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 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 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.

New Travelling Wave Solutions of Burgers Equation with Finite Transport Memory

2010 ◽  
Vol 65 (8-9) ◽  
pp. 633-640 ◽  
Author(s):  
Rathinasamy Sakthivel ◽  
Changbum Chun ◽  
Jonu Lee
Keyword(s):  
Evolution Equations ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Diffusion Processes ◽  
Initial Conditions ◽  
Travelling Wave ◽  
Nonlinear Evolution Equations ◽  
Wave Solutions ◽  
Wide Range ◽  
Finite Memory

The nonlinear evolution equations with finite memory have a wide range of applications in science and engineering. The Burgers equation with finite memory transport (time-delayed) describes convection-diffusion processes. In this paper, we establish the new solitary wave solutions for the time-delayed Burgers equation. The extended tanh method and the exp-function method have been employed to reveal these new solutions. Further, we have calculated the numerical solutions of the time-delayed Burgers equation with initial conditions by using the homotopy perturbation method (HPM). Our results show that the extended tanh and exp-function methods are very effective in finding exact solutions of the considered problem and HPM is very powerful in finding numerical solutions with good accuracy for nonlinear partial differential equations without any need of transformation or perturbation

scholarly journals Reproducing kernel approach for numerical solutions of fuzzy fractional initial value problems under the Mittag-Leffler kernel differential operator

2021 ◽  
Author(s):  
omar abu arqub ◽  
Jagdev Singh ◽  
Banan Maayah ◽  
Mohammed Alhodaly
Keyword(s):  
Reproducing Kernel ◽  
Differential Operators ◽  
Numerical Solutions ◽  
Reproducing Kernel Hilbert Space ◽  
Research Work ◽  
Three Dimensional ◽  
Characterization Theorem ◽  
Space Technique ◽  
Kernel Approach ◽  
Reproducing Kernel Theory

In this research study, fuzzy fractional differential equations in presence of the Atangana-Baleanu-Caputo differential operators are analytically and numerically treated using extended reproducing Kernel Hilbert space technique. With the utilization of a fuzzy strongly generalized differentiability form, a new fuzzy characterization theorem beside two fuzzy fractional solutions is constructed and computed. To besetment the attitude of fuzzy fractional numerical solutions; analysis of convergence and conduct of error beyond the reproducing kernel theory are explored and debated. In this tendency, three computational algorithms and modern trends in terms of analytic and numerical solutions are propagated. Meanwhile, the dynamical characteristics and mechanical features of these fuzzy fractional solutions are demonstrated and studied during two applications via three-dimensional graphs and tabulated numerical values. In the end, highlights and future suggested research work are eluded.

scholarly journals Hyperparameter Free MEE-FP based Learning for Next Generation Communication Systems

2021 ◽  
Author(s):  
Rangeet MItra ◽  
Georges Kaddoum ◽  
Daniel B. da Costa
Keyword(s):  
Communication Systems ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Unbiased Estimation ◽  
Learning Approaches ◽  
Step Size ◽  
Information Theoretic ◽  
Information Theoretic Learning ◽  
Wide Range ◽  
Non Gaussian

Information theoretic learning (ITL) criteria have emerged useful for mitigating degradations caused by unknown non-Gaussian noise processes in future wireless communication systems. Specifically, the reproducing kernel Hilbert space (RKHS) based approaches relying on ITL based learning criteria are envisioned to provide nearoptimal mitigation of unknown hardware impairments and non-Gaussian noises. Among several ITL criteria, the recent works find the minimum error entropy with fiducial points (MEE-FP) promising due to its guarantee of unbiased estimation and generalization over generic noise distributions. However, MEE-FP based learning approaches are known to depend on an accurate kernel-width initialization. Also, the optimal value of this kernelwidth is well-known to vary temporally and across deployment scenarios. To remove the dependency on kernelwidth, a hyperparameter-free MEE-FP based adaptive algorithm is derived using random-Fourier features with sampled kernel widths (RFF-SKW). In addition, a detailed convergence analysis is presented for the proposed hyperparameter-free MEE-FP, which promises a near-optimal error-floor independent of step-size and guarantees convergence for a wide range of step sizes. The promised hyperparameter-independence and improved convergence for the proposed hyperparameter-free MEE-FP are validated by computer simulations considering different case studies.

scholarly journals Hyperparameter Free MEE-FP based Learning for Next Generation Communication Systems

2021 ◽  
Author(s):  
Rangeet MItra ◽  
Georges Kaddoum ◽  
Daniel B. da Costa
Keyword(s):  
Communication Systems ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Unbiased Estimation ◽  
Learning Approaches ◽  
Step Size ◽  
Information Theoretic ◽  
Information Theoretic Learning ◽  
Wide Range ◽  
Non Gaussian

Information theoretic learning (ITL) criteria have emerged useful for mitigating degradations caused by unknown non-Gaussian noise processes in future wireless communication systems. Specifically, the reproducing kernel Hilbert space (RKHS) based approaches relying on ITL based learning criteria are envisioned to provide nearoptimal mitigation of unknown hardware impairments and non-Gaussian noises. Among several ITL criteria, the recent works find the minimum error entropy with fiducial points (MEE-FP) promising due to its guarantee of unbiased estimation and generalization over generic noise distributions. However, MEE-FP based learning approaches are known to depend on an accurate kernel-width initialization. Also, the optimal value of this kernelwidth is well-known to vary temporally and across deployment scenarios. To remove the dependency on kernelwidth, a hyperparameter-free MEE-FP based adaptive algorithm is derived using random-Fourier features with sampled kernel widths (RFF-SKW). In addition, a detailed convergence analysis is presented for the proposed hyperparameter-free MEE-FP, which promises a near-optimal error-floor independent of step-size and guarantees convergence for a wide range of step sizes. The promised hyperparameter-independence and improved convergence for the proposed hyperparameter-free MEE-FP are validated by computer simulations considering different case studies.

Progressive Mesh Densification Method for Numerical Solution of Mixed Elastohydrodynamic Lubrication

2015 ◽  
Vol 138 (2) ◽  
Author(s):  
Wei Pu ◽  
Jiaxu Wang ◽  
Dong Zhu
Keyword(s):  
Surface Roughness ◽  
Numerical Solution ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Solution Process ◽  
Elastohydrodynamic Lubrication ◽  
Operating Conditions ◽  
Machined Surface ◽  
Progressive Mesh ◽  
Wide Range

Numerical solution of mixed elastohydrodynamic lubrication (EHL) is of great importance for the study of lubrication formation and breakdown, as well as surface failures of mechanical components. However, converged and accurate numerical solutions become more difficult, and solution process with a fixed single discretization mesh for the solution domain appears to be quite slow, especially when the lubricant films and surface contacts coexist with real-machined roughness involved. Also, the effect of computational mesh density is found to be more significant if the average film thickness is small. In the present study, a set of sample cases with and without machined surface roughness are analyzed through the progressive mesh densification (PMD) method, and the obtained results are compared with those from the direct iteration method with a single fixed mesh. Besides, more numerical analyses with and without surface roughness in a wide range of operating conditions are conducted to investigate the influence of different compound modes in order to optimize the PMD procedure. In addition, different initial conditions are used to study the effect of initial value on the behaviors of this transient solution. It is observed that, no matter with or without surface roughness considered, the PMD method is stable for transient mixed EHL problems and capable of significantly accelerating the EHL solution process while ensuring numerical accuracy.

Similarity solutions and viscous gravity current adjustment times

2019 ◽  
Vol 874 ◽  
pp. 285-298 ◽  
Author(s):  
Thomasina V. Ball ◽  
Herbert E. Huppert
Keyword(s):  
Differential Equation ◽  
Similarity Solution ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Gravity Current ◽  
Similarity Solutions ◽  
Partial Differential ◽  
Initial Shape ◽  
Wide Range

A wide range of initial-value problems in fluid mechanics in particular, and in the physical sciences in general, are described by nonlinear partial differential equations. Recourse must often be made to numerical solutions, but a powerful, well-established technique is to solve the problem in terms of similarity variables. A disadvantage of the similarity solution is that it is almost always independent of any specific initial conditions, with the solution to the full differential equation approaching the similarity solution for times $t\gg t_{\ast }$, for some $t_{\ast }$. But what is $t_{\ast }$? In this paper we consider the situation of viscous gravity currents and obtain useful formulae for the time of approach, $\unicode[STIX]{x1D70F}(p)$, for a number of different initial shapes, where $p$ is the percentage disagreement between the radius of the current as determined by the full numerical solution of the governing partial differential equation and the similarity solution normalised by the similarity solution. We show that for any initial shape of volume $V,\unicode[STIX]{x1D70F}\propto 1/(\unicode[STIX]{x1D6FD}V^{1/3}\unicode[STIX]{x1D6FE}_{0}^{8/3}p)$ (as $p\downarrow 0$), where $\unicode[STIX]{x1D6FD}=g\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}/(3\unicode[STIX]{x1D707})$, with $g$ representing the acceleration due to gravity, $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$ the density difference between the gravity current and the ambient, $\unicode[STIX]{x1D707}$ the dynamic viscosity of the fluid that makes up the gravity current and $\unicode[STIX]{x1D6FE}_{0}$ the initial aspect ratio. This framework can used in many other situations, including where it is not an initial condition (in time) that is studied but one valid for specified values at a special spatial coordinate.

scholarly journals A Robust Numerical Solution of the Stochastic Collection–Breakup Equation for Warm Rain

2007 ◽  
Vol 46 (9) ◽  
pp. 1480-1497 ◽  
Author(s):  
Olivier P. Prat ◽  
Ana P. Barros
Keyword(s):  
Numerical Solution ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Strong Dependence ◽  
Discrete Equations ◽  
Warm Rain ◽  
Wide Range ◽  
Short Time ◽  
Rain Rates ◽  
The Impact

Abstract The focus of this paper is on the numerical solution of the stochastic collection equation–stochastic breakup equation (SCE–SBE) describing the evolution of raindrop spectra in warm rain. The drop size distribution (DSD) is discretized using the fixed-pivot scheme proposed by Kumar and Ramkrishna, and new discrete equations for solving collision breakup are presented. The model is evaluated using established coalescence and breakup parameterizations (kernels) available in the literature, and in that regard this paper provides a substantial review of the relevant science. The challenges posed by the need to achieve stable and accurate numerical solutions of the SCE–SBE are examined in detail. In particular, this paper focuses on the impact of varying the shape of the initial DSD on the equilibrium solution of the SCE–SBE for a wide range of rain rates and breakup kernels. The results show that, although there is no dependence of the equilibrium DSD on initial conditions for the same rain rate and breakup kernel, there is large variation in the time that it takes to reach steady state. This result suggests that, in coupled simulations of in-cloud motions and microphysics and for short time scales (&lt;30 min) for which transient conditions prevail, the equilibrium DSD may not be attainable except for very heavy rainfall. Furthermore, simulations for the same initial conditions show a strong dependence of the dynamic evolution of the DSD on the breakup parameterization. The implication of this result is that, before the debate on the uniqueness of the shape of the equilibrium DSD can be settled, there is critical need for fundamental research including laboratory experiments to improve understanding of collisional mechanisms in DSD evolution.

Sign in / Sign up

Export Citation Format

Share Document