Caputo Differential Operator Research Articles

Numerical Investigation of Chemical Schnakenberg Mathematical Model

Schnakenberg model is known as one of the influential model used in several biological processes. The proposed model is an autocatalytic reaction in nature that arises in various biological models. In such kind of reactions, the rate of reaction speeds up as the reaction proceeds. It is because when a product itself acts as a catalyst. In fact, model endows fractional derivatives that got great advancement in the investigation of mathematical modeling with memory effect. Therefore, in the present paper, the authors develop a scheme for the solution of fractional order Schnakenberg model. The proposed model describes an auto chemical reaction with possible oscillatory behavior which may have several applications in biological and biochemical processes. In this work, the authors generalized the concept of integer order Schnakenberg model to fractional order Schnakenberg model. We provided the approximate solution for the underlying generalized nonlinear Schnakenberg model in the sense of Caputo differential operator via Laplace Adomian decomposition method (LADM). Furthermore, we established the general scheme for the considered model in the form of infinite series by the aforementioned technique. The consequent results obtained by the proposed technique ensure that LADM is an effective and accurate techniques to handle nonlinear partial differential equations as compared to the other available numerical techniques. Finally, the obtained numerical solution is visualized graphically by MATLAB to describe the dynamics of desired solution.

Chaotic Behavior Analysis of a New Incommensurate Fractional-Order Hopfield Neural Network System

This study intends to examine different dynamics of the chaotic incommensurate fractional-order Hopfield neural network model. The stability of the proposed incommensurate-order model is analyzed numerically by continuously varying the values of the fractional-order derivative and the values of the system parameters. It turned out that the formulated system using the Caputo differential operator exhibits many rich complex dynamics, including symmetry, bistability, and coexisting chaotic attractors. On the other hand, it has been detected that by adapting the corresponding controlled constants, such systems possess the so-called offset boosting of three variables. Besides, the resultant periodic and chaotic attractors can be scattered in several forms, including 1D line, 2D lattice, and 3D grid, and even in an arbitrary location of the phase space. Several numerical simulations are implemented, and the obtained findings are illustrated through constructing bifurcation diagrams, computing Lyapunov exponents, calculating Lyapunov dimensions, and sketching the phase portraits in 2D and 3D projections.

Analysis of fractional‐order nonlinear dynamic systems under Caputo differential operator

The current study presents a detailed analysis of two crucial real‐world problems under the Caputo fractional derivative in order to deliver some desired results for the ecosystem. In view of the fact that memory effect plays a vital role in the application, we utilize an advantageous non‐local fractional operator to investigate and analyze a mathematical model of the planktonic ecosystem and biological system for the ecosystem on Planet GLIA‐2. On the other hand, theoretical and numerical results are given for the model created for phytoplankton, which is of great importance in preventing global warming, and the biological model. Existence and uniqueness are discussed for the solutions of both models with the help of the fixed‐point theorem under the Caputo operator. Additionally, the first‐order convergent numerical technique which is accurate, conditionally stable, and convergent in obtaining the solution of fractional‐order nonlinear systems of ordinary differential equations is utilized to simulate the two governing models. Numerical simulations including different values of arbitrary orderρindicate the righteousness of the conditions for phytoplankton, Jancor, Murrot, and Vekton populations to develop.

Reproducing kernel approach for numerical solutions of fuzzy fractional initial value problems under the Mittag–Leffler kernel differential operator

In this research study, fuzzy fractional differential equations in presence of the Atangana–Baleanu–Caputo differential operators are analytically and numerically treated using extended reproducing kernel Hilbert space technique. With the utilization of a fuzzy strongly generalized differentiability form, a new fuzzy characterization theorem beside two fuzzy fractional solutions is constructed and computed. To besetment the attitude of fuzzy fractional numerical solutions, analysis of convergence and conduct of error beyond the reproducing kernel theory are explored and debated. In this tendency, three computational algorithms and modern trends in terms of analytic and numerical solutions are propagated. Meanwhile, the dynamical characteristics and mechanical features of these fuzzy fractional solutions are demonstrated and studied during two applications via three‐dimensional graphs and tabulated numerical values. In the end, highlights and future suggested research work are eluded.

Modeling of measles epidemic with optimized fractional order under Caputo differential operator

Abstract Memory is an important characteristic of an epidemic. One of such memory dependent and highly contagious viral diseases is measles that is also responsible for more than 140,000 deaths in 2018 in various regions of Asia and Africa. In order to better understand the transmission dynamics of measles, we have developed a new epidemiological model while considering both integer and fractional order operators and presented comparison. The Caputo fractional model has a unique solution with the positively invariant region. On the basis of basic reproduction number R 0 , stability analysis is discussed and sensitivity of parameters is investigated using PRCC global technique. Not only parameters but fractional order χ is also optimized via nonlinear least-squares approach with availability of statistical data obtained from WHO. Various simulations in terms of time series plots, 3D meshes and contours are carried out to observe effects of parameters on dynamics of the epidemic wherein it is said to be persistent for χ → 0 demonstrating the role being played by Caputo fractional derivative towards measles dynamics.

Periodic dynamics of rubella epidemic under standard and fractional Caputo operator with real data from Pakistan

Memory effects of epidemics play a vital role in mathematical models of infectious diseases. In this research study, an epidemiological SEIR (Susceptible, Exposed, Infectious, Removed) type model for the rubella epidemic has been proposed via classical and fractional order Caputo differential operators while assuming the periodic transmission rate β(t). The Caputo model has been investigated for the existence and uniqueness of its solutions via fixed point theory while the unique non-negative solution remains bounded within the biologically feasible region. Later, the non-fixed biological parameters of the classical and the Caputo model are obtained via nonlinear least squares fitting technique taking the real monthly cases for the rubella epidemic in Pakistan for the period 2010–2012. The performance rate of the Caputo model is 35% higher than that of the model with integer order derivative. The numerical simulations are obtained under different cases and it is highly suggested that the infectious rate σ must be controlled as much as possible to eradicate the rubella epidemic.

New explicit and accelerated techniques for solving fractional order differential equations

This paper proposes new direct and acceleration numerical methods for solving Fractional order Differential Equations (FDEs). For the Caputo differential operator with fractional order 0 < ν < 1, we rewrite the FDE as an equivalent integral form circumventing the derivative of the solution by the integral by parts. We obtain a discrete formulation directly from the integral form using the Lagrange interpolate polynomials. We provide truncation errors depending on the fractional order ν for linear and quadratic interpolations, which are O(h2−ν) and O(h3−ν) respectively. In order to overcome a strong singular issue as ν ≈ 0 we propose the explicit predictor-corrector scheme with perturbation technique. In case of ν ≈ 1, the truncation errors are reduced by O(h) and O(h2). In order to accelerate the convergence rate, we decompose ν=ν/2+ν/2, convert the FDE into the system of FDEs, and apply the predictor-corrector scheme. Numerical tests for linear, nonlinear, variable order and time-fractional sub-diffusion problems demonstrate that the proposed methods give a prominent performance. We also compare the numerical results with other high-order explicit and implicit schema.

A comparative study of convective fluid motion in rotating cavity via Atangana–Baleanu and Caputo–Fabrizio fractal–fractional differentiations

In this paper, a fractal–fractional mathematical model of convective fluid motion in rotating cavity is investigated inside the ellipsoid with inhomogeneous external heating. The fractal–fractional differential operators namely Caputo, Caputo–Fabrizio and Atangana–Baleanu $${\mathcal{D}}_{\tau }^{{\epsilon }_{1},{\tau }_{1}}$$, $${\mathcal{D}}_{\tau }^{{\epsilon }_{2},{\tau }_{2}}$$ and $${\mathcal{D}}_{\tau }^{{\epsilon }_{3},{\tau }_{3}}$$, respectively, are used in the non-linear mathematical model of convective fluid motion in rotating cavity. The numerical algorithms have been generated in terms of newly presented fractal–fractional differential operators on the basis of Adams–Bashforth method to compute the approximate solutions explicitly. The equilibrium points and stability analysis of the fractal–fractional Atangana–Baleanu, Caputo–Fabrizio and Caputo differential operators in Caputo sense have been investigated for non-linear mathematical model of convective fluid motion in rotating cavity. The numerical solutions are simulated in three types of variations (i) presence of fractional parameter without fractal parameter, (ii) presence of fractal parameter without fractional parameter, and (iii) presence of fractal parameter as well as fractional parameter. The chaotic behavior of convective fluid motion in rotating cavity based on each fractal–fractional differential operator has been highlighted as (a) projection on the x–y plane, (b) projection on the x–z plane, (c) projection on the y–z plane and (d) projection on the $$xyz$$ plane in three dimensions.

Solvability for boundary value problems of nonlinear fractional differential equations with mixed perturbations of the second type

In this paper, we consider the solvability for boundary value problems of nonlinear fractional differential equations with mixed perturbations of the second type. The expression of the solution for the boundary value problem of nonlinear fractional differential equations with mixed perturbations of the second type is discussed based on the definition and the property of the Caputo differential operators. By the fixed point theorem in Banach algebra due to Dhage, an existence theorem for the boundary value problem of nonlinear fractional differential equations with mixed perturbations of the second type is given under mixed Lipschitz and Caratheodory conditions. As an application, an example is presented to illustrate the main results. Our results in this paper extend and improve some well-known results. To some extent, our work fills the gap on some basic theory for the boundary value problems of fractional differential equations with mixed perturbations of the second type involving Caputo differential operator.

Deterministic modeling of dysentery diarrhea epidemic under fractional Caputo differential operator via real statistical analysis

Non-Markovian characteristics, possessing memory effects and hereditary properties, play a vital role when it comes to the transmission dynamics of a disease or an epidemic within human society over the course of time. As a result, non-local operators from the field of fractional calculus are the most suitable choices to comprehend dynamics of the disease transmission. This research study is related with formulation of dysentery diarrhea dynamical nonlinear autonomous model via fractional Caputo differential operator with order τ ∈ (0, 1). Using fixed point theory, its solutions are determined to have properties satisfying existence and uniqueness conditions under the Caputo operator. In addition, the non-negative hyperoctant R+4 is positively invariant region of the proposed fractional dysentery diarrhea model. Biological parameters of the classical and the Caputo fractional model are estimated under nonlinear parameter estimation technique and the optimized value of the fractional order parameter τ in the Caputo dysentery diarrhea model is computed to be 9.995e-01. The sensitivity of the basic reproduction number R0 is thoroughly investigated for the most and the least sensitive parameter. Numerical simulations are found to be in good agreement with real statistical data and the theoretical predictions about the disease.

Mathematical modeling for the impacts of deforestation on wildlife species using Caputo differential operator

Abstract This research study shows the impacts of deforestation on wildlife species using a newly proposed four dimensional nonlinear mathematical model based upon fractional-order ordinary differential equations. Being a nonlinear model, some theorems using fixed point theory have been proved showing the existence and uniqueness properties for the solution of the fractional-order model. Using an explicit version of Adams–Bashforth–Moulton method devised for the fractional-order ordinary differential equations with convergence order p = min ( 1 + τ , 2 ) , where τ is the order of the differential equations used in the model; some numerical simulations in the form of graphical illustrations have been carried out depicting the better performance of the fractional-order model for being capable enough to capture all the history information of the system under consideration which is a phenomenon not found in the classical (integer-order) differential equations. Varying values for both the fractional-order parameter τ and parameters of the model itself are used during the required numerical simulations.

Analysis of 3D IS-LM macroeconomic system model within the scope of fractional calculus

A mathematical model providing an asymptotic description of macro-economic system is considered in this work. The system in general deals with performance, behaviour, decision-making of an economy as a whole and also the structure. Due to the complexities of this system, a more complex mathematical model is requested. In this work, we considered the extension of the model using some non-local differential operators and the stochastic approach where the given parameters are converted to normal distributions. We have presented the conditions of existence of uniquely exact solutions of the system using the fixed-point theorem approach. Each model is solved numerical via a newly introduced modified Adams-Bashforth for fractional differential equations. We presented numerical simulations for different values of fractional order. The models with the Atangana-Baleanu and Caputo differential operators provided us with new attractors.

Generalized Spline Method for Integro-Differential Equations of Fractional Order

In This paper generalized spline method and Caputo differential operator is applied to solve linear fractional integro-differential equations of the second kind. Comparison of the applied method with exact solutions reveals that the method is tremendously effective.

The existence of a positive solution for nonlinear fractional differential equations with integral boundary value conditions

In this paper, first, we consider the existence of a positive solution for the nonlinear fractional differential equation boundary value problem urn:x-wiley:mma:media:mma4105:mma4105-math-0002 where 0≤λ &lt; 1,CDα is the Caputo's differential operator of order α, and f:[0,1] × [0,∞)→[0,∞) is a continuous function. Using some cone theoretic techniques, we deduce a general existence theorem for this problem. Then, we consider two following more general problems for arbitrary α, 1≤n &lt; α≤n + 1:Problem 1: urn:x-wiley:mma:media:mma4105:mma4105-math-0003 where , 0≤λ &lt; k + 1;Problem 2: urn:x-wiley:mma:media:mma4105:mma4105-math-0004 where 0≤λ≤α and Dα is the Riemann–Liouville fractional derivative of order α.For these problems, we give existence results, which improve recent results in the literature. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Existence and uniqueness of global solutions of caputo-type fractional differential equations

Abstract In this paper we consider initial value problems for fractional differential equations involving Caputo differential operators. By establishing a new property of the Mittag-Leffler function and using the Schauder fixed point theorem, we obtain new sufficient conditions for the existence and uniqueness of global solutions of initial value problems.

Existence of the Solution for System of Coupled Hybrid Differential Equations with Fractional Order and Nonlocal Conditions

This paper is motivated by some papers treating the fractional hybrid differential equations with nonlocal conditions and the system of coupled hybrid fractional differential equations; an existence theorem for fractional hybrid differential equations involving Caputo differential operators of order1&lt;α≤2is proved under mixed Lipschitz and Carathéodory conditions. The existence and uniqueness result is elaborated for the system of coupled hybrid fractional differential equations.

A Kamenev-type oscillation result for a linear [formula omitted]-order fractional differential equation

We investigate the eventual sign changing for the solutions of the linear equation x(α)′+q(t)x=0,t⩾0, when the functional coefficient q satisfies the Kamenev-type restriction limsupt→+∞1tε∫t0t(t-s)εq(s)ds=+∞ for some ε>2,t0>0. The operator x(α) is the Caputo differential operator and α∈(0,1).

The Fractional Differential Polynomial Neural Network for Approximation of Functions

In this work, we introduce a generalization of the differential polynomial neural network utilizing fractional calculus. Fractional calculus is taken in the sense of the Caputo differential operator. It approximates a multi-parametric function with particular polynomials characterizing its functional output as a generalization of input patterns. This method can be employed on data to describe modelling of complex systems. Furthermore, the total information is calculated by using the fractional Poisson process.

On a fractional differential equation with infinitely many solutions

We present a set of restrictions on the fractional differential equation , , where and , that leads to the existence of an infinity of solutions (a continuum of solutions) starting from . The operator is the Caputo differential operator.