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 Analytical Solution of Two-Dimensional Sine-Gordon Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Alemayehu Tamirie Deresse ◽  
Yesuf Obsie Mussa ◽  
Ademe Kebede Gizaw
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Test Problems ◽  
Two Dimensional ◽  
Sine Gordon Equation ◽  
Science And Engineering ◽  
Transform Method ◽  
Differential Transform ◽  
Good Agreement

In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear sine-Gordon equations subject to appropriate initial conditions. Some lemmas which help us to solve the governing problem using the proposed method are proved. This scheme has the advantage of generating an analytical approximate solution or exact solution in a convergent power series form with conveniently determinable components. The method considers the use of the appropriate initial conditions and finds the solution without any discretization, transformation, or restrictive assumptions. The accuracy and efficiency of the proposed method are demonstrated by four of our test problems, and solution behavior of the test problems is presented using tables and graphs. Further, the numerical results are found to be in a good agreement with the exact solutions and the numerical solutions that are available in literature. We have showed the convergence of the proposed method. Also, the obtained results reveal that the introduced method is promising for solving other types of nonlinear partial differential equations (NLPDEs) in the fields of science and engineering.

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.

One-Dimensional Nonlinear Parametric Instability of Inhomogeneous Plasma: Time Domain Problem

2021 ◽  
Vol 24 (3) ◽  
pp. 272-279
Author(s):  
N. V. Gerasimenko ◽  
F. M. Trukhachev ◽  
E. Z. Gusakov ◽  
L. V. Simonchik ◽  
A. V. Tomov
Keyword(s):  
Initial Conditions ◽  
Inhomogeneous Plasma ◽  
Parametric Instability ◽  
Analytical Models ◽  
Dimensional Model ◽  
One Dimensional ◽  
Proposed Model ◽  
Wide Range ◽  
Spatial And Temporal Characteristics ◽  
Good Agreement

A numerical one-dimensional model of convective parametric instability of inhomogeneous plasma is developed. By using this model, a numerical solution describing spatial and temporal characteristics of interacting waves is obtained. The results obtained are in a good agreement with known analytical models and substantially generalize them. In particular, an important advantage of the proposed model is the possibility of varying initial conditions, analyzing behavior of the system in the presence of incident wave fluctuations that is important for the future study of the absolute instability mode. The model is also provides possibility to simulate absolute parametric instability with a wide range of controllable parameters, as well as to study interacting wave transients.

scholarly journals Novel Numerical Investigations of Fuzzy Cauchy Reaction–Diffusion Models via Generalized Fuzzy Fractional Derivative Operators

2021 ◽  
Vol 5 (4) ◽  
pp. 151
Author(s):  
Manar A. Alqudah ◽  
Rehana Ashraf ◽  
Saima Rashid ◽  
Jagdev Singh ◽  
Zakia Hammouch ◽  
...  
Keyword(s):  
Initial Conditions ◽  
Hybrid Approach ◽  
Reaction Diffusion ◽  
Approximate Solutions ◽  
Vital Role ◽  
Reaction Diffusion Equations ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Proposed Model ◽  
Generalized Hukuhara Differentiability

The present research correlates with a fuzzy hybrid approach merged with a homotopy perturbation transform method known as the fuzzy Shehu homotopy perturbation transform method (SHPTM). With the aid of Caputo and Atangana–Baleanu under generalized Hukuhara differentiability, we illustrate the reliability of this scheme by obtaining fuzzy fractional Cauchy reaction–diffusion equations (CRDEs) with fuzzy initial conditions (ICs). Fractional CRDEs play a vital role in diffusion and instabilities may develop spatial phenomena such as pattern formation. By considering the fuzzy set theory, the proposed method enables the solution of the fuzzy linear CRDEs to be evaluated as a series of expressions in which the components can be efficiently identified and generating a pair of approximate solutions with the uncertainty parameter λ∈[0,1]. To demonstrate the usefulness and capabilities of the suggested methodology, several numerical examples are examined to validate convergence outcomes for the supplied problem. The simulation results reveal that the fuzzy SHPTM is a viable strategy for precisely and accurately analyzing the behavior of a proposed model.

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.

scholarly journals Homotopy Perturbation Method

2017 ◽  
pp. 24-61
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Perturbation Method ◽  
Exact Solutions ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Homotopy Perturbation ◽  
Wide Range ◽  
Transfer Problems

The homotopy perturbation method (HPM) is employed to compute an approximation to the solution of the system of nonlinear differential equations governing on the problem. It has been attempted to show the capabilities and wide-range applications of the homotopy perturbation method in comparison with the previous ones in solving heat transfer problems. The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations.

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.

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 Dynamics Analysis of Unbalanced Motorized Spindles Supported on Ball Bearings

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Junfeng Liu ◽  
Tao Lai ◽  
Xiaoan Chen
Keyword(s):  
High Speed ◽  
Magnetic Force ◽  
Numerical Solutions ◽  
Dynamic Properties ◽  
Lyapunov Stability Theory ◽  
Force Model ◽  
Proposed Model ◽  
Dynamic Operating ◽  
Motorized Spindles ◽  
Good Agreement

This paper presents an improved dynamic model for unbalanced high speed motorized spindles. The proposed model includes a Hertz contact force model which takes into the internal clearance and an unbalanced electromagnetic force model based on the energy of the air magnetic field. The nonlinear characteristic of the model is analysed by Lyapunov stability theory and numerical analysis to study the dynamic properties of the spindle system. Finally, a dynamic operating test is carried out on a DX100A-24000/20-type motorized spindle. The good agreement between the numerical solutions and the experimental data indicates that the proposed model is capable of accurately predicting the dynamic properties of motorized spindles. The influence of the unbalanced magnetic force on the system is studied, and the sensitivities of the system parameters to the critical speed of the system are obtained. These conclusions are useful for the dynamic design of high speed motorized spindles.

Two-Dimensional Transient Solutions for Crossflow Heat Exchangers With Neither Gas Mixed

1987 ◽  
Vol 109 (2) ◽  
pp. 281-286 ◽  
Author(s):  
G. Spiga ◽  
M. Spiga
Keyword(s):  
Heat Exchangers ◽  
Initial Conditions ◽  
Transient Behavior ◽  
Computational Time ◽  
Two Dimensional ◽  
Transform Method ◽  
Wide Range ◽  
The Laplace Transform ◽  
Inlet Conditions ◽  
Conductance Ratio

The two-dimensional transient behavior of gas-to-gas crossflow heat exchangers is investigated, solving by analytical methods the thermal balance equations in order to determine the transient distribution of temperatures in the core wall and in both the unmixed gases. Assuming large wall capacitance, the general solutions are deduced by the Laplace transform method and are presented as integrals of modified Bessel functions on space and time, for a transient response with any arbitrary initial and inlet conditions, in terms of the number of transfer units, capacity rate and conductance ratio. Specializing the entrance temperature and assuming constant initial conditions, the most meaningful transient conditions (such as step, ramp, and exponential responses) have been simulated and the relevant solutions, expressed by means of either integrals or series, have been accurately computed with extremely low computational time. The temperature responses are then presented in graphic form for a wide range of the number of transfer units.

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