# Computing Robin Problem on Unbounded Simply Connected Domain via an Integral Equation with the Generalized Neumann Kernel

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## Abstract

A Robin problem is a mixed problem with a linear combination of Dirichlet and Neumann D-N conditions. The aim of this paper are presents a new boundary integral equation BIE method for the solution of unbounded Robin boundary value problem BVP in the simply connected domain. The method show how to reformulate the Robin boundary value problem BVP as Riemann-Hilbert problem RHP which lead to the system of integral equation, and the related differential equations are also created that give rise to unique solutions. Numerical results on several tests regions by the Nyström method NM with the trapezoidal rule TR are presented to clarify the solution technique for the Robin problem when the boundaries are sufficiently smooth.

#### Keywords

Robin problem, Riemann-Hilbert problem, Integral equation, Generalized Neumann kernel, Simply connected region.#### References

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[2] W. Fang, Z. Suxing. Numerical recovery of Robin boundary from boundary measurements for the Laplace equation.Journal of Computational and Applied Mathematics, 224:573-580, 2009.

[3] K. Gustafson, A. Takehisa. The third boundary condition was it Robin’s?The Mathematical Intelligencer, 20:63-71, 1998.

[4] I. N. Sneddon.Mixed Boundary Value Problems In Potential Theory. North-Holland. 1966

[5] M. M. S. Nasser, A. H. M. Murid, M. Ismail, E. M. A. Alejaily. Boundary integral equations with the generalized Neumann kernel for Laplace’s equation in multiply connected regions. Appl. Math. Comput, 217:4710-4727, 2011.

[6] S. A. A. Alhatemi, A. H. M. Murid, M. M. S. Nasser. A boundary integral equation with the generalized Neumann kernel for a mixed boundary value problem in unbounded multiply connected regions.Boundary Value Problems, 1: 1-17, 2013.

[7] M. M. S. Nasser. Numerical conformal mapping via a boundary integral equation with the generalized Neumann kernel. SIAM Journal on Scientific Computing, 31:1695-1715, 2009.

[8] R. Wegmann, A. H. M. Murid, M. M. S. Nasser. The Riemann-Hilbert problem and the generalized Neumann kernel.Journal of Computational and Applied Mathematics, 182:388-415, 2005.

[9] R. Wegmann, M. M. S. Nasser.The Riemann-Hilbert problem and the generalized Neumann kernel on multiply connected regions.Journal of Computational and Applied Mathematics, 214:36-57, 2008.

[10] T. Petrila. Complex value boundary element method for some mixed boundary value problems. Studia Univ, Babes-Bolyai, Informatica, 44:37-42, 1999.

[11] R. M. M. Mattheij, S. W Rienstra, J. H. M. ten Thije Boonkkamp. Partial Differential Equations Modeling, Analysis, Computation.SIAM. 2005

[12] S. Salsa.Partial Differential Equations in Action From Modeling to Theory.Springer Science and Business Media. 2008

[13] M. M. S. Nasser. The Riemann-Hilbert problem and the generalized Neumann kernel on unbounded multiply connected regions. The University Researcher Journal, 20: 47-60, 2009.

[14] F. D. Gakhov. Boundary Value Problem. Oxford Pergamon Press. 1966

[15] S. H. H. Al-Shatri, A. H. M. Murid, M. Ismail, M. I. Muminov. Solving Robin problems in multiply connected regions via an integral equation with the generalized Neumann kernel.Boundary Value Problem.2016, 1-23,2016.

[16] S. H. H. Al-Shatri, A. H. M. Murid, M. Ismail. Solving Robin problems in bounded doubly connected regions via an integral equation with the generalized Neumann kernel. AIP Conference Proceedings.1750,1, p. 030004) 2016.

[17] K. E. Atkinson. The Numerical Solution of Integral Equations of the Second Kind. Cambridge: Cambridge University Press. 1997

[18] D. Gaier. Konstruktive Methoden der Konformen Abbildung.Berlin-Göttingen-Heidelberg. 1964

[19] M. Abramowitz, I. E. Stegun. Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. Courier Corporation. 1964

[20] J. Helsing, R. Ojala. On the evaluation of layer potentials close to their sources. J. Comput. Phys,227:2899-2921, 2008.

[2] W. Fang, Z. Suxing. Numerical recovery of Robin boundary from boundary measurements for the Laplace equation.Journal of Computational and Applied Mathematics, 224:573-580, 2009.

[3] K. Gustafson, A. Takehisa. The third boundary condition was it Robin’s?The Mathematical Intelligencer, 20:63-71, 1998.

[4] I. N. Sneddon.Mixed Boundary Value Problems In Potential Theory. North-Holland. 1966

[5] M. M. S. Nasser, A. H. M. Murid, M. Ismail, E. M. A. Alejaily. Boundary integral equations with the generalized Neumann kernel for Laplace’s equation in multiply connected regions. Appl. Math. Comput, 217:4710-4727, 2011.

[6] S. A. A. Alhatemi, A. H. M. Murid, M. M. S. Nasser. A boundary integral equation with the generalized Neumann kernel for a mixed boundary value problem in unbounded multiply connected regions.Boundary Value Problems, 1: 1-17, 2013.

[7] M. M. S. Nasser. Numerical conformal mapping via a boundary integral equation with the generalized Neumann kernel. SIAM Journal on Scientific Computing, 31:1695-1715, 2009.

[8] R. Wegmann, A. H. M. Murid, M. M. S. Nasser. The Riemann-Hilbert problem and the generalized Neumann kernel.Journal of Computational and Applied Mathematics, 182:388-415, 2005.

[9] R. Wegmann, M. M. S. Nasser.The Riemann-Hilbert problem and the generalized Neumann kernel on multiply connected regions.Journal of Computational and Applied Mathematics, 214:36-57, 2008.

[10] T. Petrila. Complex value boundary element method for some mixed boundary value problems. Studia Univ, Babes-Bolyai, Informatica, 44:37-42, 1999.

[11] R. M. M. Mattheij, S. W Rienstra, J. H. M. ten Thije Boonkkamp. Partial Differential Equations Modeling, Analysis, Computation.SIAM. 2005

[12] S. Salsa.Partial Differential Equations in Action From Modeling to Theory.Springer Science and Business Media. 2008

[13] M. M. S. Nasser. The Riemann-Hilbert problem and the generalized Neumann kernel on unbounded multiply connected regions. The University Researcher Journal, 20: 47-60, 2009.

[14] F. D. Gakhov. Boundary Value Problem. Oxford Pergamon Press. 1966

[15] S. H. H. Al-Shatri, A. H. M. Murid, M. Ismail, M. I. Muminov. Solving Robin problems in multiply connected regions via an integral equation with the generalized Neumann kernel.Boundary Value Problem.2016, 1-23,2016.

[16] S. H. H. Al-Shatri, A. H. M. Murid, M. Ismail. Solving Robin problems in bounded doubly connected regions via an integral equation with the generalized Neumann kernel. AIP Conference Proceedings.1750,1, p. 030004) 2016.

[17] K. E. Atkinson. The Numerical Solution of Integral Equations of the Second Kind. Cambridge: Cambridge University Press. 1997

[18] D. Gaier. Konstruktive Methoden der Konformen Abbildung.Berlin-Göttingen-Heidelberg. 1964

[19] M. Abramowitz, I. E. Stegun. Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. Courier Corporation. 1964

[20] J. Helsing, R. Ojala. On the evaluation of layer potentials close to their sources. J. Comput. Phys,227:2899-2921, 2008.

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