scholarly journals A Numerical Algorithm for the Solutions of ABC Singular Lane–Emden Type Models Arising in Astrophysics Using Reproducing Kernel Discretization Method

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 923 ◽  
Author(s):  
Omar Abu Arqub ◽  
Mohamed S. Osman ◽  
Abdel-Haleem Abdel-Aty ◽  
Abdel-Baset A. Mohamed ◽  
Shaher Momani
Keyword(s):  
Convergence Analysis ◽  
Fractional Order ◽  
Numerical Algorithm ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Discretization Method ◽  
Fractional Models ◽  
Discretization Technique ◽  
And Mathematics

This paper deals with the numerical solutions and convergence analysis for general singular Lane–Emden type models of fractional order, with appropriate constraint initial conditions. A modified reproducing kernel discretization technique is used for dealing with the fractional Atangana–Baleanu–Caputo operator. In this tendency, novel operational algorithms are built and discussed for covering such singular models in spite of the operator optimality used. Several numerical applications using the well-known fractional Lane–Emden type models are examined, to expound the feasibility and suitability of the approach. From a numerical viewpoint, the obtained results indicate that the method is intelligent and has several features stability for dealing with many fractional models emerging in physics and mathematics, using the new presented derivative.

scholarly journals PIECEWISE OPTIMAL FRACTIONAL REPRODUCING KERNEL SOLUTION AND CONVERGENCE ANALYSIS FOR THE ATANGANA–BALEANU–CAPUTO MODEL OF THE LIENARD’S EQUATION

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040007
Author(s):  
SHAHER MOMANI ◽  
OMAR ABU ARQUB ◽  
BANAN MAAYAH
Keyword(s):  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Reproducing Kernel Hilbert Space ◽  
Weakly Nonlinear ◽  
Analytical Technique ◽  
Wide Range ◽  
Fractional Models ◽  
Weakly Nonlinear Waves ◽  
Fractional Solution

In this paper, an attractive reliable analytical technique is implemented for constructing numerical solutions for the fractional Lienard’s model enclosed with suitable nonhomogeneous initial conditions, which are often designed to demonstrate the behavior of weakly nonlinear waves arising in the oscillating circuits. The fractional derivative is considered in the Atangana–Baleanu–Caputo sense. The proposed technique, namely, reproducing kernel Hilbert space method, optimizes numerical solutions bending on the Fourier approximation theorem to generate a required fractional solution with a rapidly convergent form. The influence, capacity, and feasibility of the presented approach are verified by testing some applications. The acquired results are numerically compared with the exact solutions in the case of nonfractional derivative, which show the superiority, compatibility, and applicability of the presented method to solve a wide range of nonlinear fractional models.

scholarly journals System of Time Fractional Models for COVID-19: Modeling, Analysis and Solutions

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 787
Author(s):  
Olaniyi Iyiola ◽  
Bismark Oduro ◽  
Trevor Zabilowicz ◽  
Bose Iyiola ◽  
Daniel Kenes
Keyword(s):  
Fractional Order ◽  
Recovery Rate ◽  
Death Rate ◽  
Numerical Solutions ◽  
Complex Nature ◽  
Uniqueness Of Solutions ◽  
Fractional Equations ◽  
Effective Contact ◽  
Proposed Model ◽  
Fractional Models

The emergence of the COVID-19 outbreak has caused a pandemic situation in over 210 countries. Controlling the spread of this disease has proven difficult despite several resources employed. Millions of hospitalizations and deaths have been observed, with thousands of cases occurring daily with many measures in place. Due to the complex nature of COVID-19, we proposed a system of time-fractional equations to better understand the transmission of the disease. Non-locality in the model has made fractional differential equations appropriate for modeling. Solving these types of models is computationally demanding. Our proposed generalized compartmental COVID-19 model incorporates effective contact rate, transition rate, quarantine rate, disease-induced death rate, natural death rate, natural recovery rate, and recovery rate of quarantine infected for a holistic study of the coronavirus disease. A detailed analysis of the proposed model is carried out, including the existence and uniqueness of solutions, local and global stability analysis of the disease-free equilibrium (symmetry), and sensitivity analysis. Furthermore, numerical solutions of the proposed model are obtained with the generalized Adam–Bashforth–Moulton method developed for the fractional-order model. Our analysis and solutions profile show that each of these incorporated parameters is very important in controlling the spread of COVID-19. Based on the results with different fractional-order, we observe that there seems to be a third or even fourth wave of the spike in cases of COVID-19, which is currently occurring in many countries.

Numerical solution of time-fractional three-species food chain model arising in the realm of mathematical ecology

2020 ◽  
Vol 13 (02) ◽  
pp. 2050011 ◽  
Author(s):  
Ved Prakash Dubey ◽  
Rajnesh Kumar ◽  
Devendra Kumar
Keyword(s):  
Fractional Order ◽  
Food Chain ◽  
Fractional Derivatives ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Time Derivative ◽  
Chain Model ◽  
Homotopy Analysis ◽  
Food Chain Model ◽  
Derivatives Of

This research paper implements the fractional homotopy analysis transform technique to compute the approximate analytical solution of the nonlinear three-species food chain model with time-fractional derivatives. The offered technique is a fantastic blend of homotopy analysis method (HAM) and Laplace transform (LT) operator and has been used fruitfully in the numerical computation of various fractional differential equations (FDEs). This paper involves the fractional derivatives of Caputo style. The numerical solutions of this selected fractional-order food chain model are evaluated by making use of the associated initial conditions. It is revealed by the adopting procedure that the more desirable estimation of the solution can be easily acquired through the calculation of some number of iteration terms only — a fact which authenticates the easiness and soundness of the suggested hybrid scheme. The variations of fractional order of time derivative on the solutions for different specific cases have been depicted through graphical presentations. The outcomes demonstrated through the graphs expound that the adopted scheme is very fantastic and accurate.

An Efficient Computational Technique for Fractional Model of Generalized Hirota–Satsuma-Coupled Korteweg–de Vries and Coupled Modified Korteweg–de Vries Equations

2020 ◽  
Vol 15 (7) ◽  
Author(s):  
P. Veeresha ◽  
D. G. Prakasha ◽  
Devendra Kumar ◽  
Dumitru Baleanu ◽  
Jagdev Singh
Keyword(s):  
Fractional Order ◽  
Water Waves ◽  
Initial Conditions ◽  
Computational Method ◽  
Computational Technique ◽  
Suggested Technique ◽  
The Future ◽  
De Vries ◽  
Fractional Models ◽  
The Future Method

Abstract The aim of the present investigation to find the solution for fractional generalized Hirota–Satsuma coupled Korteweg–de-Vries (KdV) and coupled modified KdV (mKdV) equations with the aid of an efficient computational scheme, namely, fractional natural decomposition method (FNDM). The considered fractional models play an important role in studying the propagation of shallow-water waves. Two distinct initial conditions are choosing for each equation to validate and demonstrate the effectiveness of the suggested technique. The simulation in terms of numeric has been demonstrated to assure the proficiency and reliability of the future method. Further, the nature of the solution is captured for different value of the fractional order. The comparison study has been performed to verify the accuracy of the future algorithm. The achieved results illuminate that, the suggested computational method is very effective to investigate the considered fractional-order model.

Unified Galerkin- and DAE-Based Approximation of Fractional Order Systems

2010 ◽  
Vol 6 (2) ◽  
Author(s):  
Satwinder Jit Singh ◽  
Anindya Chatterjee
Keyword(s):  
Fractional Order ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Fractional Order Systems ◽  
Order Operator ◽  
Differential Algebraic Equation ◽  
Initial Value ◽  
Numerical Examples ◽  
Excellent Accuracy

We consider numerical solutions of nonlinear multiterm fractional integrodifferential equations, where the order of the highest derivative is fractional and positive but is otherwise arbitrary. Here, we extend and unify our previous work, where a Galerkin method was developed for efficiently approximating fractional order operators and where elements of the present differential algebraic equation (DAE) formulation were introduced. The DAE system developed here for arbitrary orders of the fractional derivative includes an added block of equations for each fractional order operator, as well as forcing terms arising from nonzero initial conditions. We motivate and explain the structure of the DAE in detail. We explain how nonzero initial conditions should be incorporated within the approximation. We point out that our approach approximates the system and not a specific solution. Consequently, some questions not easily accessible to solvers of initial value problems, such as stability analyses, can be tackled using our approach. Numerical examples show excellent accuracy.

scholarly journals Using Reproducing Kernel for Solving a Class of Fractional Order Integral Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Zhiyuan Li ◽  
Meichun Wang ◽  
Yulan Wang ◽  
Jing Pang
Keyword(s):  
Fractional Order ◽  
Kernel Function ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Jacobi Polynomials ◽  
Kernel Method ◽  
Schmidt Orthogonalization ◽  
Integral Differential Equations ◽  
First Time ◽  
Reproducing Kernel Function

This paper is devoted to the numerical scheme for a class of fractional order integrodifferential equations by reproducing kernel interpolation collocation method with reproducing kernel function in the form of Jacobi polynomials. Reproducing kernel function in the form of Jacobi polynomials is established for the first time. It is implemented as a reproducing kernel method. The numerical solutions obtained by taking the different values of parameter are compared; Schmidt orthogonalization process is avoided. It is proved that this method is feasible and accurate through some numerical examples.

scholarly journals Numerical solutions of fuzzy equal width models via generalized fuzzy fractional derivative operators

2021 ◽  
Vol 7 (2) ◽  
pp. 2695-2728
Author(s):  
Rehana Ashraf ◽  
◽  
Saima Rashid ◽  
Fahd Jarad ◽  
Ali Althobaiti ◽  
...  
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Proposed Model ◽  
Wide Range ◽  
Fractional Models ◽  
The Subject ◽  
Good Agreement ◽  
Numerical Approaches

<abstract><p>The Shehu homotopy perturbation transform method (SHPTM) via fuzziness, which combines the homotopy perturbation method and the Shehu transform, is the subject of this article. With the assistance of fuzzy fractional Caputo and Atangana-Baleanu derivatives operators, the proposed methodology is designed to illustrate the reliability by finding fuzzy fractional equal width (EW), modified equal width (MEW) and variants of modified equal width (VMEW) models with fuzzy initial conditions (ICs). In cold plasma, the proposed model is vital for generating hydro-magnetic waves. We investigated SHPTM's potential to investigate fractional nonlinear systems and demonstrated its superiority over other numerical approaches that are accessible. Another significant aspect of this research is to look at two significant fuzzy fractional models with differing nonlinearities considering fuzzy set theory. Evaluating various implementations verifies the method's impact, capabilities, and practicality. The level impacts of the parameter $ \hbar $ and fractional order are graphically and quantitatively presented, demonstrating good agreement between the fuzzy approximate upper and lower bound solutions. The findings are numerically examined to crisp solutions and those produced by other approaches, demonstrating that the proposed method is a handy and astonishingly efficient instrument for solving a wide range of physics and engineering problems.</p></abstract>

scholarly journals Numerical solvability of generalized Bagley–Torvik fractional models under Caputo–Fabrizio derivative

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shatha Hasan ◽  
Nadir Djeddi ◽  
Mohammed Al-Smadi ◽  
Shrideh Al-Omari ◽  
Shaher Momani ◽  
...  
Keyword(s):  
Reproducing Kernel ◽  
Initial Conditions ◽  
Reproducing Kernel Hilbert Space ◽  
Real Life ◽  
Approximate Solutions ◽  
Kernel Functions ◽  
Point Of View ◽  
The Novel ◽  
Life Problems ◽  
Fractional Models

AbstractThis paper deals with the generalized Bagley–Torvik equation based on the concept of the Caputo–Fabrizio fractional derivative using a modified reproducing kernel Hilbert space treatment. The generalized Bagley–Torvik equation is studied along with initial and boundary conditions to investigate numerical solution in the Caputo–Fabrizio sense. Regarding the generalized Bagley–Torvik equation with initial conditions, in order to have a better approach and lower cost, we reformulate the issue as a system of fractional differential equations while preserving the second type of these equations. Reproducing kernel functions are established to construct an orthogonal system used to formulate the analytical and approximate solutions of both equations in the appropriate Hilbert spaces. The feasibility of the proposed method and the effect of the novel derivative with the nonsingular kernel were verified by listing and treating several numerical examples with the required accuracy and speed. From a numerical point of view, the results obtained indicate the accuracy, efficiency, and reliability of the proposed method in solving various real life problems.

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