Caputo Sense Research Articles

A New analytical Modeling for Fractional Telegraph Equation Arising in Electromagnetic

In this article, the He’s variation iteration method (VIM) and Elzaki integral transform are proposed to analyze the time-fractional telegraph equations arising in electromagnetics. The Caputo sense is used to describe fractional derivatives. One of the advantages of this technique is that there is neither need to compute the Lagrange multiplier by calculating the integration in recurrence relations or via taking the convolution theorem. Further, to decrease nonlinear computational terms, the Adomian polynomial is identified with the homotopy perturbation method (HPM). The proposed method is applied to some examples of linear and nonlinear fractional telegraph equations. The solutions obtained by the new computational technique indicate that this method is efficient and facilitates the process of solving time fractional differential equations.

Adaptation of Sumudu Transform Iterative Method for Solving Fractional Integro-Differential Equations

In this paper, we discussed a new Sumudu transform iterative method and successfully applied on linear and nonlinear fractional integro-differential equations. The obtained results are compatible with those that have been obtained by other methods. The fractional derivative here is described in the Caputo sense. The proposed method is found to be powerful and efficient.

Numerical solution of Atangana–Baleanu–Caputo time-space fractional diffusion equation

In this article, the time-space fractional diffusion equation is solved by using the fractional operator in Atangana–Baleanu–Caputo (ABC) sense based on the Mittag-Leffler function involving non-singular and non-local kernels. This study mainly develop the numerical method for time-space linear and non-linear ABC fractional diffusion equation by utilising centred difference approximation with second-order accuracy of Riesz fractional derivative in space and trapezoidal formula for fractional integral approximation for a system of ABC integral equations. Also, we examine the stability and show that the proposed scheme converges at the rate of having space mesh size h and time step size t. Numerical simulations are presented based on the presented method. We finished the article with a conclusion.

On the non-uniqueness of the solution to a boundary value problem of heat conduction with a load in the form of a fractional derivative

The paper deals with the second boundary value problem for the loaded heat equation in the first quadrant. The loaded term contains a fractional derivative in the Caputo sense of an order α, 2<α<3. The boundary value problem is reduced to an integro-differential equation with a difference kernel by inverting the differential part. It is proved that a homogeneous integro-differential equation has at least one non-zero solution. It is shown that the solution of the homogeneous boundary value problem corresponding to the original boundary value problem is not unique, and the load acts as a strong perturbation of the boundary value problem.

Novel and accurate Gegenbauer spectral tau algorithms for distributed order nonlinear time-fractional telegraph models in multi-dimensions

This paper sheds light on the numerical solutions of the multi-dimensional distributed-order (DO) nonlinear time-fractional telegraph equations (TFTEs) models. We present novel and accurate shifted Gegenbauer spectral tau schemes for solving the aforementioned models. In the proposed schemes, the known and unknown functions are approximated in terms of the shifted Gegenbauer polynomials (SGPs) by using the Kronecker product structure in time and space. The matrix representation of the DO fractional derivative of SGPs in the Caputo sense has been derived. Also, novel operational matrices of the multiplication of shifted Gegenbauer functions in multi-dimension have been constructed to remove the obstacle that facing the use of the tau method in the presence of nonlinear term or variable coefficients or both of them. The proposed method takes full advantage of the nonlocal nature of DO fractional differential operators. The presented method is applied on six miscellaneous test examples to illustrate its robustness and effectiveness for smooth and non smooth solutions.

Comparative Analysis of Correlation and Kaplan–Yorke Dimensions for Discrete-Time Fractional Systems

The aim of this paper is to investigate the discrete-time fractional systems from the following aspects. First, the discrete-time fractional unified system in Caputo sense is established with the help of Euler’s discretization method. Furthermore, the dynamic behaviors of the discrete-time fractional Lü system (DFLS) which is deemed as a representative for unified system are observed. Then, the correlation dimension ([Formula: see text]) and Kaplan–Yorke dimension ([Formula: see text]) of the DFLS are evaluated by the aid of Grassberger–Procaccia algorithm and the Lyapunov exponent spectrum, respectively. Finally, the intrinsic connections between [Formula: see text] and [Formula: see text] are analyzed by the statistical modeling idea when the DFLS is in chaotic vibrations. The main results show that [Formula: see text] shares a positive correlation with [Formula: see text] for the chaotic DFLS, while the differences between [Formula: see text] and [Formula: see text] are not only related to the ratio of the largest and smallest Lyapunov exponents, but also closely tied up with the fractional order [Formula: see text] itself.

Numerical and quantitative analysis of HIV/AIDS model with modified Atangana-Baleanu in Caputo sense derivative

Fractional calculus plays an important role in the development of control strategies, the study of the dynamical transmission of diseases, and some other real-life problems nowadays. The time-fractional HIV/AIDS model is examined using a novel method in this paper. Based on the Atangana-concept Baleanu’s of a derivative in the Caputo sense, the current modified fractional derivative operator uses singular and non-local kernels. This new modified fractional operator is given a numerical approximation and applied to the HIV/AIDS model. In the presence of this novel operator, we present some significant analysis for the HIV/AIDS epidemic model. The uniqueness and stability criteria of the model have been demonstrated using the Picard successive approximation approach and Banach’s fixed point theory. The Laplace Adomian decomposition method (LADM) was used to obtain the numerical solution for the modified Atangana-Baleanu Caputo derivative model. The convergence analysis is verified for the proposed scheme. Finally, numerical results and simulations are derived with the proposed scheme for HIV/AIDS model. On the dynamics of HIV/AIDS transmission, the effects of many biological parameters are examined.

Memory effect on prey–predator dynamics: Exploring the role of fear effect, additional food and anti-predator behaviour of prey

Memory over past events has great importance on the survival of species in an ecological system. It is helpful in searching favourite food, escaping from predator attacks, group defence, use of environmental protection in the future time for the species. To explore the role of different ecological effects like fear effect, additional food and anti-predator behaviour of prey in the presence memory on an ecological system, in this paper we have proposed a two-dimensional prey–predator model and formulated mathematically using a system of two fractional order differential equations. Here the Caputo sense fractional derivative is used to construct the fractional order differential equations. The existence–uniqueness, non-negativity and boundedness of system solutions have been established. The detailed mathematical and graphical analysis for the feasibility of different equilibrium points and their analytical conditions for local asymptotic stability has been studied. The sufficient parametric conditions for the global stability of non-trivial equilibrium points are investigated. The Hopf bifurcation analysis of the system around the coexistence equilibrium point is derived in terms of system parameters. Using numerical simulations we have established that the memory effect can stabilize the system from an unstable periodic situation and makes the above-considered ecological effects more positively effective.

Effective Optimized Decomposition Algorithms for Solving Nonlinear Fractional Differential Equations

Abstract In this paper, the optimized decomposition method, which was developed to solve integer-order differential equations, will be modified and extended to handle nonlinear fractional differential equations. Fractional derivatives will be considered in terms of Caputo sense. The suggested modifications design new optimized decompositions for the series solutions depending on linear approximations of the nonlinear equations. Two optimized decomposition algorithms have been introduced to obtain approximate solutions of broad classes of initial value problems (IVPs) consisting of nonlinear fractional ordinary differential equations (ODEs) and partial differential equations (PDEs). A comparative study was conducted between the proposed algorithms and the Adomian decomposition method (ADM) by means of some test illustration problems. The implemented numerical simulation results showed that the proposed algorithms give better accuracy and convergence, and reduce the complexity of computational work compared to the Adomian's approach. This confirms the belief that the optimized decomposition method will be used effectively and widely as a powerful tool in solving various fractional differential equations.

Analytical Approaches for Approximate Solution of the Time-Fractional Coupled Schrödinger–KdV Equation

In this article, we use the homotopy perturbation method and the Adomian decomposition method with the Yang transformation to discover analytical solution to the time-fractional coupled Schrödinger–KdV equation. In the Caputo sense, fractional derivatives are described. A convergent series is used to calculate the solutions of fractional PDEs. Analytical results achieved applying the homotopy perturbation and decomposition techniques are numerically calculated and represented in the form of tables and figures. The simplicity, efficacy, and high degree of accuracy of the used method are then demonstrated by comparing these solutions to the actual solutions and the results. Finally, the applied approaches are the most popular and convergent methods for solving nonlinear fractional-order partial deferential problems.

Dynamics of a fractional order mathematical model for COVID-19 epidemic transmission

To achieve the aim of immediately halting spread of COVID-19 it is essential to know the dynamic behavior of the virus of intensive level of replication. Simply analyzing experimental data to learn about this disease consumes a lot of effort and cost. Mathematical models may be able to assist in this regard. Through integrating the mathematical frameworks with the accessible disease data it will be useful and outlay to comprehend the primary components involved in the spreading of COVID-19. There are so many techniques to formulate the impact of disease on the population mathematically, including deterministic modeling, stochastic modeling or fractional order modeling etc. Fractional derivative modeling is one of the essential techniques for analyzing real-world issues and making accurate assessments of situations. In this paper, a fractional order epidemic model that represents the transmission of COVID-19 using seven compartments of population susceptible, exposed, infective, recovered, the quarantine population, recovered–exposed, and dead population is provided. The fractional order derivative is considered in the Caputo sense. In order to determine the epidemic forecast and persistence, we calculate the reproduction number R0. Applying fixed point theory, the existence and uniqueness of the solutions of fractional order derivative have been studied . Moreover, we implement the generalized Adams–Bashforth–Moulton method to get an approximate solution of the fractional-order COVID-19 model. Finally, numerical result and an outstanding graphic simulation are presented.

Modified homotopy perturbation method and its application to analytical solitons of fractional-order Korteweg–de Vries equation

BackgroundExperimentally brought to light by Russell and hypothetically explained by Korteweg–de Vries, the KDV equation has drawn the attention of several mathematicians and physicists because of its extreme substantial structure in describing nonlinear evolution equations governing the propagation of weakly dispersive and nonlinear waves. Due to the prevalent nature and application of solitary waves in nonlinear dynamics, we discuss the soliton solution and application of the fractional-order Korteweg–de Vries (KDV) equation using a new analytical approach named the “Modified initial guess homotopy perturbation.”ResultsWe established the proposed technique by coupling a power series function of arbitrary order with the renown homotopy perturbation method. The convergence of the method is proved using the Banach fixed point theorem. The methodology was demonstrated with a generalized KDV equation, and we applied it to solve linear and nonlinear fractional-order Korteweg–de Vries equations, which are in Caputo sense. The method’s applicability and effectiveness were established as a feasible series of arbitrary orders that accelerate quickly to the exact solution at an integer order and are obtained as solutions. Numerical simulations were conducted to investigate the effect of Caputo fractional-order derivatives in the dispersion and propagation of water waves by varying the order alpha on the [0,1] interval. Comparative analysis of the simulation results, which were presented graphically and discussed, reveals that the degree of freedom of the Caputo fractional-order derivative is vital to controlling the magnitude of environmental hazards associated with water waves when adjusted.ConclusionThe proposed method is recommended for obtaining convergent series solutions to fractional-order partial differential equations. We suggested that applied mathematicians and physicists investigate this work to better understand the impact of the degree of freedom posed by Caputo fractional-order derivatives in wave dispersion and propagation, as physical applications can help divert wave-related environmental hazards.

A unified model of suspension concentration distribution in sediment mixed turbulent flows using generalized fractional advection-diffusion equation

The fractional operator in a space fractional advection-diffusion equation (FADE) plays a significant role in the mixing and vertical movement of sediment particles in a sediment-laden turbulent flow under non-local effects. Turbulent flow exhibits non-local mixing properties, which leads to the non-Fickian diffusion process that cannot be captured by the traditional diffusion equation. In this work, we present a generalized FADE that includes the generalized fractional differential operator in the Caputo sense. The full analytical solution is proposed utilizing the general Laplace transformation method. This generalized solution contains weight and scale functions and includes the effects of non-locality. It has been shown that several existing famous models of suspension concentration distribution for sediment particles (including both type-I and type-II distributions) in turbulent flows can be obtained from the proposed generalized solution with proper choices of the scale and weight functions in particular. Here a total of fourteen different types of concentration distribution equations including type-I and type-II profiles are derived from the general solution. Further possible generalizations of the model are also discussed which are more useful for practical applications. It is found that the several existing sediment distribution models are equivalent up to choices of weight and scale functions. Further, we found that the scale function could be physically related to the characteristic Lagrangian length of sediment mixing. The choice of the scale and weight function for both the type-I and type-II profiles are discussed and analyzed. Finally, the model is validated with experimental data as well as field data from the Missouri River, Mississippi River, and Rio Grande conveyance channels, and in each case, satisfactory agreements are obtained. These suggest the broader applicability of the present study.

Probabilistic solutions of fractional differential and partial differential equations and their Monte Carlo simulations

The work in this paper is four-fold. Firstly, we introduce an alternative approach to solve fractional ordinary differential equations as an expected value of a random time process. Using the latter, we present an interesting numerical approach based on Monte Carlo integration to simulate solutions of fractional ordinary and partial differential equations. Thirdly, we show that this approach allows us to find the fundamental solutions for fractional partial differential equations (PDEs), in which the fractional derivative in time is in the Caputo sense and the fractional in space one is in the Riesz–Feller sense. Lastly, using Riccati equation, we study families of fractional PDEs with variable coefficients which allow explicit solutions. Those solutions connect Lie symmetries to fractional PDEs.

A Computational Scheme for the Numerical Results of Time-Fractional Degasperis–Procesi and Camassa–Holm Models

This article presents an idea of a new approach for the solitary wave solution of the modified Degasperis–Procesi (mDP) and modified Camassa–Holm (mCH) models with a time-fractional derivative. We combine Laplace transform (LT) and homotopy perturbation method (HPM) to formulate the idea of the Laplace transform homotopy perturbation method (LHPTM). This study is considered under the Caputo sense. This proposed strategy does not depend on any assumption and restriction of variables, such as in the classical perturbation method. Some numerical examples are demonstrated and their results are compared graphically in 2D and 3D distribution. This approach presents the iterations in the form of a series solutions. We also compute the absolute error to show the effective performance of this proposed scheme.

Modeling and analysis of monkeypox disease using fractional derivatives

The frequency of monkeypox outbreaks and the extent of the projected outbreaks in human populations have both steadily increased. This paper proposes Atangana-Baleanu fractional-order derivatives define in Caputo sense to investigate the kinetics of Monkeypox transmission in Ghana. We determine the stability of the recommended model's equilibrium points and basic reproduction number. The solution's existence and originality, as well as the model's Hyers-Ullam stability, are proven. The models basic reproduction number was found to be R0 = 0.1940. The numerical simulation showed the fractional operator had an influence on the various compartments of the model. The dynamics of the disease in the community were shown to be influenced by fractional-order derivatives, and infections were eradicated within the first five (5) days when π = 0.2.

Abstract fractional inverse source problem of order $$0<\alpha <1$$ in a Banach space

Given an abstract Cauchy problem with time-fractional derivative in the Caputo sense of order \(0<\alpha <1\) in a Banach space with an inhomogeneous term \(\varphi (t)\vartheta \), we investigate the inverse problem of determining \(\vartheta \) from it and an overdetermination in the form of an integral. Assume that the operator of evolution A generates an analytic \(C_0\)-semigroup, we give the uniqueness and unique existence of the solution in terms of the eigenvalues and spectrum of A. And via the approach of functional calculus, the representation of the solution is given.

A re-scaling spectral collocation method for the nonlinear fractional pantograph delay differential equations with non-smooth solutions

The convergence analysis of the spectral methods of the fractional differential equations is generally carried out on the assumption that the underlying solution is sufficiently smooth. Due to the limited smoothing property of the solution operator, these methods fail to achieve spectral accuracy. This work aims to study a spectral collocation approach for the fractional nonlinear pantograph delay differential equations with nonsmooth solutions. The fractional-order derivative is considered in the Caputo sense. An auxiliary transformation is adapted to match the singularity in the corresponding solution and to maximize the convergence rate of the proposed scheme. Therefore, the solution of the resulting equation will possess better regularity and then the numerical method can achieve the spectral accuracy, which serves as an improvement compared with the existing results in the literature. The spectral convergence rate for the proposed approach is discussed in the weighted L2-norm and the L∞-norm. Finally, numerical results are given to confirm our theoretical analysis.

Numerical solutions to the fractional-order wave equation

New numerical solution to the linear fractional-order wave equation is presented. The Liouville–Caputo sense fractional-derivative operator and Crank–Nicholson finite difference method (CN-FDM) algorithm are employed. The stability of the present technique is considered by the fractional Von Neumann stability analysis method. Special example as an application of the method is provided. The obtained results are examined to check the derived stability condition of the proposed algorithm. Computational results indicate that the present numerical algorithm is efficient and applicable for the problem under study and many other problems.

Impact of quarantine on fractional order dynamical model of Covid-19

In this paper, a Covid-19 dynamical transmission model of a coupled non-linear fractional differential equation in the Atangana-Baleanu Caputo sense is proposed. The basic dynamical transmission features of the proposed system are briefly discussed. The qualitative as well as quantitative results on the existence and uniqueness of the solutions are evaluated through the fixed point theorem. The Ulam-Hyers stability analysis of the suggested system is established. The two-step Adams-Bashforth-Moulton (ABM) numerical method is employed to find its numerical solution. The numerical simulation is performed to accesses the impact of various biological parameters on the dynamics of Covid-19 disease.