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Journal of Applied Sciences
  Year: 2017 | Volume: 17 | Issue: 10 | Page No.: 491-501
DOI: 10.3923/jas.2017.491.501
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A Proposed Adaptive Inverse Multiquadric Shape Parameter Applied with the Dual Reciprocity BEM to Nonlinear and Coupled PDE

Sayan Kaennakham and Nissaya Chuathong

Background and Objective: The effect of shape parameter is known to play a crucial role in determining the final results of a collocation-based numerical method. This is also the case for boundary element method where radial basis functions are used to collocate the non-homogeneous term. However, finding an optimal shape parameter is known not to be simple particularly when dealing with complex PDE problems. This investigation was carried out focusing on three purposes. Firstly, it is to propose a new form of shape parameter contained in the inverse-multiquadric RBF that behaves both linearly and exponentially. Secondly, it is to integrate the effect of the local phenomena of the problem at hand into the mechanism of the proposed shape via. the local Reynolds number (Re). Thirdly, the methodology of dual reciprocity boundary element method is studied, applied and computationally implemented to one of the most challenging types of PDEs, ‘Burgers equations’, famous for its rich in transient, couple and nonlinear phenomena. Materials and Methods: The study began with gathering mostly-used and proposed forms of shape parameter and analyzing the general aspects in terms of their affectiveness. A new form of shape parameter that was hoped to alleviate the drawbacks commonly found when using those previously proposed shape parameter. The investigation then moved on applying the methodology of the dual reciprocity boundary element method in conjunction with the newly formulated shape parameter to one of the most challenging forms of PDE namely Burgers’ equations. The overall effectiveness of the was evaluated by comparing the results against both their analytical solutions and other numerical works when available in literature. Results: Main findings of this work are as follows. Firstly, the method has successfully been applied to Burgers’ equations using inverse multiquadric radial basis function at relatively high Reynolds number. Secondly, it is found from all the results obtained in this work that the proposed shape parameter can outperform the fixed ones and certainly deserves further investigation. Lastly, when compared with other numerical works, the accuracy lies in an acceptable level and moreover, gets better when the problem become advective dominated, at high Reynolds number. Conclusion: It is found in this work that with its ability to adapt itself locally, the proposed choice of variable shape provides reasonable solutions while requiring only the upper and lower bounding values. This makes choosing the suitable shape much more effective and simpler, particularly when the flow reaches the stage of instability, high Reynolds number.
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How to cite this article:

Sayan Kaennakham and Nissaya Chuathong, 2017. A Proposed Adaptive Inverse Multiquadric Shape Parameter Applied with the Dual Reciprocity BEM to Nonlinear and Coupled PDE. Journal of Applied Sciences, 17: 491-501.

DOI: 10.3923/jas.2017.491.501






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