American Journal of Applied Mathematics and Statistics
ISSN (Print): 2328-7306 ISSN (Online): 2328-7292 Website: Editor-in-chief: Mohamed Seddeek
Open Access
Journal Browser
American Journal of Applied Mathematics and Statistics. 2016, 4(2), 37-42
DOI: 10.12691/ajams-4-2-2
Open AccessArticle

Solving the Laplace’s Equation by the FDM and BEM Using Mixed Boundary Conditions

Hassan Ghassemi1, , Saeid Panahi1 and Ahmad Reza Kohansal2

1Department of Maritime Engineering, Amirkabir University of Technology, Tehran, Iran

2Department of Shipbuilding, Faculty of Engineering, Persian Gulf University, Bousher, Iran

Pub. Date: May 07, 2016

Cite this paper:
Hassan Ghassemi, Saeid Panahi and Ahmad Reza Kohansal. Solving the Laplace’s Equation by the FDM and BEM Using Mixed Boundary Conditions. American Journal of Applied Mathematics and Statistics. 2016; 4(2):37-42. doi: 10.12691/ajams-4-2-2


This paper presents to solve the Laplace’s equation by two methods i.e. the finite difference method (FDM) and the boundary element method (BEM). The body is ellipse and boundary conditions are mixed. In the BEM, the integration domain needs to be discretized into small elements. The boundary integral equation derived using Green’s theorem by applying Green’s identity for any point in the surface. The methods are applied to examine an example of square domain with mixed boundary condition. Both types of numerical models are computed and compared with analytical solution. The results obtained agree perfectly with those obtained from exact solution.

Mixed boundary condition Boundary element Method Finite Difference Method

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit


[1]  Katsikadelis, J. T., Boundary element, theory and application, Elsevier publication, 2002.
[2]  Sauter S.A., Schwab C., “Boundary Element Method,” Springer-Verlag Berlin, 2011.
[3]  Selvadurai A.P.S., “Partial Differential Equations in Mechanics 1: Fundamentals, Laplace’s Equation, Diffusion Equation, Wave Equation,” Springer-Verlag Berlin Heidelberg, First edition, 2000.
[4]  Gilbarg D., Trudinger N.S., “Elliptic Partial Differential Equations of Second Order,” Springer-Verlag Berlin Heidelberg New York, 2001.
[5]  Pozrikidis C., “A Practical Guide to Boundary Element Methods with the Software Library BEMLIB,” Chapman & Hall/CRC, 2002.
[6]  Beer G., Smith I., Duenser C., “The Boundary Element Method with Programming,” Springer Wien Newyork, 2008.
[7]  Aliabadi M. H., Wen P. H., “Boundary Element Methods in Engineering and Sciences,” Imperial College Press, 2011.
[8]  Masataka T., Thomas A. Cruse, “Boundary Element Methods in Applied Mechanics,” Pergamon press, 1988.
[9]  Heinz Antes, “A Short Course on Boundary Element Methods,” lecture notes, Technische Universitat Braunschweig, 2010.
[10]  Ramachandran P. A., “Boundary Element Methods in Transport Phenomena,” Washington University, 1994.
[11]  Carrier G.F., Pearson C.E., “Partial Differential Equations: Theory and Technique”, 2nd edition, New York Academic press, 1988.
[12]  Banerjee P.K., “The Boundary Element Methods in Engineering,” McGraw-Hill, London, 1994.
[13]  Ghassemi, H., Bakhtiari M., Boundary element methods, Amirkabir University of Technology publication, 2016. (in Persian).
[14]  Qian Z., Fu C.L., Li Z.P., “Two regularization methods for a Cauchy problem for the Laplace equation,” Mathematical Analysis and Applications, May 2007.
[15]  Morales M., Rodolfo Diaz R.A., Herrera W. J., “Solutions of Laplace’s equation with simple boundary conditions, and their applications for capacitors with multiple symmetries,” Journal of electrostatics, 78 (2015) 31-45.
[16]  Lesnic D., Elliott L. and Ingham D.B., “An iterative boundary element method for solving numerically the Cauchy problem for the Laplace equation,” Engineering Analysis with Boundary Elements, Vol-20, Issue 2, September 1997.
[17]  Ren Q., Chan C.L., “Analytical evaluation of the BEM singular integrals for 3D Laplace and Stokes flow equations using coordinate transformation,” Engineering Analysis with Boundary Elemens, Vol-53, pp1-8, April 2015.
[18]  Li Z.C., Zhang L.P., Wei Y., Lee M.G., Chiang J.Y., “Boundary methods for Dirichlet problems of Laplace’s equation in elliptic domains with elliptic holes,” Engineering Analysis with Boundary Elements, Vol-61, December 2015.
[19]  Lee M.G., Li Z.C., Huang H.T., Chiang J.Y., “Neumann problems of Laplace’s equation in circular domains with circular holes by methods of field equations,” Engineering Analysis with Boundary Elements, Vol-51, Feb 2015.
[20]  Yang S., “The Neumann problem of Laplace’s equation in semionvex domains,” Nonlinear Analysis: Theory, Methods & Applications, Vol-133, March 2016.
[21]  Gao X.W., “Evaluation of regular and singular domain integrals with boundary-only discretization-theory and Fortran code,” Journal of Computational and Applied Mathematics, 175 (2005) 265-290.
[22]  Ghassemi H., Kohansal A.R., “Numerical evaluation of various levels of singular integrals arising in BEM and its application in hydrofoil analysis”, Applied Mathematics and Computation. 213 (2009) 277-289.
[23]  Jun L., Beer G., Meek J., “Efficient evaluation of integrals of order 1/r, 1/r2, 1/r3 using Gauss method,” Eng. Anal. 2 (1985) 118-123.