Caputo Sense Research Articles

A new Technique For Solving Fractional Nonlinear Equations By Sumudu Transform and Adomian Decomposition Method

A novel technique Sumudu transform Adomian decomposition method (STADM), is employed to handle some kinds of nonlinear time-fractional equations. We demonstrate that this method finds the solution without discretization or restrictive assumptions. This method is efficient, simple to implement, and produces good results. The fractional derivative is described in the Caputo sense. The solutions are obtained using STADM, and the results show that the suggested technique is valid and applicable and provides a more refined convergent series solution. The MATLAB software carried out all the computations and graphics. Moreover, a graphical representation was made for the solution of some examples. For integer and fractional order problems, solution graphs are shown. The results confirmed that the accuracy of this technique converges to the integer order of the issues.

Boundary state feedback control for semilinear fractional-order reaction diffusion systems

This paper investigates the stabilization results for a semilinear time fractional-order reaction diffusion partial differential equations using backstepping method. The parabolic system is addressed with time fractional-order of (0, 1) in the Caputo sense. More general type of actuation setup which can be expressed as Dirichlet, Neumann and Robin type boundary actuation's is considered. The main aim is to achieve the stabilization of the considered system using an invertible transformation through the stability of target system which is verified by Mittag-Leffler stability based on (Q, S, R) dissipativity theory and linear matrix inequality (LMI) technique. The explicit solutions of kernel functions are found by method of successive approximation and to be design the boundary control law for the closed loop system. Finally, the proposed results are validated through numerical examples.

A HIGH ORDER NUMERICAL SCHEME FOR FRACTAL-FRACTIONAL LASER SYSTEM WITH CHAOS STUDY

The primary goal of this research is to look into the dynamical behavior of a fractional and fractal-fractional (FF) order laser chaotic model. The model is first considered with Caputo’s fractional derivative. An iterative technique based on the Laplace transform and its inverse aspect is used to achieve a particular solution for the fractional model. The existence of a unique solution is examined for the fractional laser chaotic model. Next, the laser chaotic model is addressed with a FF approach in the Caputo sense. The approximate results are achieved for the proposed fractional and FF models by using the Atangana–Seda numerical methods. The dynamical structures are plotted at various values of fractional and FF parameters.

To study the transmission dynamic of SARS-CoV-2 using nonlinear saturated incidence rate

In this work, we construct a new SARS-CoV-2 mathematical model of SQIR type. The considered model has four compartments as susceptible S, quarantine Q, infected I and recovered R. Here saturated nonlinear incidence rate is used for the transmission of the disease. We formulate our model first and then the disease-free and endemic equilibrium (EE) are calculated. Further, the basic reproduction number is computed via the next generation matrix method. Also on using the idea of Dulac function, the global stability for the proposed model is discussed. By using the Routh–Hurwitz criteria, local stability is investigated. Through nonstandard finite difference (NSFD) scheme, numerical simulations are performed. Keeping in mind the significant importance of fractional calculus in recent time, the considered model is also investigated under fractional order derivative in Caputo sense. Finally, numerical interpretation of the model by using various fractional order derivatives are provided. For fractional order model, we utilize fractional order NSFD method. Comparison with some real data is also given.

Ritz approximate method for solving delay fractional optimal control problems

In this paper, a numerical method based on the Müntz–Legendre polynomials for solving a class of time-delay fractional optimal control problems (TDFOCPs) is presented. The concept of fractional derivatives is considered in the Caputo sense. The key point of the applied method is that the initial and boundary conditions are imposed upon the approximations of state and control functions. First, the unknown state or control and their delayed functions are approximated by the Müntz–Legendre polynomials basis using the Ritz method then the next one is calculated through the given fractional differential equation. By substituting the estimated values in the cost function, the TDFOCP is converted to an unconstrained linear or nonlinear optimization problem. The convergence of the suggested method is extensively argued and several illustrative examples are presented to show the applicability and effectiveness of the method. It is shown that only a few terms of Müntz–Legendre polynomials are needed for obtaining the high accuracy and satisfactory results. The obtained results are compared with the available results in the literature to demonstrate the superiority of the applied approach. Furthermore, the approximated solutions agree with exact solutions and as α approaches an integer value, the method provides solution for the integer-order optimal control problem.

Homotopi pertürbasyon Elzaki dönüşümü yöntemi ile doğrusal olmayan zaman-kesirli kısmi diferansiyel denklemler için yeni yaklaşık analitik çözümler

Some nonlinear time-fractional partial differential equations are solved by homotopy perturbation Elzaki transform method. The fractional derivatives are defined in the Caputo sense. The applications are examined by homotopy perturbation Elzaki transform method. Besides, the graphs of the solutions are plotted in the MAPLE software. Also, absolute error comparison of homotopy perturbation Elzaki transform method and homotopy perturbation Sumudu transform method solutions with the exact solution of nonlinear time-fractional partial differential equations is presented. In addition, this absolute error comparison is indicated in the tables. The novelty of this article is the first analysis of both the gas dynamics equation of Caputo fractional order and the Klein-Gordon equation of Caputo fractional order via this method. Thus, homotopy perturbation Elzaki transform method is quick and effective in obtaining the analytical solutions of time-fractional partial differential equations.

Numerical Investigation of the One-dimensional Fractional Rayleigh-Stokes Equation in Caputo Sense by Kansa RBF Method

Numerical Investigation of the One-dimensional Fractional Rayleigh-Stokes Equation in Caputo Sense by Kansa RBF Method

Fractional-View Analysis of Space-Time Fractional Fokker-Planck Equations within Caputo Operator

In this article, we investigate the fractional-order Fokker-Planck equations with the help of the Yang transform decomposition method (YTDM). The YTDM combines Yang transform, Adomian decomposition method, and Adomian polynomials into one method. In the Caputo sense, fractional derivatives of space and time are studied. The convergent series form solution demonstrates the method’s efficiency in resolving several types of fractional differential equations. Compared to other methods of finding approximate and exact solutions for nonlinear partial differential equations, this technique is more efficient and time-consuming.

A NOVEL ANALYTICAL APPROACH FOR SOLVING TIME-FRACTIONAL DIFFUSION EQUATIONS

This article investigates the approximate analytical solutions of the time-fractional diffusion equations using a novel analytical approach, namely the Sumudu transform iterative method. The time-fractional derivatives are considered in the Caputo sense. The analytical solutions are found in closed form, in terms of Mittag-Leffler functions. Furthermore, the findings are shown graphically, and the solution graphs demonstrate a strong relationship between the approximate and exact solutions.

Technique for Solving the Bagley-Torvik Equation via Integer-Order Differential Equations

This paper proposes an analytical technique for solving the Bagley-Torvik equation (BTE) in the Caputo sense. The main idea of this technique is based on reformulating the considered problem as a system of linear FDEs of half-order. Then the resulting system is transformed into a set of integer-order differential equations. In such a transformation, the singularity terms are removed from the FDE system. So, the solution of the BTE can be obtained via solving this system. Finally, two examples are given to demonstrate the efficiency of the proposed technique.

Modeling the impact of the vaccine on the COVID-19 epidemic transmission via fractional derivative.

To achieve the goal of ceasing the spread of COVID-19 entirely it is essential to understand the dynamical behavior of the proliferation of the virus at an intense level. Studying this disease simply based on experimental analysis is very time consuming and expensive. Mathematical modeling might play a worthy role in this regard. By incorporating the mathematical frameworks with the available disease data it will be beneficial and economical to understand the key factors involved in the spread of COVID-19. As there are many vaccines available globally at present, henceforth, by including the effect of vaccination into the model will also support to understand the visible influence of the vaccine on the spread of COVID-19 virus. There are several ways to mathematically formulate the effect of disease on the population like deterministic modeling, stochastic modeling or fractional order modeling etc. Fractional order derivative modeling is one of the fundamental methods to understand real-world problems and evaluate accurate situations. In this article, a fractional order epidemic model S_{p}E_{p}I_{p}Er_{p}R_{p}D_{p}Q_{p}V_p on the spread of COVID-19 is presented. S_{p}E_{p}I_{p}Er_{p}R_{p}D_{p}Q_{p}V_p consists of eight compartments of population namely susceptible, exposed, infective, recovered, the quarantine population, recovered-exposed, and dead population. The fractional order derivative is considered in the Caputo sense. For the prophecy and tenacity of the epidemic, we compute the reproduction number R_0. Using fixed point theory, the existence and uniqueness of the solutions of fractional order derivative have been studied. Furthermore, we are using the generalized Adams–Bashforth–Moulton method, to obtain the approximate solution of the fractional-order COVID-19 model. Finally, numerical results and illustrative graphic simulation are given. Our results suggest that to reduce the number of cases of COVID-19 we should reduce the contact rate of the people if the population is not fully vaccinated. However, to tackle the issue of reducing the social distancing and lock down, which have very negative impact on the economy as well as on the mental health of the people, it is much better to increase the vaccine rate and get the whole nation to be fully vaccinated.

INVESTIGATING FRACTAL-FRACTIONAL MATHEMATICAL MODEL OF TUBERCULOSIS (TB) UNDER FRACTAL-FRACTIONAL CAPUTO OPERATOR

This paper discussed a new operator known as a fractal-fractional (FF), considered in the Caputo sense. We have investigated the fractional mathematical model of Tuberculosis (TB) disease under FF Caputo derivative. We have provided the existence and uniqueness for the appropriate system by using Banach and Leray–Schauder theorem and for the stability of the model, we have used the Ulam–Hyers approach. Applying the methods of basic theorems of FF calculus (FFC) and the iterative numerical techniques of fractional Adams–Bashforth method for approximate solution. For the simulation of the model, we have considered different values for fractional order [Formula: see text] and fractal dimension [Formula: see text] and compared the results with integer order for real data. The FFC technique is applied as a beneficial technique to know about the real-world problem and also to control the whole world situation of the aforesaid pandemic in the different continents and territories of the world. This new operator, FF in the form of Caputo derivative, gives better results than ordinary integer order. Several results have been discussed by taking the different fractal dimensions and arbitrary order.

An efficient computational approach for the solution of time-space fractional diffusion equation

ABSTRACT The main aim of this paper is to construct an efficient recursive algorithm to solve a time-space fractional Poisson’s equation which can be treated as a time-space fractional diffusion equation in two dimensions. The fractional derivatives in both time and space are defined in the Caputo sense. A homotopy perturbation method is introduced to approximate the solution, and a comparison is made between the exact and the approximate solutions. In addition, we present a procedure for solving higher-order fractional Poisson’s equations. In this case, the equation is converted to a system of fractional differential equations in which the order of the time derivatives is less than or equal to one. The convergence analysis is carried out, and an apriori bound of the solution is obtained for the present problem. Numerical examples are provided and the experimental evidence proves the effectiveness of the proposed method.

On the numerical study of fractional and non-fractional model of nonlinear Duffing oscillator: a comparison of integer and non-integer order approaches

ABSTRACT This manuscript analyzes nonlinear Duffing oscillators via integer and non-integer order differential operators known as the Caputo, Caputo–Fabrizio, and the Atangana–Baleanu taken in the Caputo sense. Application of the fixed-point theory shows that the fractional-order nonlinear Duffing oscillator does have a unique solution subject to appropriate initial conditions. Furthermore, the fractional-order parameters α,β and γ∈]0,1] for the Caputo, Caputo–Fabrizio, and the Atangana–Baleanu operators, respectively, when kept at variation, revealed exciting changes in the amplitude and the frequency responses for the fractional displacement function of the nonlinear oscillator. Lastly, numerical simulations have been carried out and presented in graphical illustrations to observe the effects of each parametric value of the nonlinear Duffing oscillator under the classical, the Caputo, the Caputo–Fabrizio, and the Atangana–Baleanu fractional differential operators wherein the Markovian nature of the oscillator is retained when α=β=γ=1. Based upon the comparative mathematical analysis of the oscillator, it has been shown that the Atangana–Baleanu fractional-order operator possessing non-locality and non-singular convolution kernel has the better capability to retain the memory effects of the system.

Numerical Investigation of Nonlinear Shock Wave Equations with Fractional Order in Propagating Disturbance

The symmetry design of the system contains integer partial differential equations and fractional-order partial differential equations with fractional derivative. In this paper, we develop a scheme to examine fractional-order shock wave equations and wave equations occurring in the motion of gases in the Caputo sense. This scheme is formulated using the Mohand transform (MT) and the homotopy perturbation method (HPM), altogether called Mohand homotopy perturbation transform (MHPT). Our main finding in this paper is the handling of the recurrence relation that produces the series solutions after only a few iterations. This approach presents the approximate and precise solutions in the form of convergent results with certain countable elements, without any discretization or slight perturbation theory. The numerical findings and solution graphs attained using the MHPT confirm that this approach is significant and reliable.

Anomalous transport for multispecies reactive system with first order decay: time-fractional model

The prediction of pollutant migration and its concentration variation in the subsurface hydrology is vitally requisite for the assessment and restorative treatment of polluted groundwater system. Pollutant migration for the multispecies reactive system cannot be reliably investigated by classical form of convection-dispersion equation (CDE), due to the presence of more than one reactive species. This paper establishes a time-fractional model for multispecies reactive system under the first order sequential reaction network to understand the anomalous or non-Fickian migration phenomenon for reactive pollutants. At present, most of the fractional models are presented for the classical CDE to investigate migration phenomenon for single species system, not for the multispecies reactive system due to the complexity of the modelled framework. The impact of fractional derivative model is analysed for variable dependent migration parameters and constant parameters, both for the multispecies reactive migration phenomenon. The fractional derivative is expressed in the Caputo sense and to handle the non-linearity of problem, Homotopy perturbation method (HPM) is adopted. The advantage of this method, to get the solutions, is that the HPM is independent of small parameters required for the deformation process as used in other existing perturbation techniques, which make it much more convenient to use for non-linear systems. The impact of the fractional derivative index and other migration parameters is graphically depicted for the reactive species and significant influence of fractional term is observed. The derived solutions are then validated by using the corresponding solutions obtained by other existing well-established methods to ensure the convergence of the HPM solutions. As there do not exist any solutions for such fractional models for multispecies reactive system, this study may be advantageous to convey better understanding for the anomalous or non-Fickian migration for reactive pollutants and their remediation planning in the groundwater resources.

Application of fractional optimal control theory for the mitigating of novel coronavirus in Algeria

In this paper, we investigate the dynamics of novel coronavirus infection (COVID-19) using a fractional mathematical model in Caputo sense. Based on the spread of COVID-19 virus observed in Algeria, we formulate the model by dividing the infected population into two sub-classes namely the reported and unreported infective individuals. The existence and uniqueness of the model solution are given by using the well-known Picard–Lindelöf approach. The basic reproduction number R0 is obtained and its value is estimated from the actual cases reported in Algeria. The model equilibriums and their stability analysis are analyzed. The impact of various constant control parameters is depicted for integer and fractional values of α. Further, we perform the sensitivity analysis showing the most sensitive parameters of the model versus R0 to predict the incidence of the infection in the population. Further, based on the sensitivity analysis, the Caputo model with constant controls is extended to time-dependent variable controls in order obtain a fractional optimal control problem. The associated four time-dependent control variables are considered for the prevention, treatment, testing and vaccination. The fractional optimality condition for the control COVID-19 transmission model is presented. The existence of the Caputo optimal control model is studied and necessary condition for optimality in the Caputo case is derived from Pontryagin’s Maximum Principle. Finally, the effectiveness of the proposed control strategies are demonstrated through numerical simulations. The graphical results revealed that the implantation of time-dependent controls significantly reduces the number of infective cases and are useful in mitigating the infection.

Computational simulations for solving nonlinear composite oscillation fractional

In this document, we are keen on showing the rough scientific answers for fragmentary differential conditions by utilizing the Vieta-Lucas polynomial procedure. The fractional derivatives have represented in the Caputo sense. The implementation identified with the Vieta-Lucas strategy has been produced for differential conditions to the degree of admittance to estimated scientific arrangements of frameworks of partial differential conditions. The arrangements of our model conditions are determined as concurrent series with effectively process-able segments. A few models are tackled as representations, utilizing emblematic calculation. The mathematical outcomes show that the methodology is not difficult to carry out and exact with implementation to frameworks of fragmentary differential conditions. The strategy presented a hopeful device for addressing numerous direct and nonlinear partial differential conditions.

A robust analytical approach to the generalized Burgers–Fisher equation with fractional derivatives including singular and non-singular kernels

This paper presents a new heuristic approach for the approximate solution of the generalised Burgers–Fisher problem with two distinct derivative senses: Caputo fractional derivative (C) with a singular kernel and Atangana–Baleanu fractional derivative in Caputo sense (ABC) with a non-singular (Mittag–Leffler kernel) kernel. The proposed method is based on the η-homotopy analysis method with a new integral transform, and a well-proven algorithm that quickly provides the approximate solutions and maintains excellent accuracy. Comparative studies have been carried out to check the consistency of the proposed technique with previous methodologies in the literature. Finally, the convergence analysis of the study has been derived and studied the solutions’ behaviour in two different derivative senses having fractional-order μ→1.

On the eventual periodicity of fractional order dispersive wave equations using RBFS and transform

In this research work, let’s focus on the construction of numerical scheme based on radial basis functions finite difference (RBF-FD) method combined with the Laplace transform for the solution of fractional order dispersive wave equations. The numerical scheme is then applied to examine the eventual periodicity of the proposed model subject to the periodic boundary conditions. The implementation of proposed technique for high order fractional and integer type nonlinear partial differential equations (PDEs) is beneficial because this method is local in nature, therefore it yields and resulted in sparse differentiation matrices instead of full and dense matrices. Only small dimensions of linear systems of equations are to be solved for every center in the domain and hence this procedure is more reliable and efficient to solve large scale physical and engineering problems in complex domain. Laplace transform is utilized for obtaining the equivalent time-independent equation in Laplace space and also valuable to handle time-fractional derivatives in the Caputo sense. Application of Laplace transform avoids the time steeping procedure which commonly encounters the time instability issues. The solution to the transformed model is then obtained by computing the inversion of Laplace transform with an appropriate contour in a complex space, which is approximated by trapezoidal rule with high accuracy. Also since the Laplace transform operator is linear, it cannot be used to transform non-linear terms therefore let’s use a linearization approach and an appropriate iterative scheme. The proposed approach is tasted for some nonlinear fractional order KdV and Burgers equations. The capacity, high order accuracy and efficiency of our approach are demonstrated using examples and resultsRBFs Methods