Abstract
This paper introduces two novel numerical algorithms for the efficient solution of coupled systems of nonlinear boundary value problems. The methods are benchmarked against existing methods by finding dual solutions of the highly nonlinear system of equations that model the flow of a viscoelastic liquid of Oldroyd-B type in a channel of infinite extent. The methods discussed here are the spectral relaxation method and spectral quasi-linearisation method. To verify the accuracy and efficiency of the proposed methods a comparative evaluation of the performance of the methods against established numerical techniques is given.
Highlights
Exact solutions to a wide class of problems in engineering and science are generally available only for a limited range of problems
In addition to the classical numerical methods, such as those based on finite differences, finite elements, and finite volume techniques, there is currently a wide variety of methods for nonlinear equations, such as, among others, linearisation methods [1,2,3,4] and the transform methods of Fokas [5,6,7]
This paper introduces two novel techniques based on a combination of linearisation techniques and spectral methods and that allow for simple and straightforward integration of systems of ordinary differential equations on finite and infinite domains
Summary
Exact solutions to a wide class of problems in engineering and science are generally available only for a limited range of problems. For this reason the quest for new techniques and the improvement of existing techniques for finding solutions of nonlinear equations are an ongoing challenge in engineering and science. In addition to the classical numerical methods, such as those based on finite differences, finite elements, and finite volume techniques, there is currently a wide variety of methods for nonlinear equations, such as, among others, linearisation methods [1,2,3,4] and the transform methods of Fokas [5,6,7]. We present an overview of these techniques and provide a comparative evaluation of the two methods against results in the literature
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