain results for Ulam-type stability for differential equations with a piecewise constant order of the Ψ-tempered Caputo fractional derivative.

A semi-analytical approach via yang transform on fractional-order navier-stokes equation

In this current, the applications of the Yang transformation technique are taken under consideration to deal with the non-linear fractional Navier–Stokes equation and fractional coupled Navier–Stokes equation. The suggested method produces approximate-analytical solutions in the form of a series that are correspondingly dependent on fractional-order derivative values and have modest, comprehensible mechanics and easy implementation. The Caputo fractional derivative is employed, and the numerical scheme’s stability and convergence are examined. Numerical examples demonstrate the analytical solution of the technique and it is examined that the proposed techniques are robust, efficient and reduce the number of numerical computations. The current technique’s results are compatible with the theoretical analysis, and the suggested technique can be extended to solve numerous higher-order non-linear dynamics.

Existence and stability results for implicit impulsive convex combined Caputo fractional differential equations

This paper deals with the existence and uniqueness results for a class of impulsive implicit fractional initial value problems of the convex combined Caputo fractional derivative. The arguments are based on Banach's contraction principle, Schauder's and Mönch's fixed point theorems. We will also establish the Ulam stability and give some examples to illustrate our results.

Two linear finite difference schemes based on exponential basis for two-dimensional time fractional Burgers equation

In present paper, we propose tailored finite point method (TFPM) based on exponential basis for solving two-dimensional time fractional Burgers equation. We use the L1 and L1-2 discretization formulas for the Caputo fractional derivative in time direction and use the TFPM to approximate the spatial derivatives. This method with the help of exponential functions fit the properties of the local solution in time and space simultaneously which act as basis functions in the frame of TFPM. We focus on constructing two linear implicit finite difference schemes, i.e. a five-point node centered scheme and a four-point cell-centered scheme based on exponential functions for deriving the numerical solution of given problem. The stability and convergence of fully discrete schemes based on five-point node centered are discussed. At last, we use some numerical examples to show the accuracy and efficiency of the presented schemes.

On the Relationship Between the Pontryagin Maximum Principle and the Hamilton–Jacobi–Bellman Equation in Optimal Control Problems for Fractional-Order Systems

We consider the optimal control problem of minimizing the terminal cost functional for a dynamical system whose motion is described by a differential equation with Caputo fractional derivative. The relationship between the necessary optimality condition in the form of Pontryagin’s maximum principle and the Hamilton–Jacobi–Bellman equation with so-called fractional coinvariant derivatives is studied. It is proved that the costate variable in the Pontryagin maximum principle coincides, up to sign, with the fractional coinvariant gradient of the optimal result functional calculated along the optimal motion.

Optimal Feedback in a Linear–Quadratic Optimal Control Problem for a Fractional-Order System

For a dynamical system described by a linear differential equation with a Caputo fractional derivative, we consider an optimal control problem of minimizing a quadratic terminal–integral performance functional. We propose and justify the construction of optimal feedback (optimal control synthesis) that generates the corresponding optimal control for any initial state of the system.

Insights into time fractional dynamics in the Belousov-Zhabotinsky system through singular and non-singular kernels

In the realm of nonlinear dynamics, the Belousov-Zhabotinsky reaction system has long held the fascination of researchers. The Belousov-Zhabotinsky system continues to be an active area of research, offering insights into the fundamental principles of nonlinear dynamics in complex systems. To deepen our understanding of this intricate system, we introduce a pioneering approach to tackle the time fractional Belousov-Zhabotinsky system, employing the Caputo and Atangana-Baleanu Caputo fractional derivatives with the double Laplace method. The solution we obtained is in the form of series which helps in investigating the accuracy of the proposed method. The primary advantage of the proposed technique lies in the low amount of calculations required and produce high degree of precision in the solutions. Furthermore, the existence and uniqueness of the solution are investigated thereby enhancing the overall credibility of our study. To visually represent our results, we present a series of 2D and 3D graphical representations that vividly illustrate the behavior of the model and the impact of changing the fractional order derivative and the time on the obtained solutions.

Unconditional superconvergence analysis of an energy-stable L1 scheme for coupled nonlinear time-fractional prey-predator equations with nonconforming finite element

The constrained nonconforming rotated Q1 (CNQ1rot) finite element is applied to develop an energy-stable L1 fully-discrete scheme through the Caputo fractional derivative approximation for the coupled nonlinear time-fractional prey-predator equations, and the unconditional superconvergence behavior is investigated. In which a novel high-accuracy estimate for the CNQ1rot element is given with the help of the Bramble-Hilbert (B-H) lemma, and the energy stability of the numerical solution is confirmed. Both of them are critical for demonstrating the unique solvability of the proposed scheme via the Brouwer fixed point theorem, and deducing the superclose estimation without any limitations between the spatial division size h and the time step τ. In addition, the above high-accuracy estimate can be extended to anisotropic meshes. Then, the unconditional superconvergence result is obtained by introducing the interpolation post-processing operator. Lastly, the theoretical findings are supported by some numerical experiments.

Application of Yang homotopy perturbation transform approach for solving multi-dimensional diffusion problems with time-fractional derivatives

In this paper, we aim to present a powerful approach for the approximate results of multi-dimensional diffusion problems with time-fractional derivatives. The fractional order is considered in the view of the Caputo fractional derivative. In this analysis, we develop the idea of the Yang homotopy perturbation transform method (YHPTM), which is the combination of the Yang transform (YT) and the homotopy perturbation method (HPM). This robust scheme generates the solution in a series form that converges to the exact results after a few iterations. We show the graphical visuals in two-dimensional and three-dimensional to provide the accuracy of our developed scheme. Furthermore, we compute the graphical error to demonstrate the close-form analytical solution in the comparison of the exact solution. The obtained findings are promising and suitable for the solution of multi-dimensional diffusion problems with time-fractional derivatives. The main advantage is that our developed scheme does not require assumptions or restrictions on variables that ruin the actual problem. This scheme plays a significant role in finding the solution and overcoming the restriction of variables that may cause difficulty in modeling the problem.

A Jacobi spectral method for calculating fractional derivative based on mollification regularization

In this article, we construct a Jacobi spectral collocation scheme to approximate the Caputo fractional derivative based on Jacobi–Gauss quadrature. The convergence analysis is provided in anisotropic Jacobi-weighted Sobolev spaces. Furthermore, the convergence rate is presented for solving Caputo fractional derivative with noisy data by invoking the mollification regularization method. Lastly, numerical examples illustrate the effectiveness and stability of the proposed method.

Comparative Analysis and Definitions of Fractional Derivatives

Fractional Calculus (FC) has emerged as a valuable tool in various fields. This study explores the historical development of (FC) and examines prominent definitions regarding Fractional Derivatives (FD), such as the Riemann-Liouville, Grunwald-Letnikov, Caputo Fractional Derivative, Katugampula derivatives, Caputo Fractional Derivative, Caputo-Fabrizio Fractional Derivative and as well as Atangana-Baleanu Fractional Derivative. It critically evaluates their strengths, weaknesses and implications on (FD) equations. The findings contribute to establishing a clearer understanding of Fractional Derivatives (FD) and guiding their appropriate use in practical scenarios. It investigates the impact of different definitions on the properties and behaviors of (FD), providing valuable insights for researchers.