Order Derivative Operators Research Articles

Approximate Solution and Analysis of Smoking Epidemic Model with Caputo Fractional Derivatives

In this paper, an epidemic model is presented to describe the dynamics of drugs usage among the adults. The Caputo fractional derivative operator of order $$\phi \in (0, 1]$$ is employed to obtain the system of fractional differential equations. Smoking is the large problem in the entire world. Despite the overwhelming facts about smoking, it is still a bad habit which is widely spread and socially accepted. A non-linear mathematical model is established to study and assess the dynamics of smoking and its impact on public health in a community. The analysis of two different states, disease free and endemic which means the disease dies out or persist in a population has been made by presenting sensitivity analysis. The stability of the model is provided by threshold or reproductive number. The Laplace Adomian decomposition method has been employed to solve smoking model which has five compartments, potential smoker P, occasional smokers O, smokers S, temporarily quit smokers Q and permanently quit smokers L. Finally, some numerical results are presented which shows the effect of fractional parameter $$\phi $$ on our obtained solutions.

A class of fractional differential hemivariational inequalities with application to contact problem

In this paper, we study a class of generalized differential hemivariational inequalities of parabolic type involving the time fractional order derivative operator in Banach spaces. We use the Rothe method combined with surjectivity of multivalued pseudomonotone operators and properties of the Clarke generalized gradient to establish existence of solution to the abstract inequality. As an illustrative application, a frictional quasistatic contact problem for viscoelastic materials with adhesion is investigated, in which the friction and contact conditions are described by the Clarke generalized gradient of nonconvex and nonsmooth functionals, and the constitutive relation is modeled by the fractional Kelvin–Voigt law.

On interpolation functions for the number of k-ary Lyndon words associated with the Apostol–Euler numbers and their applications

The aim of this paper is to construct interpolation functions for the numbers of the k-ary Lyndon words which count n digit primitive necklace class representative on the set of the k-letter alphabet. By using the unified zeta-type function and the unification of the Apostol-type numbers which are defined by Ozden et al. (Comput Math Appl 60:2779–2787, 2010), we give an alternating series for the numbers of the k-ary Lyndon words, $$L_{k}\left( n\right) $$ in terms of the Apostol–Euler numbers and Frobenius–Euler numbers. We investigate various properties of these functions. Furthermore, applying higher order derivative operator to the interpolation functions for the Lyndon words, we derive ODEs including Stirling-type numbers, the Apostol–Euler numbers, the unified zeta-type functions and also combinatorial sums. By using recurrence relation of the Apostol–Euler numbers, we give computation algorithms for computing not only the Apostol–Euler numbers but also the interpolation functions of the numbers $$L_{k}\left( n\right) $$ . We also give some remarks, observations and computations for sums of infinite series including these interpolation functions.

A New Integer Order Approximation Table for Fractional Order Derivative Operators

There is considerable interest in the study of fractional order calculus in recent years because real world can be modelled better by fractional order differential equation. However, computing analytical time responses such as unit impulse and step responses is still a difficult problem in fractional order systems. Therefore, advanced integer order approximation table which gives satisfying results is very important for simulation and realization. In this paper, an integer order approximation table is presented for fractional order derivative operators (sa) where α ϵ R and 0 < α < 1 using exact time response functions computed with inverse Laplace transform solution technique. Thus, the time responses computed using the given table are almost the same with exact solution. The results are also compared with some well-known integer order approximation methods such as Oustaloup and Matsuda. It has been shown in numerical examples that the proposed method is very successful in comparison to Oustaloup’s and Matsuda’s methods.

Fractional order linear time invariant system stabilization by brute-force search

Fractional calculus increases their applications in system design and analysis problems because of providing more realistic modeling of real systems. Owing to computational complexity of fractional calculus, the computer-aided design and analysis methods are required for engineering applications of fractional order systems. This study presents a numerical method for parametric robust stabilization of fractional order systems by employing single-parameter perturbation. This method implements a fractional order perturbation strategy on the basis of brute-force search technique for system stabilization problems. In order to meet a predefined minimum argument root design specification, the proposed algorithm searches for a desired placement of the minimum argument characteristic root within the first Riemann sheet by performing iterative perturbations of the fractional order. This approach can provide a straightforward numerical solution for robust stabilization problems of fractional order systems by employing an order perturbation scheme. Moreover, a possible utilization of a fractional order derivative operator as a system stabilizer is theoretically discussed. Illustrative examples show the utilization of the proposed stabilization algorithms for computer-aided fractional order system design applications.

General fractional calculus operators containing the generalized Mittag-Leffler functions applied to anomalous relaxation

In this paper, we address a family of the general fractional calculus operators of Wiman and Prabhakar types for the first time. The general Mittag-Leffler function to structure the kernel functions of the fractional order derivative operators and their Laplace integral transforms are considered in detail. The formulations are as the mathematical tools proposed to investigate the anomalous relaxation.

Necessary conditions for convergence of difference schemes for fractional-derivative two-point boundary value problems

A fractional-derivative two-point boundary value problem of the form \({\tilde{D}}^\delta u=f\) on (0, 1) with Dirichlet boundary conditions is studied. Here \({\tilde{D}}^\delta \) is a Caputo or Riemann–Liouville fractional derivative operator of order \(\delta \in (1,2)\). The discretisation of this problem by an arbitrary difference scheme is examined in detail when u or f is a polynomial. For any convergent difference scheme, it is proved rigorously that the entries of the associated matrix must satisfy certain identities. It is shown that some of these identities are not satisfied by certain well-known schemes from the research literature; this clarifies the type of problem to which these schemes can be applied successfully. The effects of the special boundary condition \(u(0)=0\) and the special right-hand-side condition \(f(0)=0\) are also investigated. This leads, under certain circumstances, to a sharpening of a recently-published finite difference scheme convergence result of two of the authors.

Transition Curve Analysis of Linear Fractional Periodic Time-Delayed Systems Via Explicit Harmonic Balance Method

This paper presents a technique to obtain the transition curves of fractional periodic time-delayed (FPTD) systems based on a proposed explicit harmonic balance (EHB) method. This method gives the analytical Hill matrix of FPTD systems explicitly with a symbolic computation-free algorithm. Furthermore, all linear operations on Fourier basis vectors including fractional order derivative operators and time-delayed operators for a linear FPTD system are obtained. This technique is illustrated with parametrically excited simple and double pendulum systems, with both time-delayed states and fractional damping.

On a boundary-localized Higgs boson in 5D theories.

In the context of a simple five-dimensional (5D) model with bulk matter coupled to a brane-localized Higgs boson, we point out a non-commutativity in the 4D calculation of the mass spectrum for excited fermion towers: the obtained expression depends on the choice in ordering the limits, N rightarrow infty (infinite Kaluza–Klein tower) and epsilon rightarrow 0 (epsilon being the parameter introduced for regularizing the Higgs Dirac peak). This introduces the question of which one is the correct order; we then show that the two possible orders of regularization (called I and II) are experimentally equivalent, as both can typically reproduce the measured observables, but that the one with less degrees of freedom (I) could be uniquely excluded by future experimental constraints. This conclusion is based on the exact matching between the 4D and 5D analytical calculations of the mass spectrum – via regularizations of type I and II. Beyond a deeper insight into the Higgs peak regularizations, this matching brings another confirmation of the validity of the 5D mixed formalism. All the conclusions, deduced from regularizing the Higgs peak through a brane shift or a smoothed square profile, are expected to remain similar in realistic models with a warped extra-dimension. The complementary result of the study is that the non-commutativity disappears, both in the 4D and the 5D calculations, in the presence of higher order derivative operators. For clarity, the 4D and 5D analytical calculations, matching with each other, are presented in the first part of the paper, while the second part is devoted to the interpretation of the results.

Computational imaging from non-uniform degradation of staggered TDI thermal infrared imager.

For the Time Delay Integration (TDI) staggered line-scanning thermal infrared imager, a Computational Imaging (CI) approach is developed to achieve higher spatial resolution images. After a thorough analysis of the causes of non-uniform image displacement and degradation for multi-channel staggered TDI arrays, the study aims to approach one-dimensional (1D) sub-pixel displacement estimation and superposition of images from time-division multiplexing scanning lines. Under the assumption that a thermal image is 2D piecewise C(2) smooth, a sparse-and-smooth deconvolution algorithm with L1-norm regularization terms combining the first and second order derivative operators is proposed to restore high frequency components and to suppress aliasing simultaneously. It is theoretically and experimentally demonstrated, with simulation and airborne thermal infrared images, that this is a state-of-the-art practical CI method to reconstruct clear images with higher frequency components from raw thermal images that are subject to instantaneous distortion and blurring.

Reaction-Advection-Diffusion Equations with Space Fractional Derivatives and Variable Coefficients on Infinite Domain

Abstract In this paper we consider a space fractional reaction-advection-diffusion equation which is actually a semi-linear Cauchy problem with a spatial fractional derivative operator of order α, 0 &lt; α &lt; 2. We establish and prove assertions concerning the existence and uniqueness of solution within certain Colombeau space. The proofs are given in the case when the left Liouville fractional derivative is involved, but the results are also valid in the case of the right Liouville fractional derivative as well as for the Riesz fractional derivative. The solutions are obtained by using the theory of generalized uniformly continuous semigroups of operators. We have also proved that the non-regularized and corresponding regularized operators, as well as solutions of non-regularized and corresponding regularized equations are L2−associated.

On the sectorial property of the Caputo derivative operator

In this note, we establish the sectorial property of the Caputo fractional derivative operator of order α∈(1,2) with a zero Dirichlet boundary condition.

Higher derivatives and power spectrum in effective single field inflation

We study next-to-leading corrections to the effective action of the curvature perturbation obtained by integrating out the coupled heavy isocurvature perturbation. These corrections result from including higher order derivative operators, weighted by the mass scale of the heavy physics, in the effective theory expansion. We find that the correction terms are suppressed by the ratio of the Hubble parameter to the heavy mass scale. The corresponding corrections to the power spectrum of the curvature perturbation are presented for a simple illustrative example.

New Insights into the Fractional Order Diffusion Equation Using Entropy and Kurtosis.

Fractional order derivative operators offer a concise description to model multi-scale, heterogeneous and non-local systems. Specifically, in magnetic resonance imaging, there has been recent work to apply fractional order derivatives to model the non-Gaussian diffusion signal, which is ubiquitous in the movement of water protons within biological tissue. To provide a new perspective for establishing the utility of fractional order models, we apply entropy for the case of anomalous diffusion governed by a fractional order diffusion equation generalized in space and in time. This fractional order representation, in the form of the Mittag–Leffler function, gives an entropy minimum for the integer case of Gaussian diffusion and greater values of spectral entropy for non-integer values of the space and time derivatives. Furthermore, we consider kurtosis, defined as the normalized fourth moment, as another probabilistic description of the fractional time derivative. Finally, we demonstrate the implementation of anomalous diffusion, entropy and kurtosis measurements in diffusion weighted magnetic resonance imaging in the brain of a chronic ischemic stroke patient.

Discrete dynamics of one dimensional Collatz like integral value transformations

In Discrete dynamics and number theory, one of the most important challenging conjectures is Collatz conjecture. Some basic algebraic and analytical properties of Collatz like integral value transformations (IVTs) in the context of discrete dynamical system over \(\mathbb {N}_0\) are adumbrated. The dynamical maps associated to the dynamical systems are everywhere continuous but nowhere differentiable (smooth) in domain \(\mathbb {N}_0\). Under such Collatz like IVTs, for any initial point \(X_0\)\(\in \)\(\mathbb {N}_0\) the dynamical systems are convergent and converge eventually to a single point attractor, zero. In other words, the image sets associated to the Collatz like dynamical maps in each iteration converge to a single point image set consisting the point zero. The rate of convergence of the Collatz like dynamical system with respect to some property is also described. It is observed that the \(l_2\)-norm of the ith order derivative operator for all Collatz like map is a decreasing convergent sequence which converges to zero.

Gauge-flation confronted with Planck

Gauge-flation is a recently proposed model in which inflation is driven solely by a non-Abelian gauge field thanks to a specific higher order derivative operator. The nature of the operator is such that it does not introduce ghosts. We compute the cosmological scalar and tensor perturbations for this model, improving over an existing computation. We then confront these results with the Planck data. The model is characterized by the quantity γ ≡ g2Q2/H2 (where g is the gauge coupling constant, Q the vector vev, and H the Hubble rate). For γ < 2, the scalar perturbations show a strong tachyonic instability. In the stable region, the scalar power spectrum ns is too low at small γ, while the tensor-to-scalar ratio r is too high at large γ. No value of γ leads to acceptable values for ns and r, and so the model is ruled out by the CMB data. The same behavior with γ was obtained in Chromo-natural inflation, a model in which inflation is driven by a pseudo-scalar coupled to a non-Abelian gauge field. When the pseudo-scalar can be integrated out, one recovers the model of Gauge-flation plus corrections. It was shown that this identification is very accurate at the background level, but differences emerged in the literature concerning the perturbations of the two models. On the contrary, our results show that the analogy between the two models continues to be accurate also at the perturbative level.

Stable Computation of Differentiation Matrices and Scattered Node Stencils Based on Gaussian Radial Basis Functions

Radial basis function (RBF) approximation has the potential to provide spectrally accurate function approximations for data given at scattered node locations. For smooth solutions, the best accuracy for a given number of node points is typically achieved when the basis functions are scaled to be nearly flat. This also results in nearly linearly dependent basis functions and severe ill-conditioning of the interpolation matrices. Fornberg, Larsson, and Flyer recently generalized the RBF-QR method to provide a numerically stable approach to interpolation with flat and nearly flat Gaussian RBFs for arbitrary node sets in up to three dimensions. In this work, we consider how to extend this method to the task of computing differentiation matrices and stencil weights in order to solve partial differential equations. The expressions for first and second order derivative operators as well as hyperviscosity operators are established, numerical issues such as how to deal with non-unisolvency are resolved, and the accuracy and computational efficiency of the method are tested numerically. The results indicate that using the RBF-QR approach for solving PDE problems can be very competitive compared with using the ill-conditioned direct solution approach or using variable precision arithmetic to overcome the conditioning issue.

Improved Total Variation-Type Regularization Using Higher Order Edge Detectors

We present a novel deconvolution approach for accurately restoring piecewise smooth signals from blurred data. There are two separate stages. The first stage uses higher order total variation (TV) restorations to obtain an estimate of the location of jump discontinuities from the blurred data. In the second stage the estimated jump locations are used to determine the local orders of a variable order TV restoration. The method replaces the first order derivative approximation used in standard TV by a variable order derivative operator. Smooth segments as well as jump discontinuities are restored, while the staircase effect typical for standard first order TV regularization is avoided. Compared to first order TV, signal restorations are more accurate representations of the true signal, as measured in a relative $l^2$-norm. The method can also be used to obtain an accurate estimation of the locations and sizes of the true jump discontinuities. The approach is independent of the algorithm used for the standard TV problem and is, consequently, readily incorporated into existing TV restoration codes.

A fractional adaptation law for sliding mode control

AbstractThis paper presents a novel parameter tuning law that forces the emergence of a sliding motion in the behavior of a multi‐input multi‐output nonlinear dynamic system. Adaptive linear elements are used as controllers. Standard approach to parameter adjustment employs integer order derivative or integration operators. In this paper, the use of fractional differentiation or integration operators for the performance improvement of adaptive sliding mode control systems is presented. Hitting in finite time is proved and the associated conditions with numerical justifications are given. The proposed technique has been assessed through a set of simulations considering the dynamic model of a two degrees of freedom direct drive robot. It is seen that the control system with the proposed adaptation scheme provides (i) better tracking performance, (ii) suppression of undesired drifts in parameter evolution, (iii) a very high degree of robustness and improved insensitivity to disturbances and (iv) removal of the controller initialization problem. Copyright © 2008 John Wiley &amp; Sons, Ltd.

Real-time application of critical dimension measurement of TFT-LCD pattern using a newly proposed 2D image-processing algorithm

A critical dimension measurement system for TFT-LCD patterns has been implemented in this study. To improve the measurement accuracy, an imaging auto-focus algorithm, fast pattern-matching algorithm, and precise edge detection algorithm with subpixel accuracy have been developed and implemented in the system. The optimum focusing position can be calculated using the image focus estimator. The two-step auto-focusing technique has been newly proposed for various LCD patterns, and various focus estimators have been compared to select a stable and accurate one. Fast pattern matching and subpixel edge detection have been developed for measurement. The new approach, called NEMC, is based on edge detection for the selection of influential points; in this approach, points having a strong edge magnitude are only used in the matching procedure. To accelerate pattern matching, point correlation and an image pyramid structure are combined. Edge detection is the most important technique in a vision inspection system. A two-stage edge detection algorithm has been introduced. In the first stage, a first order derivative operator such as the Sobel operator is used to place the edge points and to find the edge directions using a least-square estimation method with pixel accuracy. In the second stage, an eight-connected neighborhood of the estimated edge points is convolved with the LoG (Laplacian of Gaussian) operator, and the LoG-filtered image can be modeled as a continuous function using the facet model. The measurement results of the various patterns are finally presented. The developed system has been successfully used in the TFT-LCD manufacturing industry, and repeatability of less than 30 nm (3 σ) can be obtained with a very fast inspection time.