Fractional Derivative Method Research Articles

A numerical solution of time-fractional coupled Korteweg-de Vries equation by using spectral collection method

Abstract In this paper, we present a numerical method proficient for solving a system of time–fractional partial differential equations. For this sake, we use spectral collection method based on shifted Chebyshev polynomials in space and finite difference method for time fractional derivative. Furthermore, we employ this method to solve the time-fractional coupled Korteweg–de Vries (KdV) equations with appropriate initial and boundary conditions.

A space‐time spectral method for one‐dimensional time fractional convection diffusion equations

In this paper, we propose a space‐time spectral method for solving a class of time fractional convection diffusion equations. Because both fractional derivative and spectral method have global characteristics in bounded domains, we propose a space‐time spectral‐Galerkin method. The convergence result of the method is proved by providing a priori error estimate. Numerical results further confirm the expected convergence rate and illustrate the versatility of our method. Copyright © 2016 John Wiley & Sons, Ltd.

A wavelet method for numerical fractional derivative with noisy data

Numerical fractional differentiation is a classical ill-posed problem in the sense that a small perturbation in the data can cause a large change in the fractional derivative. In this paper, we consider a wavelet regularization method for solving a reconstruction problem for numerical fractional derivative with noise. A Meyer wavelet projection regularization method is given, and the Hölder-type stability estimates under both apriori and aposteriori regularization parameter choice rules are obtained. Some numerical examples show that the method works well.

Fractional order entropy: New perspectives

The entropy is a measurement of disorder or a measurement of heterogeneous information and it is known that amount of information cannot be negative. There are many methods about entropies by applied derivative to a probability distribution functions. In this paper, we applied fractional order derivative to probability distribution function, and this fractional order derivative is different than many of fractional order derivative methods in literature. Tsallis entropy was obtained by applying fractional order derivative methods in literature. In this paper, a new method for fractional order derivative was applied to probability distribution functions and two new definitions for entropy computation were obtained. One of them concludes in negative values for some cases, it is known that entropy cannot be negative, so, we selected the other definition (Definition 2) for entropy computation. The obtained new definition for entropy was applied to four probability distribution functions and obtained results were compared with Shannon entropies. The experimental results illustrated that the proposed method is superior to Shannon entropy method and the better entropy values were obtained for orders in interval (0,1).

Legendre Wavelet Operational Matrix of fractional Derivative through wavelet-polynomial transformation and its Applications in Solving Fractional Order Brusselator system

In this paper we propose the wavelet operational method based on shifted Legendre polynomial to obtain the numerical solutions of nonlinear fractional-order chaotic system known by fractional-order Brusselator system. The operational matrices of fractional derivative and collocation method turn the nonlinear fractional-order Brusselator system to a system of algebraic equations. Two illustrative examples are given in order to demonstrate the accuracy and simplicity of the proposed techniques.

Generalisation of Taylor's Formula and Differential Transform Method for Composite Fractional Derivative

In this paper, we first establish a generalized Taylor's formula for composite fractional derivative and then develop the generalized differential transform method for composite fractional derivative. As an application, we solve fractional Fokker–Planck Equation (FFPE) with space derivative considered as composite fractional derivative and time derivative considered as Caputo fractional derivative.

Fractional derivative formalism for non-destructive insulation diagnosis by polarization–depolarization current measurements

The fractional derivative method is considered as a possible formalism for the non-destructive insulation diagnosis by polarization and depolarization current measurements. The underlying motive is based on non-Debye relaxation processes clearly observed in long-time relaxation experiments with oil–paper, electrolytic and ultra-capacitors, transformers and other systems. Modes of relaxation at different time scales depend on charging prehistory and are sensitive to the state of a system. Fractional formalism provides useful parameters characterizing the state of the insulation system, which can be extracted from polarization–depolarization current measurements. Adequacy of the technique is examined by comparison of theoretical and experimental results for oil–paper, and ultra-capacitors.

Interpreting data on solar cosmic ray fluxes via the fractional derivative method

Solar cosmic ray propagation through the interplanetary magnetic field is considered as a random process of particles traveling along magnetic lines at a finite velocity of free motion and with a free path distributed according to an inverse power law. The propagator is presented as a sum of direct (nonscattered) flux (singular part of solution) and multiple scattered flux (regular part). In the long-time asymptotic, the regular part is described by an equation with a fractional-order derivative. Using analytical expressions for the propagator, we numerically calculate fluxes of energetic particles accelerated by shock waves generated by solar flares. The presented model is in better agreement with Ulysses and Voyager 2 data than the Perri-Zimbardo model and may therefore be recommended for use in interpreting the results of further experiments.

Tsallis Distribution-Based Fractional Derivative Method for Lamb Wave Signal Recovery

A new fractional derivative method based on the Tsallis distribution function is developed to recover the Lamb wave signals from the noisy Lamb wave signals. After the fractional derivative of the amplitude spectrum of the noisy Lamb signal at different derivative orders, the quartic polynomial function between the peak amplitude and the derivative order as well as that between the peak frequency and the derivative order is proposed based on the Tsallis distribution function. Then, the characteristic parameters of the amplitude spectrum are deduced by using the proposed polynomial relationship. Finally, the noise-free Lamb wave signal is recovered based on the characteristic parameters. Simulated results indicate that the Lamb wave signals can be recovered in the case of the white noise, the transient noise and the sine wave signal. Experimental results confirm the validity of the method. Consequently, the developed method can recover the Lamb wave signals effectively.

An approximate method for numerically solving multi-dimensional delay fractional optimal control problems by Bernstein polynomials

In this paper, we present a method for solving a class of fractional optimal control problems with time delay. The method is based upon the Bernstein polynomial basis. The main reason of using this technique is the efficiency and simple application of Bernstein polynomials in dealing with dynamical systems, which contains constant delay. In this work, we use the fractional derivative in Caputo sense and explain our method for fractional derivative of order \(0 < \alpha \le 1\). Some numerical examples are given to illustrate the effectiveness of our method.

粘弾性体の分数階微分有限変形理論と非線形有限要素法 : 減衰マトリックスと運動方程式の定式化(機械力学,計測,自動制御)

Fractional derivative finite deformation theory and nonlinear finite element method were developed for evaluating dynamic characteristics of viscoelastic bodies used in shock and vibration reduction. Fractional differential orders from the static and dynamic testing of viscoelastic material with small strain were obtained. Then a fractional derivative constitutive model for finite deformation theory was constructed and nonlinear finite element equations of motion, involving a newly developed viscoelastic damping matrix as well as standard mass and stiffness matrices, were formulated. Based on this work, a nonlinear finite element computer program for the evaluation of vibration characteristics of viscoelastic products was developed. Simple numerical examples were computed and compared with the experimental results. Both results are in good agreement.

Band shape determination with robust estimator based on continuous wavelet transform

The paper focuses on an alternative approach that allows one to identify overlapping band shapes with the help of the continuous wavelet transform (CWT). We show that less number of special points for determining a band shape is required unlike the fractional derivative spectrometry method [S.S. Kharintsev, M.Kh. Salakhov, Spectrochim. Acta Part A, 60 (2004) 2125]. Besides, the CWT-based derivative spectrometry can be successfully utilized in a case of complex spectra corrupted with a white and/or high-frequency noise. The power of this method is illustrated on model examples and experimental spectra of 1,2-diphenylethane in crystalline and melted phase.