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ARTICLE

A LARGE PARAMETER SPECTRAL PERTURBATION METHOD FOR NONLINEAR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS THAT MODELS BOUNDARY LAYER FLOW PROBLEMS

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a School of Mathematics,Statistics and Computer Sciences, University of KwaZulu-Natal, Pietermaritzburg, Private Bag X01 Scottsville-3209, South Africa
b DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), Private Bag 3, Wits 2050, Johannesburg, South Africa
c Department of Mathematics, Amity University, Kolkata, Newtown- 700135,West Bengal, India
∗ University of Swaziland, Private Bag 4, Kwaluseni, Matsapha M201, Swaziland
† Corresponding author. Email: sabya.mondal.2007@gmail.com

Frontiers in Heat and Mass Transfer 2017, 9, 1-13. https://doi.org/10.5098/hmt.9.36

Abstract

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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