Caputo Operator Research Articles

EXISTENCE AND UNIQUENESS FOR FRACTIONAL DIFFERENTIAL EQUATIONS WITH MULTIPOINT AND MULTI-TERM INTEGRAL BOUNDARY CONDITIONS

We investigated the existence and uniqueness of solutions to a nonlinear fractional differential equation containing the Caputo operator supplemented with separated multipoint and multi-term integral boundary conditions. The existence results are obtained by utilizing the Banach contraction principle and the well-known fixed point theorem of Krasnoselskii for a sum of contraction and compact mappings in Banach spaces. Some examples are also presented to illustrate the main results.

A Hybrid Interpolation Method for Fractional PDEs and Its Applications to Fractional Diffusion and Buckmaster Equations

This study presents a novel numerical method to solve PDEs with the fractional Caputo operator. In this method, we apply the Newton interpolation numerical scheme in Laplace space, and then, the solution is returned to real space through the inverse Laplace transform. The Newton polynomial provides good results as compared to the Lagrangian polynomial, which is used to construct the Adams–Bashforth method. This procedure is used to solve fractional Buckmaster and diffusion equations. Finally, a few numerical simulations are presented, ensuring that this strategy is highly stable and quickly converges to an exact solution.

Analysis of Huanglongbing disease model with a novel fractional piecewise approach

Huanglongbing (yellow dragon disease), often known as citrus greening, is one of the world's most destructive citrus illnesses. It is caused by Candidatus Liberibacter asiaticus, a bacterial disease that spreads through the tree canopy, causing the tree to degrade and eventually die. The aim of this paper is to study the dynamics of the Huanglongbing disease model using the novel piecewise derivative approach in the sense of singular and non-singular kernels. The singular kernel operator is in the sense of Caputo, while the non-singular kernel is the Atangana-Baleanu Caputo operator. The existence and uniqueness of the solution with piecewise derivatives are examined for the aforementioned problem. The suggested problem's approximate solution is obtained using the piecewise numerical iterative scheme based on the Newton polynomial approach. The numerical simulation for the piecewise derivable problem under consideration is presented using the data for various fractional orders. From the simulations, it is observed that when the vaccination rate is high then the number of exposed and infected citrus trees is small as compared to the lower vaccination rates.

COMPUTATIONAL ANALYSIS OF FRACTIONAL ORDER IMPERFECT TESTING INFECTION DISEASE MODEL

These days the whole world is facing a serious problem of infectious maladies and how to control the endemic of these diseases. Testing correctly is one of the most important procedures in preventing the spread of infectious diseases, as incorrect testing can turn a susceptible person into an infected person. In this paper, we study the dynamics of imperfect testing and diagnostics of infectious diseases model by replacing the integer order derivative in the sense of fractional order Atangana–Baleanu operator coupled with Caputo operator. This AB-fractional operator is the generalization of classical derivative and gives more data of the factors of the nonlocal dynamical frameworks. Fixed point theorems have been used for the verification of existence results and Picard’s stability technique utilized for the stability study of a fractional order imperfect testing infectious disease (ITID) model. Finally, numerical computations are implemented for the fractional order ITID model to illustrate the results graphically.

Chaos control and characterization of brushless DC motor via integral and differential fractal-fractional techniques

ABSTRACT Although brushless DC motors have been preferably used in wide industrial applications on the basis of superior properties and stable operation, the stability of brushless DC motors is found to be dependent on their chaotic characteristics and initial operating conditions. Owing to such characteristics, this article presents the chaos control and chaotic characteristics of brushless DC motors through newly defined mathematical techniques, so-called fractal-fractional differentiations. Mathematical modeling of the governing equations of the brushless DC motor has been developed by means of Caputo, Caputo-Fabrizio and Atangana-Baleanu fractal-fractional differential operators. The sensitive initial conditions and vibrant parameters have been analyzed for the comparative performance and assessment of the brushless DC motor. The numerical simulations of fractal-fractionalized governing equations of the brushless DC motor have been performed by means of the Adams-Bashforth-Moulton method using MATLAB package. Finally, the graphical illustrations have been underlined for the proposed Caputo, Caputo-Fabrizio and Atangana-Baleanu fractal-fractional differential operators subject to the chaotic behaviors investigated for the brushless DC motor.

Novel Analysis of Fuzzy Fractional Emden-Fowler Equations within New Iterative Transform Method

The analytical behavior of fractional differential equations is often puzzling and difficult to predict under uncertainty. It is crucial to develop a robust, extensive, and extremely successful theory to address these problems. An application of fuzzy fractional differential equations can be found in applied mathematics and engineering. Using the iterative transform technique, the study determines the analytic solution of fractional fuzzy Emden-Fowler equation in the sense of the Caputo operator, which is applied to evaluate the physical model range in several scientific and engineering disciplines. The derived solutions to the fractional fuzzy Emden-Fowler equations are more generic and applicable to a broader range of problems. Through the translation of fractional fuzzy differential equations into equivalent crisp systems of fractional differential equations, we obtain a parametric description of the solutions. The graphical and numerical representation demonstrates the symmetry among the upper and lower fuzzy solution representations in their simplest form, which may aid in the comprehension of artificial intelligence, control system models, computer science, image processing, quantum optics, medical science, physics, measure theory, stochastic optimization theory, biology, mathematical finance, and other domains, as well as nonfinancial evaluation.

The fractional view analysis of the Navier-Stokes equations within Caputo operator

In this research article the residual power series method is implemented for the solution of the Navier-stokes equations with two and three dimensions having the initial conditions. Caputo operator is used for the fractional derivative. The formulation is made in general form and then applied to the specific problems to check the validity of the suggested method. The solution of some numerical examples of the Navier-stokes equations are presented for both fractional and integer orders of the problems. The comparison of the obtained and exact solution is provided by using 2D and 3D plots of the solutions which confirmed the higher accuracy of the proposed method. Tables are drawn to show the results obtained, exact solutions and absolute error of the suggested method for each problem. It is investigated that as the number of terms increase in the series solution of the problems, the obtained solutions have the closed contact with actual solutions of each problem. From the calculated values it is cleared that the method is accurate and easy and therefore can be extended further to linear and nonlinear problems.

The Fractional Investigation of Some Dynamical Systems With Caputo Operator

In the present work, an Elzaki transformation is combined with a decomposition technique for the solutions of fractional dynamical systems. The targeted problems are related to the systems of fractional partial differential equations. Fractional differential equations are useful for more accurate modeling of various phenomena. The Elzaki transform decomposition method is implemented in a very simple and straightforward manner to solve the suggested problems. The proposed technique requires fewer calculations and needs no discretization or parametrization. The derivative of fractional order is represented in a Caputo form. To show the conclusion, which is drawn from the results, some numerical examples are considered for their approximate analytical solution. The series solutions to the targeted problems are obtained having components with a greater rate of convergence toward the exact solutions. The new results are represented by using tables and graphs, which show the sufficient accuracy of the present method as compared to other existing techniques. It is shown through graphs and tables that the actual and approximate results are very close to each other, which shows the applicability of the presented method. The fractional-order solutions are in best agreement with the dynamics of the given problems and provide infinite choices for an optimal solution to the suggested mathematical model. The novelty of the present work is that it applies an efficient procedure with less computational cost and attains a higher degree of accuracy. Furthermore, the proposed technique can be used to solve other nonlinear fractional problems in the future, which will be a scientific contribution to research society.

Fractional model of smoking with relapse and harmonic mean type incidence rate under Caputo operator

In this manuscript, we have analyzed a new time non-integer order smoking model endowed with Caputo fractional type derivative operator. The model is composed of four compartmental agents of various smokers named as a potential smoker, Light smoker, Smoker and Quit smoker. The first attempt is about the qualitative study of the considered problem along with the existing result and the derivation of the unique solution by the application of the fixed point theory. Secondly, for the semi-analytical solution of the smoking model, a very famous technique of the Generalized Adams–Bashforth Moulton method is used. Numerical simulation for different integer and non-integer order laying between (0, 1) has been drawn using Matlab. For different initial data, the result of simulation depicts the solution of the considered model is stable and shows the convergence of a single point. Smaller fractional orders show good results in the framework of stability.

A Survey on Recent Results on Lyapunov-Type Inequalities for Fractional Differential Equations

This survey paper is concerned with some of the most recent results on Lyapunov-type inequalities for fractional boundary value problems involving a variety of fractional derivative operators and boundary conditions. Our work deals with Caputo, Riemann-Liouville, ψ-Caputo, ψ-Hilfer, hybrid, Caputo-Fabrizio, Hadamard, Katugampola, Hilfer-Katugampola, p-Laplacian, and proportional fractional derivative operators.

Complex Dynamic Behaviour of Food Web Model with Generalized Fractional Operator

We apply a new generalized Caputo operator to investigate the dynamical behaviour of the non-integer food web model (FWM). This dynamical model has three population species and is nonlinear. Three types of species are considered in this population: prey species, intermediate predators, and top predators, and the top predators are also divided into mature and immature predators. We calculated the uniqueness and existence of the solutions applying the fixed-point hypothesis. Our study examines the possibility of obtaining new dynamical phase portraits with the new generalized Caputo operator and demonstrates the portraits for several values of fractional order. A generalized predictor–corrector (P-C) approach is utilized in numerically solving this food web model. In the case of the nonlinear equations system, the effectiveness of the used scheme is highly evident and easy to implement. In addition, stability analysis was conducted for this numerical scheme.

An approximate analytical view of physical and biological models in the setting of Caputo operator via Elzaki transform decomposition method

This article offers a well-organized and novel algorithm for solving time-fractional Fornberg–Whitham, Klein–Gordon equation and biological population models occurring from physics and engineering. The Elzaki (E)-transformation and decomposition process are combined in this algorithm. To evaluate the numerical outcomes of fractional-order partial differential equations, the E-transform decomposition method is generated in series form and nonlinearity terms are decayed. To demonstrate the feasibility of the proposed approach, numerical algorithms and examples are illustrated via graphs and tables. Moreover, it is viewed that the solutions of the new methodology are in strong correlation with the exact findings. Numerical simulations were carried out to ensure that the proposed methods are precise, as shown by the exact solutions resolving complex nonlinear problems.

On the Analytical Treatment for the Fractional-Order Coupled Partial Differential Equations via Fixed Point Formulation and Generalized Fractional Derivative Operators

High-dimensional fractional equation investigation is a cutting-edge discipline with considerable pragmatic and speculative consequences in engineering, epidemiology, and other scientific disciplines. In this study, a hybrid Jafari transform mixed with the Adomian decomposition method for obtaining the analytical solution to Burgers’ problem is provided. Burgers’ equation is a vital mathematical expression that appears in a variety of computational modelling fields, including fluid mechanics, nonlinear acoustics, gas dynamics, and traffic flow. By considering a hybrid transform, semianalytical techniques are constructed for the Caputo and Atangana-Baleanu fractional derivative operators. Besides that, existence and uniqueness analyses are carried out with the aid of the Banach contraction-fixed point theory. To obtain the models’ findings, we employed the Jafari transform on fractional-order Burger equations (BEs), supplemented by the inverse Jafari transform. The projected findings for the fractional BEs have been depicted visually. Ultimately, numerical figures are provided to validate the practicality and efficacy. The solution obtained by employing the supplied methodologies has been validated to have the appropriate rate of convergence to the precise solution. The main advantage of the suggested method is the relatively small number of computations performed. It can also be used to address fractional-order scientific issues in a multitude of fields.

Deeper investigation of modified epidemiological computer virus model containing the Caputo operator

The main aim of this paper is to analyze the modified epidemiological Susceptible-Infected-Removed model including an antidotal population compartment A (SIRA). The fractional natural decomposition method (FNDM) and variational iteration method (VIM) are applied to the governing model. This model is used to explain the wave behaviors of the infection virus arising in computer science. The SIRA model uses the Caputo derivative satisfying the initial conditions. Moreover, numerical investigations and strain conditions for the optimal values of parameters to minimize the effect of computer virus are also reported. The Lipschitz condition theorem and the Banach space are considered to present the uniqueness with the Caputo operator. Furthermore, various wave distributions of the nature of virus's are also extracted in plots.

Analytical Investigation of Fractional-Order Korteweg–De-Vries-Type Equations under Atangana–Baleanu–Caputo Operator: Modeling Nonlinear Waves in a Plasma and Fluid

This article applies the homotopy perturbation transform technique to analyze fractional-order nonlinear fifth-order Korteweg–de-Vries-type (KdV-type)/Kawahara-type equations. This method combines the Zain Ul Abadin Zafar-transform (ZZ-T) and the homotopy perturbation technique (HPT) to show the validation and efficiency of this technique to investigate three examples. It is also shown that the fractional and integer-order solutions have closed contact with the exact result. The suggested technique is found to be reliable, efficient, and straightforward to use for many related models of engineering and several branches of science, such as modeling nonlinear waves in different plasma models.

Delsarte equation for Caputo operator of fractional calculus

A fractional order variant of the Delsarte equation is investigated involving the Caputo differential derivative. Solvability of the resulting fractional hyperbolic Cauchy problem is achieved in the sense of distributions. A regularity result shows that the solution may be a function of time. Rigorous Delsarte representations are established. The symmetry between the fractional operators acting on space and time, induced by the Delsarte equation, opens the door to new type of fractional partial differential equations.

An Efficient Technique of Fractional-Order Physical Models Involving ρ-Laplace Transform

In this article, the ρ-Laplace transform is paired with a new iterative method to create a new hybrid methodology known as the new iterative transform method (NITM). This method is applied to analyse fractional-order third-order dispersive partial differential equations. The suggested technique procedure is straightforward and appealing, and it may be used to solve non-linear fractional-order partial differential equations effectively. The Caputo operator is used to express the fractional derivatives. Four numerical problems involving fractional-order third-order dispersive partial differential equations are presented with their analytical solutions. The graphs determined that their findings are in excellent agreement with the precise answers to the targeted issues. The solution to the problems at various fractional orders is achieved and found to be correct while comparing the exact solutions at integer-order problems. Although both problems are the non-linear fractional system of partial differential equations, the present technique provides its solution sophisticatedly. Including both integer and fractional order issues, solution graphs are carefully drawn. The fact that the issues’ physical dynamics completely support the solutions at both fractional and integer orders is significant. Moreover, despite using very few terms of the series solution attained by the present technique, higher accuracy is observed. In light of the various and authentic features, it can be customized to solve different fractional-order non-linear systems in nature.

Mathematical analysis and numerical simulation of time-fractional host-parasitoid system with Caputo operator

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology

A Nonlocal Boundary Value Problem with the Frankl Condition for an Equation of Mixed Parabolic-Hyperbolic Type with the Fractional Gerasimov–Caputo Operator

A Nonlocal Boundary Value Problem with the Frankl Condition for an Equation of Mixed Parabolic-Hyperbolic Type with the Fractional Gerasimov–Caputo Operator

Mathematical analysis of an extended SEIR model of COVID-19 using the ABC-fractional operator

This paper aims to suggest a time-fractional SPEPIPAIPSPHPRP model of the COVID-19 pandemic disease in the sense of the Atangana–Baleanu–Caputo operator. The proposed model consists of six compartments: susceptible, exposed, infected (asymptomatic and symptomatic), hospitalized and recovered population. We prove the existence and uniqueness of solutions to the proposed model via fixed point theory. Furthermore, a stability analysis in the context of Ulam–Hyers and the generalized Ulam–Hyers criterion is also discussed. For the approximate solution of the suggested model, we use a well-known and efficient numerical technique, namely the Toufik–Atangana numerical scheme, which validates the importance of arbitrary order derivative ϑ and our obtained theoretical results. Finally, a concise analysis of the simulation is proposed to explain the spread of the infection in society.