Fractional Order Differential Operator Research Articles

Improving a Tikhonov regularization method with a fractional-order differential operator for the inverse black body radiation problem

Tikhonov regularization is an usual method to solve an ill-posed problem, recommended when the input data are contaminated with noise. However, in some cases, the use of this technique is not sufficient to provide good solutions. In this work, an improvement of the Tikhonov regularization method was proposed and tested in the inverse black body radiation problem. The method proposed consisted in including the norm of the fractional-order derivative of the solution in the original functional proposed by Tikhonov. In the present framework, the regularized solution depends on the regularization parameter, λ, and on the fractional derivative order, α. For α assuming real value between 0 and 2, the solution obtained is more precise than those from the usual Tikhonov regularization method.

Guest Editorial Introduction to the Special Section on Nonlinear Fractional-Order Circuits and Systems: Advanced Analysis and Effective Implementation

Nowadays, fractional order differential operators, as a generalization for classical differential operators, have established their key-role in modeling, analysis, and implementation of specific circuits and systems in which one typically faces nonlinear behaviors. It enforces to exploit analysis and implementation methods covering simultaneously “fractionality” and “nonlinearity” aspects. This special section, entitled “Nonlinear Fractional Order Circuits and Systems: Advanced Analysis and Effective Implementation,” aims at introducing some of these methods.

Nonlinear Fractional-Order Circuits and Systems: Motivation, A Brief Overview, and Some Future Directions

In recent years, fractional-order differential operators, and the dynamic models constructed based on these generalized operators have been widely considered in design and practical implementation of electrical circuits and systems. Simultaneously, facing with fractional-order dynamics and the nonlinear ones in electrical circuits and systems enforces us to use more advanced tools (in comparison to those commonly used in design and analysis of linear fractional-order/nonlinear integer-order circuits and systems) for their analysis, design, and implementation. Discussing on such a motivation, this tutorial paper aims to provide an overview on the recent achievements in proposing effective tools for analysis and design of nonlinear fractional-order circuits and systems. Moreover, some open problems, which can specify future directions for continuing research works on the aforementioned subject, are discussed.

Multispectral Image Fusion Using Fractional-Order Differential and Guided Filtering

Remote sensing satellites can provide a large number of multispectral images. However, due to the limitations of optical sensors embedded in satellites, the spatial resolution of multispectral images is relatively low. Pansharpening aims to combine high-resolution panchromatic and multi-spectral images to generate high-resolution multi-spectral images. In this paper, we propose a pansharpening method based on a component substitution framework. We use fractional-order differential operators and guided filter to balance the spectral distortion and spatial information loss that occur when remote sensing image fusion. Fractional-order differentiation can better define the detailed map, and the guided filter can enhance the spectral information of the detailed map. Experiments show that the proposed method in this paper can better combine the spectral information and spatial information, as well as obtain satisfactory results in both subjective visual perception and objective object evaluation.

Spectral solutions of fractional differential equation modelling electrohydrodynamics flow in a cylindrical conduit

In this paper, we have studied the fractional order model of the singular nonlinear differential equation governing the electrohydrodynamics flow in a circular cylindrical conduit. It is known that the integer order differential operator is a local operator but the fractional order differential operator is non-local, hence, the next state of a system depends not only upon its existing state but also upon all of its historical states. The effect of various parameters such as strength of nonlinearity α, the Hartmann electric number Ha and the influence of fractional order derivatives β on the velocity profiles are investigated. The velocity profiles are bounded above by 11+α is re-established by the spectral method of solutions. Error analysis has been computed to assess the absolute error and residual error of the present results. The influence of fractional order derivative and its effect on the centreline velocity is analysed in detail. With the help of several comparison tables and figures, it is shown that the spectral collocation technique is a very efficient and robust method of solution compared to many other solution procedures available in the literature.

Mathematical Analysis of Implicit Impulsive Switched Coupled Evolution Equations

In the present manuscript a novel type of implicit switched coupled evolution system is studied. The underlying system is formulated with fractional order differential operator while incorporating impulses in the solution. Reducing the proposed model into fixed point problem, results for the existence and uniqueness of solution are exhibited. Also, we established results related to Hyers–Ulam type stability of the solution. An illustrative example is also solved to support the obtained theoretical results.

Research on Coal-Rock Fracture Image Edge Detection Based on Tikhonov Regularization and Fractional Order Differential Operator

Aiming at the conventional image edge detection algorithm, the first-order differential edge detection method is easy to lose the image details and the second-order differential edge detection method is more sensitive to noise. To deal with the problem, the Tikhonov regularization method is adopted to reconstruct the input coal-rock infrared images, so as to reduce the noise interference, and then, the reconstructed image is transformed by gray level. Finally, we consider the frequency characteristics and long memory properties of fractional differential, the classical first-order Sobel and second-order Laplacian edge detection algorithms are extended to fractional order pattern, and a new pattern of fractional order differential image edge detection is constructed to realize the coal-rock fracture edge features identification. The results show that, compared with integer order differential, the error rate and omission rate of fractional order differential algorithm are smaller, the quality factor is larger, and the execution time and memory footprint are smaller. From the point of view of location criteria and location accuracy, the fractional order differential algorithm is better than the integer order. In addition, the proposed method is compared with Canny algorithm, B-spline wavelet transform, and multidirection fuzzy morphological edge detection method, can detect more coal-rock fracture infrared image edge details, and is more robust to noise.

On one sum of Hankel-Clifford integral transforms of Whittaker functions

In [11], the authors considered the realization T of SO (2, 2)-representation in a space of homogeneous functions on 2×4-matrices. In this sequel, we aim to compute matrix elements of the identical operator T ( e ) and representation operator T (g) for an appropriate g with respect to the mixed basis related to two different bases in the SO (2, 2)-carrier space and evaluate some improper integrals involving a product of Bessel-Clifford and Whittaker functions. The obtained result can be rewritten in terms of Hankel-Clifford integral transforms and their analogue. The first and the second Hankel-Clifford transforms introduced by Hayek and Perez–Robayna, respectively, play an important role in the theory of fractional order differential operators (see, e.g., [6, 8]). The similar result have been derived recently by the authors for the regular Coulomb function in [12].

Green's formula for integro-differential operators

In this report we establish Green's formula for an integro-differential operator, and apply it to describe a class of self-adjoint fractional order differential operators. A found symmetric fractional order Caputo–Riemann–Liouville type operator can be considered as a fractional analogue of the classical Sturm–Liouville operator in some sense.

Recovering an unknown source in a fractional diffusion problem

A standard inverse problem is to determine a source which is supported in an unknown domain D from external boundary measurements. Here we consider the case of a time-independent situation where the source is equal to unity in an unknown subdomain D of a larger given domain Ω and the boundary of D has the star-like shape, i.e.∂D={q(θ)(cos⁡θ,sin⁡θ)⊤:θ∈[0,2π]}. Overposed measurements consist of time traces of the solution or its flux values on a set of discrete points on the boundary ∂Ω. The case of a parabolic equation was considered in [6]. In our situation we extend this to cover the subdiffusion case based on an anomalous diffusion model and leading to a fractional order differential operator. We will show a uniqueness result and examine a reconstruction algorithm. One of the main motives for this work is to examine the dependence of the reconstructions on the parameter α, the exponent of the fractional operator which controls the degree of anomalous behaviour of the process. Some previous inverse problems based on fractional diffusion models have shown considerable differences between classical Brownian diffusion and the anomalous case.

Image Denoising Based on Adaptive Fractional Order with Improved PM Model

In order to improve the image quality, in this paper, we propose an improved PM model. In the proposed model, we introduce two novel diffusion coefficients and a residual error term and replace the integer differential operator with the fractional differential operator in the PM model. The diffusion coefficients can be used effectively for edge detection and noise removal. The residual error term can help to prevent image distortion. Fractional order differential operator has a good characteristic that it can enhance image texture information while removing image noise. Additionally, in the two new diffusion coefficients, a novel method is proposed for automatically setting parameter k, and it does not need to do any experiments to get the value of k. For the computing fractional order diffusion coefficient, we employ the discrete Fourier transform, and an iterative scheme is carried out in the frequency domain. In the proposed model, not only is the integer differential operator replaced with the fractional differential operator, but also the order of the fractional differentiation is determined adaptively with the local variance. Comparing with some existing models, the experimental results show that the proposed algorithm can not only better suppress noise, but also better preserve edge and texture information. Moreover, the running time is greatly reduced.

Parallel solvers for fractional power diffusion problems

SummaryMathematical models with fractional‐order differential operators are computationally expensive due to the non‐local nature of these operators. In this work, we construct and investigate parallel solvers for problems described by fractional powers of elliptic operators, like fractional diffusion. Three state‐of‐the‐art approaches are used to transform the non‐local fractional‐order differential problem into local partial differential equation problems formulated in a space of higher dimension. Numerical schemes and parallel algorithms are developed for all three approaches. The resulting parallel algorithms have very different properties. We investigate the weak and strong scalability of the developed parallel algorithms and compare their parallel performance.

Variational formulation of problems involving fractional order differential operators

In this work, we consider boundary value problems involving either Caputo or Riemann-Liouville fractional derivatives of order $\alpha \in (1,2)$ on the unit interval $(0,1)$. These fractional derivatives lead to nonsymmetric boundary value problems, which are investigated from a variational point of view. The variational problem for the Riemann-Liouville case is coercive on the space $H_0^{\alpha /2}(0,1)$ but the solutions are less regular, whereas that for the Caputo case involves different test and trial spaces. The numerical analysis of these problems requires the so-called shift theorems which show that the solutions of the variational problem are more regular. The regularity pickup enables one to establish convergence rates of the finite element approximations. The analytical theory is then applied to the Sturm-Liouville problem involving a fractional derivative in the leading term. Finally, extensive numerical results are presented to illustrate the error estimates for the source problem and eigenvalue problem.

Positive Solutions for Class of State Dependent Boundary Value Problems with Fractional Order Differential Operators

We consider the following state dependent boundary-value problemD0+αy(t)-pD0+βg(t,y(σ(t)))+f(t,y(τ(t)))=0,0<t<1;y(0)=0,ηy(σ(1))=y(1),whereDαis the standard Riemann-Liouville fractional derivative of order1<α<2,0<η<1,p≤0,0<β<1,β+1-α≥0the functiongis defined asg(t,u):[0,1]×[0,∞)→[0,∞), andg(0,0)=0the functionfis defined asf(t,u):[0,1]×[0,∞)→[0,∞)σ(t),τ(t)are continuous ontand0≤σ(t),τ(t)≤t. Using Banach contraction mapping principle and Leray-Schauder continuation principle, we obtain some sufficient conditions for the existence and uniqueness of the positive solutions for the above fractional order differential equations, which extend some references.

A New Supercapacitors Fractional Order Nonlinear Model

A simple fractional order nonlinear supercapacitors (SCs) electrical model has been established, taking into account the parameter variation, diffusion control and Faraday process. Frequency responses resulting from impedance spectroscopies around various operating voltages demonstrated that a SC exhibits a nonlinear behaviour in relation to the operating voltage. The model parameters are identified using fractional vector fitting and then a nonlinear model taking voltage dependency into account is obtained. Through building fractional order differential operator module in Simulink, the numerical solution of fractional order nonlinear calculus equation can be obtained directly. The model accuracy is verified through a comparison with the charging and discharing experimental results.

Non-Local Fractional Differential-Based Approach for Image Enhancement

This study proposed an image enhancing method which is based on the non-local fractional order differential operator. In this method, a matrix form representation of discrete fractional order differentiation is introduced to enhance the digital image, which is effective to reduce the computation error caused by the traditional local approximate method of the fractional order differentiation. The proposed enhancing method is able to make effective use of the whole image information and improve the enhancing performance of the image enhancing algorithm based on the local mask. The color image enhancing strategy based on the non-local fractional differential also is given. A lot of experiments demonstrate that the proposed method is capable of enhancing gray and color image effectively.

Fractional-order [formula omitted] controller design

This paper introduces a new design method of fractional-order proportional–derivative (FOPD) and fractional-order proportional–integral–derivative (FOPID) controllers. A biquadratic approximation of a fractional-order differential operator is used to introduce a new structure of finite-order FOPID controllers. Using the new FOPD controllers, the controlled systems can achieve the desired phase margins without migrating the gain crossover frequency of the uncontrolled system. This may not be guaranteed when using FOPID controllers. The proposed FOPID controller has a smaller number of parameters to tune than its existing counterparts. A systematic design procedure is identified in terms of the desired phase and the gain margins of the controlled systems. The viability of the design methods is verified using a simple numerical example.

The Samarskii–Ionkin type problem for the fourth order parabolic equation with fractional differential operator

In the present work the Samarskii–Ionkin type non-local problem with Caputo fractional order differential operator is studied. The considered problem generalizes some previous known problems formulated for some fourth order parabolic equations. We prove the existence and uniqueness of a regular solution of the formulated problem applying the method of separation of variables.

Reparametrized ARX models for predictive control of staged and packed bed distillation columns

This work is aimed at the development of reparametrized ARX type models for high dimensional and distributed parameter systems. To keep data length small while identifying a model with ARX structure, the feasibility of reparametrizing ARX models using the fractional order differential operators and orthonormal basis filters is explored. The identified noise model is further used for developing a novel observer based MPC scheme, which explicitly uses the identified unmeasured disturbance model for the future trajectory predictions. The efficacy of the proposed modeling technique and the MPC scheme is demonstrated by conducting (a) simulation studies on a staged distillation column and (b) experimental evaluations on a laboratory scale packed bed distillation column.

Development of ARX Models for Predictive Control Using Fractional Order and Orthonormal Basis Filter Parametrization

Among various industrial vendors of model predictive control (MPC) schemes, ARX appears to be a popular choice of model structure while models for predictive control schemes are being developed. These models are, however, nonparsimonious in the number of model parameters. As a consequence, the length of the data required to keep the variance errors low while using the conventional ARX structure is significantly large. Thus, if it is possible to reparametrize the ARX model such that fewer parameters are required at the identification stage, then it is possible to reduce the length of identification data and thereby reduce the cost involved in model identification exercise. In this work, we explore the possibility of reparametrizing ARX models using the fractional-order differential operators (FO-ARX) and orthonormal basis filters (OBF-ARX). We also propose a novel approach for identification of time delay matrix from multivariate data using ARX and OBF-ARX models. The efficacy of the proposed modeling technique is demonstrated by conducting simulation studies on the benchmark Shell control problem. Analysis of the simulation results reveals that, when compared with the conventional high-order ARX structure, the FO-ARX and the OBF-ARX are better model parametrizations when the data length is less. In particular, OBF-ARX parametrization is able to estimate the time delay matrix from multivariate data quite accurately. The experimental studies establish the feasibility of using the proposed FO-ARX and OBF-ARX models for formulating MPC schemes.