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A Predictor-Corrector Method for Solving Boundary Layer Equations based on Bézier Curves
  • Vahid Ahmadi Kalkhorani,
  • Mohammad Mohammadi Aghdam
Vahid Ahmadi Kalkhorani
The Ohio State University

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Mohammad Mohammadi Aghdam
Amirkabir University of Technology
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Abstract

In this paper, a new numerical method is proposed for solving nonlinear boundary value problems (BVPs) based on Bézier curves. The well-known Falkner-Skan equation which is a third-order nonlinear BVP is considered to examine the efficiency of the presented method in terms of accuracy, convergence and stability. The presented hybrid technique takes advantage of the Method of Adjoint together with a novel predictor-collector multi-step integration procedure based on the Bernstein Polynomials and Bézier curves. In this method, first the governing equation is rewritten as a set of first-order differential equations, and the semi-infinite domain is mapped into a unit interval using appropriate coordinate transformation. The missing initial value is then determined by employing an appropriate iterative algorithm using the Method of Adjoint. Finally, the resulting initial value problem is solved using the new multi-step predictor-corrector scheme. Results of the presented method are compared with those appeared in the literature which show excellent agreement while having higher efficiency. Moreover, the accuracy and efficiency of the method are further examined by comparing predictions for the Falkner-Skan equation with five well-known multistep methods in the literature. Finally, from the stability point of view, the study also revealed that the new technique is more stable in comparison with other prominent methods.