A Bivariate Chebyshev Spectral Collocation Quasilinearization Method for Nonlinear Evolution Parabolic Equations

Motsa, S. S.; Magagula, Vusi Mpendulo; Sibanda, Precious

doi:10.1155/2014/581987

Cited by 72 publications

(34 citation statements)

References 42 publications

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“…In addition, comparison were also made against published results of Hossain and Paul (2001a,b) who used the finite difference method, series solution method and an asymptotic solution method to respectively, solve equations (6-7) and (10-11). Validation of the numerical solutions was further established by comparing the (LSPM) results with numerical approximate solutions obtained using the bivariate quasilinearisation method (BSQLM) as described by Motsa et al (2014) and Motsa and Ansari (2015). We remark that the values of all physical parameters used in this study were chosen based on the values used in the published work of Hossain and Paul (2001a,b); Saha et al (2007).…”

Section: Resultsmentioning

confidence: 99%

A Large Parameter Spectral Perturbation Method for Nonlinear Systems of Partial Differential Equations That Models Boundary Layer Flow Problems

Agbaje¹,

Motsa²,

Mondal³

et al. 2017

Frontiers in Heat and Mass Transfer

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In this work, we present a compliment of the spectral perturbation method (SPM) for solving nonlinear partial differential equations (PDEs) with applications in fluid flow problems. The (SPM) is a series expansion based approach that uses the Chebyshev spectral collocation method to solve the governing sequence of differential equation generated by the perturbation series approximation. Previously the SPM had the limitation of being used to solve problems with small parameters only. This current investigation seeks to improve the performance of the SPM by doing the series expansion about a large parameter. The new method namely the large parameter spectral perturbation method (LSPM) combines the idea of asymptotic analysis approach with numerical solution techniques. In the (LSPM), the resulting equations from the asymptotic expansion are solved numerically using the Chebyshev spectral method. The purpose of this study is to extend the existing spectral perturbation method (SPM) which was used for small parameters to be suitable for problems with large parameters. The applicability of the (LSPM), is tested on systems of earlier reported nonlinear partial differential equations that describe boundary layer fluid flow problems. The validity of the (LSPM) numerical solutions is verified by comparing with published results and the bivariate Chebyshev spectral quasilinearisation method (BSQLM) and an excellent agreement were observed. The (BSQLM) is a numerical method that blends the quasilinearisation method, the Chebyshev spectral method, and the bivariate Lagrange interpolation method. One of the advantages of this approach is that it gives results in a fraction of seconds. We remark also that simple decoupled linear systems formulas are derived for generating the solutions in the form of decoupled linear systems. Tables are generated to present error and convergence properties of the LSPM.

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Section: Resultsmentioning

confidence: 99%

A Large Parameter Spectral Perturbation Method for Nonlinear Systems of Partial Differential Equations That Models Boundary Layer Flow Problems

Agbaje¹,

Motsa²,

Mondal³

et al. 2017

Frontiers in Heat and Mass Transfer

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“…Several formulas exist for computing the derivatives when the collocating points are chosen to be Chebyshev Gauss-Lobatto points of the form (25). The method of discretization of PDEs using the bivariate approach described above has also been used in [30] who solved one equation of PDEs for different models of non-linear evolution parabolic PDEs.…”

Section: Methods Of Solution : Bsrmmentioning

confidence: 99%

Analysis of MHD Forced Convective Flow of Variable Fluid Properties over a Saturated Porous Medium with Thermal Radiation Effect

Adegbie

Fagbade

2016

IFSL

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Abstract. The present paper addresses the problem of MHD forced convective flow in a fluid saturated porous medium with Brinkman-Forchheimer model, which is an important physical phenomena in engineering applications. The paper extends the previous models to account for effects of variable fluid properties on the forced convective flow through a porous medium in the presence of radiative heat loss using bivariate spectral relaxation method (BSRM). The dynamic viscosity µ and thermal conductivity of the newtonian fluid are assumed to vary linearly respectively, with temperature whereas the contribution of thermal radiative heat loss is based on Rosseland diffussion approximation. The flow model is described and expressed in form of a highly coupled nonlinear system of partial differential equations. The method of solution BSRM as proposed by Motsa [25] seeks to decouple the original system of PDEs to form a sequence of equations that can be solved in a computationally efficient manner. BSRM is an approach that applies spectral collocation independently in all underlying independent variable is executed to obtain approximate solutions of the problem. The proposed algorithm is supposed to be a very accurate, convergent and very effective in generating numerical results. The results obtained show a significant effects of the flow control parameters on the fluid velocity and temperature respectively. Consequently, the wall shear stress and local heat transfer rate of the present paper are compared with the available results in literatures. Remarkable impacts and a good agreement are found.

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“…This is because there will be a lot of multiplication by zero which reduces the computational time and enables the matrices to be stored efficiently. Since the method combines the bivariate spectral quasilinearisation method [36], non-overlapping and overlapping multi-domain technique, for reference purposes, we shall refer to the method as the overlapping multi-domain bivariate spectral quasilinearisation method (OMD-BSQLM). The use of spectral collocation-based methods such as the OMD-BSQLM for solving systems of PDEs can be a most promising tool in the study of conjugate heat transfer problems.…”

Section: Introductionmentioning

confidence: 99%

Overlapping Multi-Domain Spectral Method for Conjugate Problems of Conduction and MHD Free Convection Flow of Nanofluids over Flat Plates

Mkhatshwa

Motsa

Sibanda

2019

MCA

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An efficient overlapping multi-domain spectral method is used in the analysis of conjugate problems of heat conduction in solid walls coupled with laminar magnetohydrodynamic (MHD) free convective boundary layer flow of copper (Cu) water and silver (Ag) water nanofluids over vertical and horizontal flat plates. The combined effects of heat generation and thermal radiation on the flow has been analyzed by imposing a magnetic field along the direction of the flow to control the motion of electrically conducting fluid in nanoscale systems. We have assumed that the nanoparticle volume fraction at the wall may be actively controlled. The dimensionless flow equations are solved numerically using an overlapping multi-domain bivariate spectral quasilinearisation method. The effects of relevant parameters on the fluid properties are shown graphically and discussed in detail. Furthermore, the variations of the skin friction coefficient, surface temperature and the rate of heat transfer are shown in graphs and tables. The findings show that the surface temperature is enhanced due to the presence of nanoparticles in the base fluid and the inclusion of the thermal radiation, heat generation and transverse magnetic field in the system. An increase in the nanoparticle volume fraction, heat generation, thermal radiation, and magnetic field parameter enhances the nanofluid velocity and temperature while reducing the heat transfer rate. The results also indicate that the Ag-water nanofluid has higher skin friction and surface temperature than the Cu-water nanofluid, while the opposite behaviour is observed in the case of the rate of heat transfer. The computed numerical results are compared with previously published results and found to be in good agreement.

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A Bivariate Chebyshev Spectral Collocation Quasilinearization Method for Nonlinear Evolution Parabolic Equations

Cited by 72 publications

References 42 publications

A Large Parameter Spectral Perturbation Method for Nonlinear Systems of Partial Differential Equations That Models Boundary Layer Flow Problems

A Large Parameter Spectral Perturbation Method for Nonlinear Systems of Partial Differential Equations That Models Boundary Layer Flow Problems

Analysis of MHD Forced Convective Flow of Variable Fluid Properties over a Saturated Porous Medium with Thermal Radiation Effect

Overlapping Multi-Domain Spectral Method for Conjugate Problems of Conduction and MHD Free Convection Flow of Nanofluids over Flat Plates

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