A novel numerical algorithm based on Galerkin–Petrov time-discretization method for solving chaotic nonlinear dynamical systems

Khan, Muhammad Sabeel; Khan, M. Ijaz

doi:10.1007/s11071-017-3964-5

Cited by 7 publications

(5 citation statements)

References 30 publications

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“…Step 3: Take Khan [39] Table I and Table II and a good agreement is found. In Table I the numerical values of the skin-friction obtained by the present approach in both the cases of Classical and Cosserat continuum are compared with the observations of Mahapatra et al, [22] Mushtaq et al, [38] Abel et al, [40] Megahed [41] and Mustafa et al [42].…”

Section: ) Numerical Results and Discussionmentioning

confidence: 86%

“…Step 2: Following Khan and Hackl, [39] obtain the discrete form of the problem in terms of the following nonlinear coupled algebraic equations…”

Section: ) Numerical Results and Discussionmentioning

confidence: 99%

“…is the fluid relaxation time and is the thermal relaxation time and are defined as The above system of differential equations along with the initial/boundary conditions as in Eq. 12, is treated numerically using the continuous Galerkin-Petrov finite element discretization scheme [39] embedded with the shooting method. The procedure is to choose appropriate initial guess for the unknown values and to shoot out the unknown initial values so that the following convergence criterion is established i.e.…”

Section: ) Introductionmentioning

confidence: 99%

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Particle rotation effects in Cosserat-Maxwell boundary layer flow with non-Fourier heat transfer using a new novel approach

et al. 2020

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In this article we use a non-classical approach to investigate different physical effects of Cosserat-Maxwell fluid flow with non-Fourier heat transfer mechanism. Furthermore, a new numerical approach is used and outlined for computing and analyzing the behavior of such flows. In particular, continuous Galerkin-Petrov discretization scheme is embedded with shooting method to get the numerical algorithm to solve the stagnation point flow of Cosserat-Maxwell fluid with Cattaneo-Christov heat transfer. The mathematical description of the physical problem is stated in the form of partial differential equations (PDEs) which govern the flow mechanism. Further, the suitable transformations are utilized to describe the governing PDEs into their respective ordinary differential equations. Numerical experiments are performed for a specific case where there are weak concentrations of the flow near the stretching surface thereby allowing the microelement to rotate and generate vortex flow near the stretching surface. Buoyancy effects

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Section: ) Numerical Results and Discussionmentioning

confidence: 86%

“…Step 2: Following Khan and Hackl, [39] obtain the discrete form of the problem in terms of the following nonlinear coupled algebraic equations…”

Section: ) Numerical Results and Discussionmentioning

confidence: 99%

Section: ) Introductionmentioning

confidence: 99%

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Particle rotation effects in Cosserat-Maxwell boundary layer flow with non-Fourier heat transfer using a new novel approach

et al. 2020

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“…In contrast to the multidomain methods that were previously mentioned, their approach is entirely numerical. Other numerical techniques utilize domain decomposition to address chaotic situations [26].…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulation for COVID-19 Model Using a Multidomain Spectral Relaxation Technique

et al. 2023

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The major objective of this work is to evaluate and study the model of coronavirus illness by providing an efficient numerical solution for this important model. The model under investigation is composed of five differential equations. In this study, the multidomain spectral relaxation method (MSRM) is used to numerically solve the suggested model. The proposed approach is based on the hypothesis that the domain of the problem can be split into a finite number of subintervals, each of which can have a solution. The procedure also converts the proposed model into a system of algebraic equations. Some theoretical studies are provided to discuss the convergence analysis of the suggested scheme and deduce an upper bound of the error. A numerical simulation is used to evaluate the approach’s accuracy and utility, and it is presented in symmetric forms.

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“…Te analysis of the chaotic characteristics of the system is crucial in elucidating the impact of parameter changes on system stability, which is essential for overcoming the quality bottleneck in deep-hole parts processing [9][10][11][12]. At present, there is no general analytical paradigm for analyzing a complex nonlinear dynamic system [13][14][15][16], which also brings great challenges to the dynamic stability analysis of complex BTA deep-hole machining system with mechanical, electrical, and hydraulic multifeld coupling.…”

Section: Introductionmentioning

confidence: 99%

The Chaotic Property of BTA Deep-Hole Machining System under the Effect of Inner Cutting Fluid

Zhang

Zhao

et al. 2023

Shock and Vibration

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To clarify the action mechanism of parameter change on system stability, the chaotic property of BTA deep-hole machining system under the effect of inner cutting fluid was analyzed. According to the kinematic characteristics of the internal cutting fluid and the equation of the moment of momentum of the system, the kinematic equation of the boring bar considering the effect of the internal fluid was established. The critical conditions of chaos were deduced according to the Hamiltonian function and Melnikov function of the plane near-Hamilton system. The mechanism of the liquid filling ratio, cutting fluid flow velocity, and frequency ratio parameters on the system’s critical instability surface is investigated. The correlation and sensitivity of influencing factors, such as filling ratio and frequency ratio, and cutting fluid flow velocity to the sensitivity of system chaos are explored. The results show that in precision machining, the change of liquid filling ratio is positively related to the stability of the system, the change of cutting fluid flow velocity is negatively correlated with the stability of the system, and the change of frequency ratio has no monotonicity effect on the stability of the system. The sensitivity of the chaotic characteristics of the system to each parameter is bounded by the filling liquid ratio h = 0.58. When 0 ≤ h ≤ 0.58, frequency ratio ω ¯ > filling ratio h > cutting fluid flow velocity V0; when 0.58 < h ≤ 1, filling ratio h > frequency ratio ω ¯ > cutting fluid flow velocity V0. These research conclusions can lay a certain theoretical foundation for the analysis, control, and optimization of the complex mechanical behavior of BTA deep-hole machining systems in engineering practice.

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A novel numerical algorithm based on Galerkin–Petrov time-discretization method for solving chaotic nonlinear dynamical systems

Cited by 7 publications

References 30 publications

Particle rotation effects in Cosserat-Maxwell boundary layer flow with non-Fourier heat transfer using a new novel approach

Particle rotation effects in Cosserat-Maxwell boundary layer flow with non-Fourier heat transfer using a new novel approach

Numerical Simulation for COVID-19 Model Using a Multidomain Spectral Relaxation Technique

The Chaotic Property of BTA Deep-Hole Machining System under the Effect of Inner Cutting Fluid

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