Non-integer Order Research Articles

The Mittag-Leffler Function for Re-Evaluating the Chlorine Transport Model: Comparative Analysis

This paper re-investigates the mathematical transport model of chlorine used as a water treatment model, when a variable order partial derivative is incorporated for describing the chlorine transport system. This model was introduced in the literature and governed by a fractional partial differential equation (FPDE) with prescribed boundary conditions. The obtained solution in the literature was based on implementing the Laplace transform (LT) combined with the method of residues and expressed in terms of regular exponential functions. However, the present analysis avoids such a method of residues, and thus a new analytical solution is introduced in this paper via Mittag-Leffler functions. Therefore, an effective approach is developed in this paper to solve the chlorine transport model with non-integer order derivative. In addition, our results are compared with several studies in the literature in case of integer-order derivative and the differences in results are explained.

Rényi entropies of the massless Dirac field on the torus

We compute the R\'enyi entropies of the massless Dirac field on the Euclidean torus (the Lorentzian cylinder at non-zero temperature) for arbitrary spatial regions. We do it by the resolvent method, i.e., we express the entropies in terms of the resolvent of a certain operator and then use the explicit form of that resolvent, which was obtained recently. Our results are different in appearance from those already existing in the literature (obtained via the replica trick), but they agree perfectly, as we show numerically for non-integer order and analytically for integer order. We also compute the R\'enyi mutual information, and find that, for appropriate choices of the parameters, it is non-positive and non-monotonic. This behavior is expected, but it cannot be seen with the simplest known R\'enyi entropies in quantum field theory because they are proportional to the entanglement entropy.

Non-integer order chaotic systems: numerical analysis and their synchronization scheme via M-backstepping technique

Non-integer order chaotic systems: numerical analysis and their synchronization scheme via M-backstepping technique

Algebraic equations and non-integer orders of fractal operators abstracted from biomechanics

Algebraic equations and non-integer orders of fractal operators abstracted from biomechanics

A novel fractal-fractional order model for the understanding of an oscillatory and complex behavior of human liver with non-singular kernel

Scientists and researchers are increasingly interested in numerical simulations of infections with non-integer orders. It is self-evident that conventional epidemiological systems can be given in a predetermined order, but fractional-order derivative systems are not stable orders. The fractional derivative proves increasingly effective in representing real-world issues when it has a non-fixed order. Various novel fractional operator notions, including special functions in the kernel, have been presented in recent decades, which transcend the constraints of prior fractional order derivatives. These novel operators have been shown to be useful in simulating scientific and technical challenges. The fractal-fractional operator is a relatively modern fractional calculus operator that has been proposed. Besides that, we propose a new technique and implement it in a human liver model and want to investigate its dynamics. In the context of this novel operator, we demonstrate certain interesting findings for the human liver model. The findings of the uniqueness and existence will be revealed. We describe modeling estimates for the proposed model using an innovative numerical method that has never been used before for a human liver model of this type. Additionally, graphical illustrations are demonstrated for both fractal and fractional orders. It is expected that the fractal-fractional approach is more invigorating and effective for epidemic models than the fractional operator.

Sobolev-Slobodeckij Spaces on Compact Manifolds, Revisited

In this manuscript, we present a coherent rigorous overview of the main properties of Sobolev-Slobodeckij spaces of sections of vector bundles on compact manifolds; results of this type are scattered through the literature and can be difficult to find. A special emphasis has been put on spaces with noninteger smoothness order, and a special attention has been paid to the peculiar fact that for a general nonsmooth domain Ω in Rn, 0<t<1, and 1<p<∞, it is not necessarily true that W1,p(Ω)↪Wt,p(Ω). This has dire consequences in the multiplication properties of Sobolev-Slobodeckij spaces and subsequently in the study of Sobolev spaces on manifolds. We focus on establishing certain fundamental properties of Sobolev-Slobodeckij spaces that are particularly useful in better understanding the behavior of elliptic differential operators on compact manifolds. In particular, by introducing notions such as “geometrically Lipschitz atlases” we build a general framework for developing multiplication theorems, embedding results, etc. for Sobolev-Slobodeckij spaces on compact manifolds. To the authors’ knowledge, some of the proofs, especially those that are pertinent to the properties of Sobolev-Slobodeckij spaces of sections of general vector bundles, cannot be found in the literature in the generality appearing here.

Direct Solution of the Black-Scholes PDE Models with Non-Integer Order

This paper presents a direct solution of Black-Scholes PDE models with non-integer order via the non-integer iterative method. The Black-Scholes PDE models of non-integer order became popular and accepted globally by option traders for the valuation of financial derivatives over the period of time and the study of its numerical approaches has a wide range of applications in the real world.. The performance of the method has been confirmed and measured.

Modeling and analysis of a fractional anthroponotic cutaneous leishmania model with Atangana-Baleanu derivative

Very recently, Atangana and Baleanu defined a novel arbitrary order derivative having a kernel of non-locality and non-singularity, known as AB derivative. We analyze a non-integer order Anthroponotic Leshmania Cutaneous (ACL) problem exploiting this novel AB derivative. We derive equilibria of the model and compute its threshold quantity, i.e. the so-called reproduction number. Conditions for the local stability of the no-disease as well as the disease endemic states are derived in terms of the threshold quantity. The qualitative analysis for solution of the proposed problem have derived with the aid of the theory of fixed point. We use the predictor corrector numerical approach to solve the proposed fractional order model for approximate solution. We also provide, the numerical simulations for each of the compartment of considered model at different fractional orders along with comparison with integer order to elaborate the importance of modern derivative. The fractional investigation shows that the non-integer order derivative is more realistic about the inner dynamics of the Leishmania model lying between integer order.

Truncation and convergence dynamics: KdV Burgers model in the sense of Caputo derivative

This study examines the time fractional KdV Burgers equation with the initial conditions by using the extended result on Caputo formula, finite difference method (FDM). For this reason, various fractional differential operators are defined and analyzed. In order to check the stability of the numerical scheme, the Fourier-von Neumann technique is used. By presenting an example of KdV Burgers equation above mentioned issues are discussed and numerical solutions of the error estimates have been found for the FDM. For the errors in $L_2$ and $L_\infty$ the method accuracy has been controlled. Moreover, the obtained results have been compared with the exact solution for different cases of non-integer order and the behavior of the potentials u is presented as a graph. The numerical results have been shown in tables.

Investigation of time-fractional SIQR Covid-19 mathematical model with fractal-fractional Mittage-Leffler kernel

In this manuscript, we investigate a nonlinear SIQR pandemic model to study the behavior of covid-19 infectious diseases. The susceptible, infected, quarantine and recovered classes with fractal fractional Atangana-Baleanu-Caputo (ABC) derivative is studied. The non-integer order ℘ and fractal dimension q in the proposed system lie between 0 and 1. The existence and uniqueness of the solution for the considered model are studied using fixed point theory, while Ulam-Hyers stability is applied to study the stability analysis of the proposed model. Further, the Adams-Bashforth numerical technique is applied to calculate an approximate solution of the model. It is observed that the analytical and numerical calculations for different fractional-order and fractal dimensions confirm better converging effects of the dynamics as compared to an integer order.

Fractal–fractional model and numerical scheme based on Newton polynomial for Q fever disease under Atangana–Baleanu derivative

Scientists and researchers are increasingly interested in mathematical modelling of infectious diseases with non-integer order. It is self-evident that a fixed order can only characterize classical models in epidemiology, but models with fractional-order derivatives are not fixed order. When a derivative has a non-fixed order, it becomes more potent in representing real-world problems. Different fresh techniques concerning fractional operators, such as the exponential decay and the Mittag-Leffler kernel, have been proposed in recent years, which transcend the constraints of prior fractional-order derivatives. These novel operators are functional in modelling problems in science and engineering—a more recent operator in fractional calculus, known as the fractal–fractional operator. Fractal–fractional operators have not been frequently utilized to explore complex dynamics of infectious disease spread in an ecosystem. In this paper, a novel fractal–fractional operator is used to study the dynamics of Q fever in livestock and ticks. Fixed point theorem is used to prove the model’s existence and uniqueness under the Atangana–Baleanu (Mittag-Leffler kernel) fractal–fractional operator. A non-linear analysis is used to determine the model’s Hyers–Ulam stability. A numerical scheme for the Q fever disease using the newly constructed numerical scheme based on Newton polynomial is presented. The numerical simulation of the model shows some exciting dynamics of the disease in an ecosystem. The fractal–fractional dynamism shows that a change in the geometric pattern of nature also affects the spread of Q fever. For example, it is noticed that the fractal dimensions and fractional order produces different asymptotic stabilities for symptomatic livestock, Coxiella burnetii in the environment, susceptible and infected ticks. It is of a view that this study will also contribute to the rich dynamics of the fractal and the fractional perspective of disease spread in nature.

The Differences Between Equivalent Foster and Cauer Circuits and Factorised Impedance

Approximations of non-integer order elements as Warburg impedance or constant phase element (CPE) are frequently made with equivalent electrical circuits. We can show that 5 various models are impedances mathematically equivalent for their frequency behavior. However, if we look closer at these models according to their various properties: model available in all cases, number of optimum, transient simulation, integration time, it seems that only one of the 5 models have all the qualities. This result is fundamental for the good choice of a model used for transient identification or for on-line identification.

Noninteger Derivative Order Analysis on Plane Wave Reflection from Electro-Magneto-Thermo-Microstretch Medium with a Gravity Field within the Three-Phase Lag Model

The present research paper illustrates how noninteger derivative order analysis affects the reflection of partial thermal expansion waves under the generalized theory of plane harmonic wave reflection from a semivacuum elastic solid material with both gravity and magnetic field in the three-phase lag model (3PHL). The main goal for this study is investigating the fractional order impact and the applications related to the orders, especially in biology, medicine, and bioinformatics, besides the integer order considering an external effect, such as electromagnetic, gravity, and phase lags in a microstretch medium. The problem fractional form was formulated, and the boundary conditions were applied. The results were displayed graphically, considering the 3PHL model with magnetic field, gravity, and relaxation time. These findings were an explicit comparison of the effect of the plane wave reflection amplitude with integer derivative order analysis and noninteger derivative order analysis. The fractional order was compared to the correspondence integer order that indicated to the difference between them and agreement with the applications in biology, medicine, and other related topics. This phenomenon has more applications in relation to the biology and biomathematics problems.

Non-integer Order Control Scheme for Pressurized Water Reactor Core Power

Tracking load changes in a pressurized water reactor (PWR) with the help of an efficient core power control scheme in a nuclear power station is very important. The reason is that it is challenging to maintain a stable core power according to the reference value within an acceptable tolerance for the safety of PWR. To overcome the uncertainties, a non-integer-based fractional order control method is demonstrated to control the core power of PWR. The available dynamic model of the reactor core is used in this analysis. Core power is controlled using a modified state feedback approach with a non-integer integral scheme through two different approximations, CRONE (Commande Robuste d'Ordre Non Entier, meaning Non-integer order Robust Control) and FOMCON (non-integer order modeling and control). Simulation results are produced using MATLAB® program. Both non-integer results are compared with an integer order PI (Proportional Integral) algorithm to justify the effectiveness of the proposed scheme. Sate-space model Core power control Non-integer control Pressurized water reactor PI controller CRONE FOMCON.

Analysis of food chain mathematical model under fractal fractional Caputo derivative.

In this article, the dynamical behavior of a complex food chain model under a fractal fractional Caputo (FFC) derivative is investigated. The dynamical population of the proposed model is categorized as prey populations, intermediate predators, and top predators. The top predators are subdivided into mature predators and immature predators. Using fixed point theory, we calculate the existence, uniqueness, and stability of the solution. We examined the possibility of obtaining new dynamical results with fractal-fractional derivatives in the Caputo sense and present the results for several non-integer orders. The fractional Adams-Bashforth iterative technique is used for an approximate solution of the proposed model. It is observed that the effects of the applied scheme are more valuable and can be implemented to study the dynamical behavior of many nonlinear mathematical models with a variety of fractional orders and fractal dimensions.

Electro‐Magnetohydrodynamic Fractional‐Order Fluid Flow with New Similarity Transformations

The fractional‐order differential equations that exist in the field of science and engineering have been studied in this paper. The 2D fluid flow problems are recommended for the classical‐ and fractional‐order analysis. It has been found that the nonlinear fractional‐order problems are more realistic than the classical models to describe the proposed flow problems since the behavior of the stress of the model problem is not linear. The similarity variable in this study has been used in fractional form to transform the modeled governing equations from partial PDEs. The acquired equations are the nonlinear ordinary differential equations (ODEs) in fractional form. The electromagnetic field has also been imposed into the fluid motion to calculate the embedded constraints in the case of classical and noninteger orders. The nonlinear problems are attempted using the FDE‐12 technique. The important physical phenomena including Nusselt number and skin friction are also calculated in case of the integer‐ and noninteger‐order problems. The electric and magnetic fields are examined and discussed in the case of fractional form and classical form.

Detection of Chaos Using Reservoir Computing Approach

Detection of chaos in time series is of utmost importance in many scientific fields. Indeed, the presence of chaos and its significance, especially in multidimensional systems, plays an essential role in the control and analysis of such systems, and in their practical use in a variety of applications. In this paper, we demonstrate a new methodology for the detection of chaos in time series using a reservoir computing (RC) paradigm called conceptor-driven network (ConDN). Case studies on the known chaotic attractors (i.e. Lorenz, Rossler, Chua) of integer (conventional) and non-integer (fractional-order) orders, as well as a physically simulated and designed spintronic device (NCVO) are used in this study to validate the proposed chaos detection approach. The proposed chaos detection approach is tested on clean and noisy time series of the mentioned attractors. It outperforms the 0-1 chaos detection test and the largest Lyapunov exponent (LLE) estimation approach especially in the high noise-level conditions. In addition, the proposed approach is capable of differentiating the time series generated by the systems whose dynamics is at the edge of chaos. The simplicity of use of the proposed chaos detection approach can be counted, as well, as one of its main advantages over traditional chaos detection methods.

FRACTIONAL ORDER LOG BARRIER INTERIOR POINT ALGORITHM FOR POLYNOMIAL REGRESSION IN THE ℓ p -NORM

Fractional calculus is the branch of mathematics that studies the several possibilities of generalizing the derivative and integral of a function to noninteger order. Recent studies found in literature have confirmed the importance of fractional calculus for minimization problems. However, the study of fractional calculus in interior point methods for solving optimization problems is still new. In this study, inspired in applications of fractional calculus in many fields, was developed the so-called fractional order log barrier interior point algorithm by replacing some integer derivatives for the corresponding fractional ones on the first order optimality conditions of Karush-Kuhn-Tucker to solve polynomial regression models in the ℓ p −norm for 1 < p < 2. Finally, numerical experiments are performed to illustrate the proposed algorithm.

Solution of fractional boundary value problems by $ \psi $-shifted operational matrices

&lt;abstract&gt;&lt;p&gt;In this paper, a numerical method is presented to solve fractional boundary value problems. In fractional calculus, the modelling of natural phenomenons is best described by fractional differential equations. So, it is important to formulate efficient and accurate numerical techniques to solve fractional differential equations. In this article, first, we introduce $ \psi $-shifted Chebyshev polynomials then project these polynomials to formulate $ \psi $-shifted Chebyshev operational matrices. Finally, these operational matrices are used for the solution of fractional boundary value problems. The convergence is analysed. It is observed that solution of non-integer order differential equation converges to corresponding solution of integer order differential equation. Finally, the efficiency and applicability of method is tested by comparison of the method with some other existing methods.&lt;/p&gt;&lt;/abstract&gt;

A hybrid analytical technique for solving nonlinear fractional order PDEs of power law kernel: Application to KdV and Fornberg-Witham equations

&lt;abstract&gt;&lt;p&gt;It is important to deal with the exact solution of nonlinear PDEs of non-integer orders. Integral transforms play a vital role in solving differential equations of integer and fractional orders. To obtain analytical solutions to integer and fractional-order DEs, a few transforms, such as Laplace transforms, Sumudu transforms, and Elzaki transforms, have been widely used by researchers. We propose the Yang transform homotopy perturbation (YTHP) technique in this paper. We present the relation of Yang transform (YT) with the Laplace transform. We find a formula for the YT of fractional derivative in Caputo sense. We deduce a procedure for computing the solution of fractional-order nonlinear PDEs involving the power-law kernel. We show the convergence and error estimate of the suggested method. We give some examples to illustrate the novel method. We provide a comparison between the approximate solution and exact solution through tables and graphs.&lt;/p&gt;&lt;/abstract&gt;