Abstract
Nonlinear partial differential equations (PDEs) modelling unsteady boundary-layer flows are solved by the spectral relaxation method (SRM) and the spectral quasilinearization method (SQLM). The SRM and SQLM are Chebyshev pseudospectral based methods that have been successfully used to solve nonlinear boundary layer flow problems described by systems of ordinary differential equations. In this paper application of these methods is extended, for the first time, to systems of nonlinear PDEs that model unsteady boundary layer flow. The new extension is tested on two problems: boundary layer flow caused by an impulsively stretching plate and a coupled four-equation system that models the problem of unsteady MHD flow and mass transfer in a porous space. Numerous simulation experiments are conducted to determine the accuracy and compare the computational performance of the proposed methods against the popular Keller-box finite difference scheme which is widely accepted as being one of the ideal tools for solving nonlinear PDEs that model boundary layer flow problems. The results indicate that the methods are more efficient in terms of computational accuracy and speed compared with the Keller-box.
Highlights
Partial differential equations (PDEs) arise in a number of physical problems, such as fluid flow, heat transfer, and biological processes
We investigated the application of the spectral relaxation method (SRM) and spectral quasilinearisation method (SQLM) in the solution of unsteady boundary layer flows that are described by systems of coupled nonlinear partial differential equations
We considered the model problems of unsteady boundary layer flow caused by an impulsively stretching sheet and the unsteady MHD flow and mass transfer in a porous space
Summary
Partial differential equations (PDEs) arise in a number of physical problems, such as fluid flow, heat transfer, and biological processes. Finding solutions of the PDEs plays a crucial role in understanding the behaviour of these problems. The PDEs modelling real-life problems are nonlinear and complex to solve exactly and various analytical and numerical methods have been employed to approximate the solutions of these problems. Many researchers in fluid mechanics have focused their attention on problems involving boundary layer flows of an incompressible fluid over a stretching surface because of their substantial applications in engineering. A large and growing body of literature has investigated problems involving steady flows. Unsteady flows are mostly defined by systems of nonlinear PDEs and are considerably more difficult to solve than steady flows problems which are often simplified into system nonlinear ODEs using the so-called similarity transformations
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