The system of equations for mixed BVP with three dirichlet boundary conditions and one neumann boundary condition

Nur Syaza Mohd Yusop

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Type :	article
Subject :	Q Science (General)
ISSN :	1312-885X
Main Author :	Nur Syaza Mohd Yusop
Additional Authors :	Nurul Akmal Mohamed
Title :	The system of equations for mixed BVP with three dirichlet boundary conditions and one neumann boundary condition

Year of Publication :

2017

Abstract :

In this paper, a system of equation for mixed Boundary Value Problem (BVP) on a square shape domain is proposed. We apply Boundary Element Method (BEM) to form a system of equations of two dimensional potential problem which satisfies Laplace equation. The mixed BVP will be reduced to Boundary Integral Equation (BIE) by using direct method which involve Green second identity representation formula. Boundary of the square shape domain is discretized into four linear boundary elements. Each of element is prescribed with specific BC value where Neumann BC is prescribed on one element of the boundary and Dirichlet BC on the other three elements of the boundary. Then, linear interpolation is used on the discretized element. However, there can be two values of Neumann BC at a corner node due to discontinuous outward normal. While, there is only one unknown value for Dirichlet BC at each node. Therefore, our first results for this considered domain yields to the underdetermined system.Hence, we apply formulation presented by Paris and Canas (1997) which applied by them on Dirichlet BVP only. Therefore, we extend the proposed method to handle the discontinuous flux problem for mixed BVP. Finally, we end up with a system of equations.

References

[1] A. Ali and C. Rajakumar, The Boundary Element Method: Applications in Sound and Vibration, A.A. Balkema, Netherlands, 2005, 1-25. [2] G. Beer, I. Smith and C. Duenser, The Boundary Element Method with Programming, Springer Wien New York, Germany, 2008. https://doi.org/10.1007/978-3-211-71576-5 [3] L. Gaul, M. Kogl and M. Wagner, Boundary Element Methods for Engineers and Scientists, Springer, New York, 2003.https://doi.org/10.1007/978-3-662-05136-8 [4] W.S. Hall, The Boundary Element Method, Kluwer Academic, United Kingdom, 1994. https://doi.org/10.1007/978-94-011-0784-6 [5] P. Hunter and A. Pullan, FEM/BEM Notes, University of Auckland, New Zealand, 2003. [6] N. A. Mohamed, Numerical Solution and Spectrum of Boundary-Domain Integral Equations, Ph.D. Thesis, Brunel University, 2013. [7] N. A. Mohamed., N. F. Mohamed., N. H. Mohamed and M. R. M. Yusof,Numerical Solution of Dirichlet Boundary-Domain Integro-Differential Equation with Less Number of Collocation Points, Applied Mathematical Sciences, 10 (2016), no. 50, 2459-2469. https://doi.org/10.12988/ams.2016.6381 [8] N. A. Mohamed., N. F. Ibrahim., M. R. M. Yusof., N. F. Mohamed and N.H. Mohamed, Implementations of Boundary–Domain Integro-Differential Equation For Dirichlet BVP With Variable Coefficient, Jurnal Teknologi, 78 (2016), no. 6–5, 71–77. [9] F. Paris and J. Canas, Boundary Element Method Fundamentals and Applications, Oxford University, United States, 1997.

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