Approximation Of Fractional Derivative Research Articles

Mitigation of numerical issues appearing in transient analyses when applying fractional derivative approximations

Due to the potential decrease of the computation time for problems with fractional order derivatives, and due to the extension of the range of applicable solvers for a given problem, the approximation of fractional derivatives (e.g., using Oustaloup’s method) is an important and frequently discussed topic. A significant problem that can occur (although one that is not discussed very often) when approximations are applied concerns the potential numerical issues that can arise when applying these approximations in a transient analysis, e.g., as shown in the paper, in a time-stepping solver. This paper examines five different methods for approximating the fractional derivative (Oustaloup’s method, a refined Oustaloup method, a modified continued fraction expansion method, Matsuda’s method and Charef’s method). An important preliminary step in the analysis was the establishment of the general form of FDAE (fractional differential-algebraic equations), the general form of the applied approximations and the general form of the DAE (differential-algebraic equations) resulting from the approximation. This allowed for a precise description of the equations according to which the transformation from the FDAE to the approximating DAE takes place. Three problems derived from studies in electrical engineering have been introduced into the analysis (a linear circuit problem, a circuit problem with nonlinear fractional elements, and a problem featuring nonlinearities and pregenerated noise). For these problems, the conditions in which numerical issues appear have been studied in detail. They are the main motivation of this study as these issues are so significant that they often result in zero-solutions or solutions tending to infinity (giving an impression as if the system is unstable). Modifications and alternatives of the transformation into the DAE, that aim at the mitigation of these numerical errors, are mentioned later. The final result is very satisfactory, where the algorithm for the transformation of the FDAE to the approximating DAE practically eliminates the most important barriers (which was the impossibility of using approximations above a certain order due to the mentioned numerical issues). The study presented in this paper is motivated by problems in electrical engineering but due to its generality it is also applicable in other fields where fractional calculus is used.

A Quasilinearization Approach for Identification Control Vectors in Fractional-Order Nonlinear Systems

This paper is concerned with solving the problem of identifying the control vector problem for a fractional multi-order system of nonlinear ordinary differential equations (ODEs). We describe a quasilinearization approach, based on minimization of a quadratic functional, to compute the values of the unknown parameter vector. Numerical algorithm combining the method with appropriate fractional derivative approximation on graded mesh is applied to SIS and SEIR problems to illustrate the efficiency and accuracy. Tikhonov regularization is implemented to improve the convergence. Results from computations, both with noisy-free and noisy data, are provided and discussed. Simulations with real data are also performed.

Numerical Algorithms for Approximation of Fractional Integrals and Derivatives Based on Quintic Spline Interpolation

Numerical algorithms for calculating the left- and right-sided Riemann–Liouville fractional integrals and the left- and right-sided fractional derivatives in the Caputo sense using spline interpolation techniques are derived. The spline of the fifth degree (the so-called quintic spline) is mainly taken into account, but the linear and cubic splines are also considered to compare the quality of the developed method and numerical calculations. The estimation of errors for the derived approximation algorithms is presented. Examples of the numerical evaluation of the fractional integrals and derivatives are executed using 128-bit floating-point numbers and arithmetic routines. For each derived algorithm, the experimental orders of convergence are calculated. Also, an illustrative computational example showing the action of the considered fractional operators on the symmetric function in the interval is presented.

Variable time steps in the numerical implementation of viscoelastic fractional models for laminated glass

Abstract We elaborate numerical approaches to calculate the rheological response of laminated glass beams, whose viscoelastic interlayer is modelled via fractional calculus. This mathematical description is very effective when the relaxation function of the polymer can be expressed by continuously connected branches of power-laws, as is the case for most materials used to laminate glass. The classical approach uses the Grünwald-Letnikov approximation of fractional derivatives, but it requires constant time steps, which would become very large to reasonably cover the entire observation time, thus losing accuracy. We propose to use the L1 algorithm with increasing time steps, which is well suited to the power law character of the relaxation function. This allows to follow the long-term creep response, providing a better approximation when needed. The method is implemented for beams laminated with viscolastic interlayers whose relaxation is described by four branches of power laws, to cover most practical cases. Numerical experiments shows its advantages over the Grünwald-Letnikov approach for characterizing the long-term structural response.

Optimal time two-mesh mixed finite element method for a nonlinear fractional hyperbolic wave model

<abstract><p>In this article, a second-order time discrete algorithm with a shifted parameter $ \theta $ combined with a fast time two-mesh (TT-M) mixed finite element (MFE) scheme was considered to look for the numerical solution of the nonlinear fractional hyperbolic wave model. The second-order backward difference formula including a shifted parameter $ \theta $ (BDF2-$ \theta $) with the weighted and shifted Grünwald difference (WSGD) approximation for fractional derivative was used to discretize time direction at time $ t_{n-\theta} $, the $ H^1 $-Galerkin MFE method was applied to approximate the spatial direction, and the fast TT-M method was used to save computing time of the developed MFE system. A priori error estimates for the fully discrete TT-M MFE system were analyzed and proved in detail, where the second-order space-time convergence rate in both $ L^2 $-norm and $ H^1 $-norm were derived. Detailed numerical algorithms with smooth and weakly regular solutions were provided. Finally, some numerical examples were provided to illustrate the feasibility and effectiveness for our scheme.</p></abstract>

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.

Analysis of Fractional Linear Multi-Step Methods of Order Four from Super-Convergence

We analyze two implicit fractional linear multi-step methods of order four for solving fractional initial value problems. The methods are derived from the Grunwald-Letnikov approximation of fractional derivative at a non-integer shift point with super-convergence. The weight coefficients of the methods are computed from fundamental G unwald weights, making them computationally efficient when compared with other known methods of order four. We also show that the stability regions are larger than that of the fractional Adams-Moulton and fractional backward difference formula methods. We present numerical results and illustrations to verify that the theoretical results obtained are indeed satisfied.

PMNN: Physical model-driven neural network for solving time-fractional differential equations

In this paper, an innovative Physical Model-driven Neural Network (PMNN) method is proposed to solve time-fractional differential equations. It establishes a temporal iteration scheme based on physical model-driven neural networks which effectively combines deep neural networks (DNNs) with interpolation approximation of fractional derivatives. Specifically, once the fractional differential operator is discretized, DNNs are employed as a bridge to integrate interpolation approximation techniques with differential equations. On the basis of this integration, we construct a neural-based iteration scheme. Subsequently, by training DNNs to learn this temporal iteration scheme, approximate solutions to the differential equations can be obtained. The proposed method aims to preserve the intrinsic physical information within the equations as far as possible. It fully utilizes the powerful fitting capability of neural networks while maintaining the efficiency of the difference schemes for fractional differential equations. The experimental results show that the PMNN maintains precision comparable to traditional methods while exhibiting superior computational efficiency. This implies the potential of PMNN in addressing large-scale problems. Moreover, when considering both error and convergence rate, PMNN consistently outperforms fPINN. Additionally, the performance of PMNN on L2−1σ surpasses that on L1 in an overall comparison. The data and code can be found at https://github.com/DouMiao1226/PMNN.

Nonlinear dynamics of fractional viscoelastic PET membranes with linearly varying density

In this paper, the nonlinear dynamics of fractional viscoelastic polyethylene terephthalate (PET) membranes with linearly varying density are elaborated. The viscoelasticities of the PET membranes are characterized with the fractional Kelvin–Voigt model, and the density distribution is considered a linear fluctuation in the lateral direction. The geometrically nonlinear formulation is established with the von-Karman theory and the Hamilton principle. The Bubnov–Galerkin method is used to solve the resulting fractional differential equations, and a refined numerical scheme combining the finite difference with a discrete Caputo-type fractional derivative approximation is developed. Comparison examples are provided to substantiate the accuracy and effectiveness of the present method. A detailed parametric analysis is performed to illuminate the effects of the fractional order, density coefficient and various system parameters on the time response of the viscoelastic PET membranes. This study paves the way for fractional modelling and vibrational analysis of viscoelastic substrates in flexible electronics manufacturing.

Numerical study on radiative MHD flow of viscoelastic fluids with distributed-order and variable-order space fractional operators

The magnetohydrodynamic (MHD) flow has been concerned widely for its widespread adoption in the field of astrophysics, electronics and many other industries over the years. The purpose of this article is to introduce the variable and distributed order space fractional models to characterize the MHD flow and heat transfer of heterogeneous viscoelastic fluids in a parallel plates. Based on the central difference approximation of Riesz space fractional derivative, the Crank–Nicolson difference scheme for the governing equations is established, and the effectiveness of the algorithm is verified by two numerical examples. We examine the effects of fractional-order model parameters on the velocity and temperature, our investigation indicates that for the constant fractional model, the larger the fractional order parameter, the smaller the velocity and temperature. The variable space fractional method can be used to describe dynamic behavior with time and space dependence, while the distributed space fractional model can describe various phenomena in which the number of differential orders varies over a certain range, characterizing their complex processes over space, and it is also more suitable for simulating the fluid flow and thermal behavior of complex viscoelastic magnetic fluid.

A high order numerical method for the variable order time-fractional reaction-subdiffusion equation

In this paper, we present a high order new numerical approximation for variable order Caputo fractional derivative of order 0<α(x,t)<1, by using the idea of interpolation. Then, by using this approximation, a numerical scheme is presented by using finite difference approach for variable order time fractional reaction-subdiffusion equation (VO-TFRSDE). The unconditionally stability of the numerical scheme is examined theoretically. The scheme is implemented on the two test problems. The numerical results are highly accurate with higher order of convergence.

Studying The Necessary Optimality Conditions and Approximates a Class of Sum Two Caputo–Katugampola Derivatives for FOCPs

In this paper, the necessary optimality conditions are studied and derived for a new class of the sum of two Caputo–Katugampola fractional derivatives of orders (α, ρ) and( β,ρ) with fixed the final boundary conditions. In the second study, the approximation of the left Caputo-Katugampola fractional derivative was obtained by using the shifted Chebyshev polynomials. We also use the Clenshaw and Curtis formula to approximate the integral from -1 to 1. Further, we find the critical points using the Rayleigh–Ritz method. The obtained approximation of the left fractional Caputo-Katugampola derivatives was added to the algorithm applied to the illustrative example so that we obtained the approximate results for the state variable x(t) and the control variable u(t) by assumed α, β ∈ (0,1) with different values for two periods of ρ &gt; 0 (ρ∈ (0,1) , ρ∈ (1,2)). In both cases, the algorithm steps show the accuracy and efficiency of the approximate results of the proposed system. An approximation to the fractal derivative was obtained and then added to the algorithm applied to the illustrative example so that we obtained the approximate results for the state variable x (t) and the control variable u (t) by imposing different values for two periods of ρ &gt; 0. In the first case we take ρ∈ (

On the order reduction of approximations of fractional derivatives: an explanation and a cure

Finite-difference based approaches are common for approximating the Caputo fractional derivative. Often, these methods lead to a reduction of the convergence rate that depends on the fractional order. In this note, we approximate the expressions in the fractional derivative components using a separate quadrature rule for the integral and a separate discretization of the derivative in the integrand. By this approach, the error terms from the Newton–Cotes quadrature and the differentiation are isolated and it is possible to conclude that the order dependent error is inevitable when the end points of the interval are included in the quadrature. Furthermore, we show experimentally that the theoretical findings carries over to quadrature rules without the end points included. Finally we show how to increase accuracy for smooth functions, and compensate for the order dependent loss.

Fault?fracture reservoir identification method based on SRGAN and reconstructed super?resolution seismic signals

To address the problem of fault?fracture reservoir identification, a new method based on super?resolution (SR) seismic signal reconstruction is established to identify faults, sliding fracture zones and induced fracture zones. First, based on a super?resolution generation countermeasure (SRGAN) deep learning method, an SR seismic signal reconstruction network framework is designed with a discriminant network (D), a generation network (G) and a visual geometry group network (V). Through the perceptual loss, objective control functions and iterative parameter updates, the nonlinear feature learning advantages of the deep network are introduced, the noise is eliminated, weak signals are recovered, and low resolution (LR) signals are restored, allowing the seismic signal to be reconstructed into an SR signal. Second, the SR seismic signal is used to extract the geometric attributes, such as the coherence based on the gradient structure tensor (GST) and the curvature based on the fractional derivative approximation (FDA). Third, principal component analysis (PCA) is used to reduce the feature dimension of the seismic attributes such as the GST coherence and FDA curvature and extract the principal components with the strongest correlations, thus eliminating redundant and residual noise interference, highlighting the spatial distribution and internal details of the fault?fracture reservoir, and allowing a fine description of the fault?fracture reservoir to be developed. Finally, this method achieves a good application effect for reconstructing SR seismic signals and identifying fault?fracture reservoirs in the Sichuan Basin of China.

Fractional derivative truncation approximation for real-time applications

Fractional derivatives are non local operators as they are well-suited for modeling long-memory phenomena such as heat transfers or fluid dynamics. However, such non-locality implies a constant knowledge of the full past of the functions to differentiate. In the context of real-time (online) system identification, such a global property may limit the implementation as calculations can become slower as time grows. Also, instead of using the full data length to compute fractional order derivatives, truncated fractional derivatives are used. However, such a truncation windowing will bring inaccuracy on the computation of fractional derivatives. This study deals with the relationship between the signal frequency content, the fractional derivative truncated approximation as well as the fractional system relaxation in order to get a good approximation of the truncated fractional derivative and consequently, to help providing real-time exploitable system identification algorithms that uses such truncated fractional derivatives.

A new fast predictor-corrector method for nonlinear time-fractional reaction-diffusion equation with nonhomogeneous terms

Time-fractional reaction-diffusion (TFRD) equation is an important fractional parabolic equation, and the research of its numerical solution has scientific significance and engineering application value. In this paper, a new fast predictor-corrector (FP-C) scheme is constructed based on fast L1 approximation of Caputo fractional derivative for solving nonlinear TFRD equation with nonhomogeneous terms. The linearized implicit difference scheme is used for the predictor step and Crank-Nicolson(C-N) scheme is used for the corrector step. Theoretical analysis proves that FP-C scheme is convergent and stable unconditionally for nonlinear TFRD equation. Numerical analysis and experiments show that the computational accuracy of FP-C scheme is $ O(\tau^{2-\alpha}+h^2) $ under the strong regularity condition and $ O(\tau^{\alpha}+h^2) $ under the weak regularity condition. Compared with the classical predictor-corrector (P-C) scheme based on standard L1 approximation, the FP-C scheme improves the computational efficiency without losing the computational accuracy. It is shown that the new FP-C scheme is an efficient method to solve nonlinear TFRD equation with nonhomogeneous terms.

Dissipativity and contractivity of the second-order averaged L1 method for fractional Volterra functional differential equations

&lt;abstract&gt;&lt;p&gt;This paper focuses on the dissipativity and contractivity of a second-order numerical method for fractional Volterra functional differential equations (F-VFDEs). Firstly, an averaged L1 method for the initial value problem of F-VFDEs is presented based on the averaged L1 approximation for Caputo fractional derivative together with an appropriate piecewise interpolation operator for the functional term. Then the averaged L1 method is proved to be dissipative with an absorbing set and contractive with an algebraic decay rate. Finally, the numerical experiments further confirm the theoretical results.&lt;/p&gt;&lt;/abstract&gt;

COMPUTATIONAL SOLUTIONS OF FRACTIONAL ELECTRIC SYMMETRIC CIRCUITS BY SUMUDU TRANSFORMATION

In this research, we study the Caputo fractional and constant proportional derivative numerical approximation of electrical symmetric circuits. It has been assumed that the derivative is in the order [Formula: see text]. For the fractional electrical symmetric circuits, the RC, LC, and RLC solutions are obtained by using the Sumudu transformation. We also compare the numerical simulation of each equation to its classical equivalent. We use a highly efficient integral transform to examine the impact of the power-law kernel. In our upcoming works, we will apply this to electrical circuits that are more intricate.

Quasi-compact fourth-order approximations for fractional derivatives and applications

The Ceylon Journal of Science is a peer-reviewed journal published quarterly by the University of Peradeniya, Sri Lanka in March, June, September and December. It is aimed at publishing high quality research articles on topics related to different disciplines in Science. The journal accepts original research articles, book reviews, reviews and mini-reviews, short communications, opinions, research notes, and commentaries and notes. The Journal has its own website https://site.pdn.ac.lk/cjs/. The journal strictly adheres to publication ethics as emphasized by the Committee on Publication Ethics (COPE). It is indexed in Sri Lanka Journals Online (SLJOL), Directory of Open Access Journals (DOAJ), Google Scholar and Zoological Records. According to the Google Scholar;H5-Index: 12H5-Median: 15According to the Exaly (1970 – 2021);Impact Factor: 0.6 (top 19%)Extended IF: 0.6 (top 19%)H-Index: 8 (top 28%)Citations/paper: 1.42

Computational Algorithm for Approximating Fractional Derivatives of Functions

This paper presents an algorithmic approach for numerically solving Caputo fractional differentiation. The trapezoidal rule was modified, the new modification was used to derive an algorithm to approximate fractional derivatives of order α &gt; 0, the fractional derivative used was based on Caputo definition for a given function by a weighted sum of function and its ordinary derivatives values at specified points. The trapezoidal rule was used in conjunction with the finite difference scheme which is the forward, backward and central difference to derive the computational algorithm for the numerical approximation of Caputo fractional derivative for evaluating functions of fractional order. The study was conducted through some illustrative examples and analysis of error.