scholarly journals Multistage Spectral Relaxation Method for Solving the Hyperchaotic Complex Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Hassan Saberi Nik ◽  
Paulo Rebelo
Keyword(s):  
Complex Systems ◽  
Lorenz System ◽  
Nonlinear Differential Equations ◽  
Permanent Magnet Synchronous Motor ◽  
Pseudospectral Method ◽  
Relaxation Method ◽  
Spectral Relaxation Method ◽  
Spectral Relaxation ◽  
Chebyshev Pseudospectral ◽  
Complex Lorenz System

We present a pseudospectral method application for solving the hyperchaotic complex systems. The proposed method, called the multistage spectral relaxation method (MSRM) is based on a technique of extending Gauss-Seidel type relaxation ideas to systems of nonlinear differential equations and using the Chebyshev pseudospectral methods to solve the resulting system on a sequence of multiple intervals. In this new application, the MSRM is used to solve famous hyperchaotic complex systems such as hyperchaotic complex Lorenz system and the complex permanent magnet synchronous motor. We compare this approach to the Runge-Kutta based ode45 solver to show that the MSRM gives accurate results.

scholarly journals Solving Hyperchaotic Systems Using the Spectral Relaxation Method

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
S. S. Motsa ◽  
P. G. Dlamini ◽  
M. Khumalo
Keyword(s):  
Dynamical Systems ◽  
Numerical Solution ◽  
Numerical Method ◽  
Pseudospectral Method ◽  
Relaxation Method ◽  
Chebyshev Pseudospectral Method ◽  
Spectral Relaxation Method ◽  
Spectral Relaxation ◽  
Chebyshev Pseudospectral ◽  
Hyperchaotic Systems

A new multistage numerical method based on blending a Gauss-Siedel relaxation method and Chebyshev pseudospectral method, for solving complex dynamical systems exhibiting hyperchaotic behavior, is presented. The proposed method, called the multistage spectral relaxation method (MSRM), is applied for the numerical solution of three hyperchaotic systems, namely, the Chua, Chen, and Rabinovich-Fabrikant systems. To demonstrate the performance of the method, results are presented in tables and diagrams and compared to results obtained using a Runge-Kutta-(4,5)-based MATLAB solver,ode45, and other previously published results.

Spectral Relaxation Method for Powell-Eyring Fluid Flow Past a Radially Stretching Heated Disk Surface in a Porous Medium

2018 ◽  
Vol 387 ◽  
pp. 575-586 ◽  
Author(s):  
K. Gangadhar ◽  
P.R. Sobhana Babu ◽  
Oluwole Daniel Makinde
Keyword(s):  
Porous Medium ◽  
Nonlinear Boundary ◽  
Nonlinear Differential Equations ◽  
Slip Flow ◽  
Relaxation Method ◽  
Disk Surface ◽  
Skin Friction Coefficient ◽  
Nonlinear Boundary Value Problems ◽  
Spectral Relaxation Method ◽  
Spectral Relaxation

In this study we use a spectral relaxation method to investigate heat transfer in axisymmetric slip flow of a MHD Powell-Eyring fluid over a radially stretching surface embedded in porous medium with viscous dissipation. The transformed governing system of nonlinear differential equations was solved numerically using the spectral relaxation method that has been proposed for the solution of nonlinear boundary layer equations. Results were obtained for the skin friction coefficient, the local Nusselt number as well as the velocity and temperature profiles for the same values of the governing physical and fluid parameters. Validation of the results was reached by the comparison with limiting cases from previous studies in the literature. We show that the proposed technique is an efficient numerical algorithm with assured convergence that serves as an alternative to common numerical methods for solving nonlinear boundary value problems. We show that the convergence rate of the spectral relaxation method is significant improved by using the method in conjunction with the successive over - relaxation method.

scholarly journals Numerical Analysis for the Synthesis of Biodiesel Using Spectral Relaxation Method

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Z. G. Makukula ◽  
S. S. Motsa ◽  
S. Shateyi
Keyword(s):  
Vegetable Oils ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Relaxation Method ◽  
Numerical Results ◽  
Reaction Rate Constants ◽  
Animal Fats ◽  
Alternative Diesel Fuel ◽  
Spectral Relaxation Method ◽  
Spectral Relaxation

Biodiesel is an alternative diesel fuel chemically defined as the mono-alkyl esters of long chain fatty acids derived from vegetable oils or animal fat. It is becoming more attractive as an alternative fuel due to the depleting fossil fuel resources. A mathematical model for the synthesis of biodiesel from vegetable oils and animal fats is presented in this study. Numerical solutions of the model are found using a spectral relaxation method. The method, originally developed for boundary value problems, is an iterative scheme based on the Chebyshev spectral collocation method developed by decoupling systems of equations using Gauss-Seidel type of techniques. The effects of the reaction rate constants and initial concentrations of the reactants on the amount of the final product are being investigated. The accuracy of the numerical results is validated by comparison with known analytical results and numerical results obtained usingode45, an efficient explicit 4th and 5th order Runge-Kutta method used to integrate both linear and nonlinear differential equations.

scholarly journals A Numerical Examination of an Unsteady Nonlinear MHD Flow in the Presence of Thermal Radiation and Heat Generation

2021 ◽  
Vol 26 (1) ◽  
pp. 1-17
Author(s):  
T.M. Agbaje ◽  
P.G.L. Leach
Keyword(s):  
Thermal Radiation ◽  
Perturbation Method ◽  
Heat Generation ◽  
Nonlinear Differential Equations ◽  
Mhd Flow ◽  
Relaxation Method ◽  
Generating Series ◽  
Spectral Relaxation Method ◽  
Spectral Relaxation ◽  
Spectral Perturbation

AbstractIn this study, the spectral perturbation method and the spectral relaxation method are used to solve the nonlinear differential equations of an unsteady nonlinear MHD flow in the presence of thermal radiation and heat generation. The SPM is mainly based on series expansion, generating series approximation coupled with the Chebyshev spectral method. The numerical results generated using the spectral perturbation method were compared with those found in the literature, and the two results were in good agreement.

scholarly journals Unsteady mixed convection flow through a permeable stretching flat surface with partial slip effects through MHD nanofluid using spectral relaxation method

Open Physics ◽  
2017 ◽  
Vol 15 (1) ◽  
pp. 323-334 ◽  
Author(s):  
Sami M. Ahamed ◽  
Sabyasachi Mondal ◽  
Precious Sibanda
Keyword(s):  
Heat Transfer ◽  
Friction Coefficient ◽  
Flat Surface ◽  
Internal Heat ◽  
Relaxation Method ◽  
Skin Friction Coefficient ◽  
Partial Slip ◽  
Spectral Relaxation Method ◽  
Flow Through ◽  
Spectral Relaxation

AbstractAn unsteady, laminar, mixed convective stagnation point nanofluid flow through a permeable stretching flat surface using internal heat source or sink and partial slip is investigated. The effects of thermophoresis and Brownian motion parameters are revised on the traditional model of nanofluid for which nanofluid particle volume fraction is passively controlled on the boundary. Spectral relaxation method is applied here to solve the non-dimensional conservation equations. The results show the illustration of the impact of skin friction coefficient, different physical parameters, and the heat transfer rate. The nanofluid motion is enhanced with increase in the value of the internal heat sink or source. On the other hand, the rate of heat transfer on the stretching sheet and the skin friction coefficient are reduced by an increase in internal heat generation. This study further shows that the velocity slip increases with decrease in the rate of heat transfer. The outcome results are benchmarked with previously published results.

On Spectral Relaxation Approach to Radiating Powell-Eyring Fluid Flow over a Stretching Disk with Newtonian Heating

2018 ◽  
Vol 387 ◽  
pp. 461-473 ◽  
Author(s):  
K. Gangadhar ◽  
D. Vijaya Kumar ◽  
S. Mohammed Ibrahim ◽  
Oluwole Daniel Makinde
Keyword(s):  
Boundary Layer ◽  
Nonlinear Boundary ◽  
Numerical Solutions ◽  
Relaxation Method ◽  
Newtonian Heating ◽  
Nonlinear Boundary Value Problems ◽  
Local Skin ◽  
Boundary Layer Equations ◽  
Spectral Relaxation Method ◽  
Spectral Relaxation

In this study we use a new spectral relaxation method to investigate an axisymmetric law laminar boundary layer flow of a viscous incompressible non-Newtonian Eyring-Powell fluid and heat transfer over a heated disk with thermal radiation and Newtonian heating. The transformed boundary layer equations are solved numerically using the spectral relaxation method that has been proposed for the solution of nonlinear boundary layer equations. Numerical solutions are obtained for the local wall temperature, the local skin friction coefficient, as well as the velocity and temperature profiles. We show that the proposed technique is an efficient numerical algorithm with assured convergence that serves as an alternative to common numerical methods for solving nonlinear boundary value problems. We show that the convergence rate of the spectral relaxation method is significantly improved by using method in conjunction with the successive over-relaxation method. It is observed that CPU time is reduced in SOR method compare with SRM method.

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