Caputo Sense Research Articles

ANALYSIS OF A NONLINEAR DYNAMICAL MODEL OF HEPATITIS B DISEASE

Mathematical epidemiology holds prime importance for comprehending the dynamics of infectious diseases. Consequently, mathematical model of hepatitis B with fractional-order derivative under Caputo sense is primarily focused in this research. The analysis of the required solution is qualitatively derived by applying the fixed-point theory approach. By perturbing the proposed model, the Ulam–Hyer’s stability techniques are further derived. To achieve the iterative series solution of the proposed system of hepatitis, the modified Euler method like Taylor’s series method is utilized. For validation and importance of the fractional operators, sufficient significant numerical results at various fractional orders are presented and compared them with the integer order. It is inferred from this research that, by using the fractional-order method, the transmission mechanism of hepatitis B disease can be acutely revealed. This study may provide positive theoretical support for the prevention and treatment of hepatitis B disease.

Modification of the Optimal Auxiliary Function Method for Solving Fractional Order KdV Equations

In this study, a new modification of the newly developed semi-analytical method, optimal auxiliary function method (OAFM) is used for fractional-order KdVs equations. This method is called the fractional optimal auxiliary function method (FOAFM). The time fractional derivatives are treated with Caputo sense. A rapidly convergent series solution is obtained from the FOAFM and is validated by comparing with other results. The analysis proves that our method is simplified and applicable, contains less computational work, and has fast convergence. The beauty of this method is that there is no need to assume a small parameter such as in the perturbation method. The effectiveness and accuracy of the method is proven by numerical and graphical results.

Numerical simulation of time-dependent influenza model with Atangana–Baleanu non-integer order derivative in Liouville–Caputo sense

Numerical simulation of time-dependent influenza model with Atangana–Baleanu non-integer order derivative in Liouville–Caputo sense

Enhancing the Accuracy of Solving Riccati Fractional Differential Equations

In this paper, we solve Riccati equations by using the fractional-order hybrid function of block-pulse functions and Bernoulli polynomials (FOHBPB), obtained by replacing x with xα, with positive α. Fractional derivatives are in the Caputo sense. With the help of incomplete beta functions, we are able to build exactly the Riemann–Liouville fractional integral operator associated with FOHBPB. This operator, together with the Newton–Cotes collocation method, allows the reduction of fractional differential equations to a system of algebraic equations, which can be solved by Newton’s iterative method. The simplicity of the method recommends it for applications in engineering and nature. The accuracy of this method is illustrated by five examples, and there are situations in which we obtain accuracy eleven orders of magnitude higher than if α=1.

Pell Collocation Pseudo Spectral Scheme for One-Dimensional Time-Fractional Convection Equation with Error Analysis

In the current manuscript, we implement the collocation method to obtain an approximate solution of one-dimensional time-fractional convection equation. The operational matrices of Pell polynomials are applied to solve the fractional partial differential equations. In the Caputo sense, we describe the time-fractional derivatives. So, this algorithm allows us to transform the fractional differential equation with initial (boundary) conditions into a system of algebraic equations in a good form. The convergence and error analysis of Pell polynomials are carefully investigated, and some examples are explored to show the comparison and efficiency of our method with others.

Accurate and efficient matrix techniques for solving the fractional Lotka–Volterra population model

This paper aims to develop accurate novel collocation techniques for solving a fractional fractional-order Lotka–Volterra (LV) predator–prey model. This type of equation has a wide variety of applications in biology to simulate the interaction between two different species. The fractional-order operator is defined in the Caputo sense. We introduce some new polynomials for solving this type of equation named the Morgan-Voyce polynomials. These new techniques named direct and quasilinear Morgan-Voyce techniques are presented and then used to solve the LV system revealing some of the new features of the model. Some of the advantages of the application of these techniques are that they are straightforward along with high accuracy. Detailed error and convergence analysis for the proposed methods are provided revealing the upper bound for these techniques. To test the accuracy of the proposed methods, the methods are tested on several examples with different values of the parameters and fractional orders and also compared with each other and with other related methods from the literature. The results are demonstrated through tables and figures and conclude that the provided technique is highly accurate compared to the other methods. The results revealed the agreement between the obtained approximate solution and the dynamics of the described model. These methods can be considered as promising to adapt to other similar models with application to different areas of engineering and biology.

Asymptotic profile for a two-terms time fractional diffusion problem

We consider the Cauchy-type problem associated to the time fractional partial differential equation: ∂tu+∂tβu-Δu=g(t,x),t>0,x∈Rnu(0,x)=u0(x),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} \\partial _t u+\\partial _t^{\\beta }u-\\varDelta u=g(t,x), &{} t>0, \\ x\\in {\\mathbb {R}}^n \\\\ u(0,x)=u_0(x), \\end{array}\\right. } \\end{aligned}$$\\end{document}with beta in (0,1), where the fractional derivative partial _t^{beta } is in Caputo sense. We provide a sufficient condition on the right-hand term g(t, x) to obtain a solution in {mathcal {C}}_b([0,infty ),H^s). We exploit a dissipative-smoothing effect which allows to describe the asymptotic profile of the solution in low space dimension.

Homotopy analysis approach to study the dynamics of fractional deterministic Lotka-Volterra model

The population dynamics of two species that is governed by the deterministic Lotka-Volterra model concerned with the interaction of predator and prey is investigated in this article as an application of homotopy analysis method. The analytical approximate solution in the form of convergent infinite series is obtained by considering the time-fractional derivatives in the Caputo sense. The simulation of obtained results exhibit the effect of variation in fractional parameters, auxiliary parameters and auxiliary linear operators on the mass concentration of both the biological species which in turn affects the structure of the system.

Dynamical Analysis of Nutrient-Phytoplankton-Zooplankton Model with Viral Disease in Phytoplankton Species under Atangana-Baleanu-Caputo Derivative

A mathematical model of the nutrient-phytoplankton-zooplankton associated with viral infection in phytoplankton under the Atangana-Baleanu derivative in Caputo sense is investigated in this study. We prove the theoretical results for the existence and uniqueness of the solutions by using Banach’s and Sadovskii’s fixed point theorems. The notion of various Ulam’s stability is used to guarantee the context of the stability analysis. Furthermore, the equilibrium points and the basic reproduction numbers for the proposed model are provided. The Adams type predictor-corrector algorithm has been applied for the theoretical confirmation to establish the approximate solutions. A variety of numerical plots corresponding to various fractional orders between zero and one are presented to describe the dynamical behavior of the fractional model under consideration.

Numerical simulation of variable-order fractional differential equation of nonlinear Lane–Emden type appearing in astrophysics

Abstract This paper suggests the Chebyshev pseudo-spectral approach to solve the variable-order fractional Lane–Emden differential equations (VOFLEDE). The variable-order fractional derivative (VOFD) is defined in the Caputo sense. The proposed method transforms the problem into a set of algebraic equations that can be solved for unknowns. Few examples are discussed to exhibit the viability and effectiveness of the approach. The present study indicates the accuracy, efficiency, and powerfulness of the Chebyshev collocation method in solving the VOFD Lane–Emden equation. Error bound and convergence analysis of the method is also discussed. It is worth noticing that using lesser collocation nodes in computation is another advantage of the technique, which eventually reduces the computational cost.

Bifurcations, stability analysis and complex dynamics of Caputo fractal-fractional cancer model

The association of cancer and immune cells has complex nature and produces chaotic behavior when it is simulated. The newly introduced operators which combine the fractal and fractional operators produce excellent and profound hidden attractors in a chaotic system which is sometimes not possible to get hidden attractors using integer order operators. The cancer model is considered under fractal fractional operator in Caputo sense. Linear stability of different equilibrium points is analyzed. The primary objective of the current paper is to analyze different bifurcations like pitch-fork, quasi, and inverse period-doubling bifurcations. Another important objective of this article is to study hidden limit cycle type chaotic structures of the cancer model via Caputo fractal-fractional operator. The existence and uniqueness of the solution and Ulam–Hyres (UH) stability are studied through the concepts of nonlinear analysis. The numerical solution is derived through the predictor-corrector method. The obtained results were presented and validated through numerical simulations. The lyapunov spectra of the state variables are presented through graphical illustration and table. Sensitivity of the state variables to the initial conditions are simulated for initial conditions 0.1 and 0.11. For various values of fractal dimensions and fractional orders, the time series oscillations and hidden limit cycles type chaotic attractors are graphically presented through MATLAB-17.

Local existence and nonexistence for fractional in time reaction–diffusion equations and systems with rapidly growing nonlinear terms

We study the fractional in time reaction–diffusion equation ∂tαu=Δu+f(u)inRN×(0,T),u(x,0)=u0(x)inRN,where 0<α<1, N≥1, T>0 and u0≥0. The fractional derivative ∂tα is meant in a generalized Caputo sense. We mainly consider the case where f has an exponential or a superexponential growth, and u0 has a singularity. We obtain integrability conditions on u0 which explicitly determine local in time existence/nonexistence of a nonnegative mild solution. Moreover, our analysis can be applied to time fractional systems.

Fractional-Order Chelyshkov Collocation Method for Solving Systems of Fractional Differential Equations

This paper presents an efficient method for solving systems of fractional differential equations by using the fractional-order Chelyshkov functions (FCHFs). The fractional derivative and the fractional integral are considered in the Caputo sense and the Riemann–Liouville sense, respectively. The proposed method is based on using the operational matrix of fractional integration for FCHFs, together with the spectral collocation method to transform the system of fractional differential equations into a system of algebraic equations. The convergence of the presented method is demonstrated. The performance of the presented method is tested through various examples of systems of fractional differential equations. A comparison with some existing methods shows that the presented method can be successfully used to solve systems of fractional differential equations with accuracy and efficiency.

Full adaptive Kalman filters for nonlinear fractional-order systems containing unknown parameters and fractional-orders

In this study, full adaptive Kalman filters are designed for continuous-time nonlinear fractional-order systems (FOSs) containing unknown parameters and fractional-orders. First, the estimated FOS is discretized using the Grünwald–Letnikov difference method to transform the fractional-order differential equation into a difference equation. Then, in terms of the nonlinear function contained in the investigated system, the Taylor expansion formula is adopted to linearize the discretized equation. Based on the method of augmented vector, an augmented state equation is established by state equations, unknown parameter equations, and fractional-order equation to achieve the state estimation. Besides, the sigmoid function is brought to ensure that the estimation of the fractional-order is performed in a suitable range. Because the covariance matrices of noises are difficult to be measured in physical systems, we also concern the problems on the state estimation, parameter estimation, and fractional-order estimation under the cases that the covariance matrix of process noise is unknown or the covariance matrix of measurement noise is unknown. Considering that the initial value can produce an error of state estimation and parameter identification for the FOS defined under the Caputo sense, the augmented vector method is used to achieve the initial value compensation. Finally, the effectiveness of the proposed algorithms is validated by four examples.

New Fractional Derivative Expression of the Shifted Third-Kind Chebyshev Polynomials: Application to a Type of Nonlinear Fractional Pantograph Differential Equations

The main goal of this paper is to develop a new formula of the fractional derivatives of the shifted Chebyshev polynomials of the third kind. This new formula expresses approximately the fractional derivatives of these polynomials in the Caputo sense in terms of their original ones. The linking coefficients are given in terms of a certain 4 F 3 1 terminating hypergeometric function. The integer derivatives of the shifted third-kind Chebyshev polynomials can be calculated using this formula after performing some reductions. To solve a nonlinear fractional pantograph differential equation with quadratic nonlinearity, the fractional derivative formula is used in conjunction with the tau technique. The role of the tau method is to convert the pantograph differential equation with its governing initial/boundary conditions into a nonlinear system of algebraic equations that can be treated with the aid of Newton’s iterative scheme. To test the method’s convergence, certain estimations are included. The proposed numerical method is demonstrated by numerical results to ensure its applicability and efficiency.

A study on the maize streak virus epidemic model by using optimized linearization-based predictor-corrector method in Caputo sense

In this article, we study the dynamics of a Maize streak virus (MSV) epidemic model by using the Caputo fractional derivative. Firstly, we define the dynamics of the given fractional-order model by checking the non-negativity and boundedness of the solution, stability of disease-free equilibrium, the existence of a unique solution, and its stability. Then we derive the numerical solution of the proposed model by using an optimized Predictor-Corrector method, which has not yet been applied to solve any kind of epidemic system until now. Our optimized method uses a linear approximation of the proposed nonlinear model to ameliorate the competence of the Predictor-Corrector schemes. To verify the correctness of our results, we plot various graphs by taking different parameter cases at various fractional-order values. Also, Caputo outputs are compared with Atangana-Baleanu-Caputo derivative outputs. Our research shows the usefulness of the proposed optimized Predictor-Corrector method in epidemic studies and for simulating the memory effects in the given MSV system. The solution methodology of the proposed system is the main novelty of this research along with other supporting analyses.

Application of the extended Chebyshev cardinal wavelets in solving fractional optimal control problems with ABC fractional derivative

In this study, two categories of fractional optimal control problems are investigated. One category is optimal control problems, which includes a fractional dynamic system, and the other category, in addition to the fractional dynamics system, includes inequality constraints. The Atangana–Baleanu fractional derivative in the Caputo sense is used to define these problems. The extended Chebyshev cardinal wavelets as an appropriate class of basis functions are introduced to construct two numerical methods for these problems. To solve these problems, at first, an operational matrix for the Atangana–Baleanu fractional integral of these wavelets is derived. Then, by approximating the fractional derivative of the state variables and control variables in terms of the extended Chebyshev cardinal wavelets, and employing the fractional integral operational matrix of these wavelets and the Lagrange multipliers method, the problems under consideration are converted into systems of algebraic equations, which can be easily solved. To examine the accuracy of the established methods, some numerical examples are solved.

On fractional order multiple integral transforms technique to handle three dimensional heat equation

In this article, we extend the notion of double Laplace transformation to triple and fourth order. We first develop theory for the extended Laplace transformations and then exploit it for analytical solution of fractional order partial differential equations (FOPDEs) in three dimensions. The fractional derivatives have been taken in the Caputo sense. As a particular example, we consider a fractional order three dimensional homogeneous heat equation and apply the extended notion for its analytical solution. We then perform numerical simulations to support and verify our analytical calculations. We use Fox-function theory to present the derived solution in compact form.

High-order numerical algorithms for the time-fractional convection–diffusion equation

In this paper, efficient methods are derived for seeking a numerical solution to the time-fractional convection–diffusion equation whose solution very likely exhibits a weak regularity at the starting time. Here, the time-fractional derivative in the Caputo sense with order in is discretized by the - methods with uniform and non-uniform meshes and the spatial derivative is approximated by the local discontinuous Galerkin (LDG) finite element methods. The fully discrete schemes for both situations are established and analyzed. It is shown that the derived schemes are numerically stable and convergent. Finally, some numerical examples are performed to testify the effectiveness of the obtained algorithm.

Numerical Solution of Linear Fractional Differential Equation with Delay Through Finite Difference Method

This article addresses a new numerical method to find a numerical solution of the linear delay differential equation of fractional order , the fractional derivatives described in the Caputo sense. The new approach is to approximating second and third derivatives. A backward finite difference method is used. Besides, the composite Trapezoidal rule is used in the Caputo definition to match the integral term. The accuracy and convergence of the prescribed technique are explained. The results are shown through numerical examples.