Boundary Integral Equation Analyses of Singular, Potential, and Biharmonic Problems pp 89-113 | Cite as

# Solution of Nonlinear Elliptic Equations with Boundary Singularities by an Integral Equation Method

## Abstract

A boundary integral equation (BIE) formulation is presented for the numerical solution of certain two dimensional nonlinear elliptic equations subject to nonlinear boundary conditions. By applying the Kirchoff transformation, all nonlinear aspects are first transferred to the boundary of the solution domain. Then the accurate solution of problems in which there are boundary singularities is demonstrated by including the analytic nature of the singular solution in only those regions nearest the singularity. Because of this, only a minor modification of the classical nonlinear BIE is required and this results in a substantial improvement in the accuracy of the numerical results throughout the entire solution domain.

The BIE has previously been applied to *either* nonlinear *or* singular problems and so the method presently described constitutes an extension in this field.

## Keywords

Boundary Integral Equation Straight Line Segment Solution Domain Nonlinear Elliptic Equation Integral Equation Method## Preview

Unable to display preview. Download preview PDF.

## References

- 1.JASWON, M.A. and SYMM, G.T., Integral Equation Methods in Potential Theory and Elastostatics, Academic Press, London, 1977.MATHGoogle Scholar
- 2.BREBBIA, C.A. and WALKER, S., Boundary Element Techniques in Engineering, Butterworth, London, 1980.MATHGoogle Scholar
- 3.BREBBIA, C.A. (Editor), Proc. 4th Int. Conf. on Boundary Element Methods in Engineering, Springer Verlag, 1981.Google Scholar
- 4.SPIEGEL, M.R., Vector Analysis, McGraw-Hill, London, 1974.Google Scholar
- 5.SYMM, G.T., Treatment of singularities in the solution of Laplace’s Equation by an integral equation method, NPL Report NAC31, 1973.Google Scholar
- 6.FAIRWEATHER, S., RIZZ0, F.J., SHIPPY, D.J. and WU, Y.S., On the numerical solution of two dimensional potential problems by an improved boundary integral equation method, J. Comput. Phys., Vol.31(1), pp.96–111, 1979.CrossRefMATHMathSciNetGoogle Scholar
- 7.INGHAM, D.B., HEGGS, P.J. and MANZOOR, M., The numerical solution of plane potential problems by improved boundary integral equation methods, J. Comput. Phys., Vol.42(1), pp.77–98, 1981.CrossRefMATHGoogle Scholar
- 8.MOTZ, H., The treatement of singularities of partial differential equations by relaxation methods, Q. appl. Math., Vol.4(4), pp.371–377, 1946.MathSciNetGoogle Scholar
- 9.WOODS, L.C., The relaxation treatment of singular points in Poisson’s Equation, Q. Jl. Mech. appl. Math., Vol.6(2), pp.163–185, 1953.CrossRefMATHMathSciNetGoogle Scholar
- 10.INGHAM, D.B., HEGGS, P.J. and MANZOOR, M., boundary integral equation analysis of transmission line singularities, IEEE Trans. Microwave Theory and Tech., Vol.MTT-29, pp.1240–1243, 1981.CrossRefGoogle Scholar
- 11.KELMANSON, M.A., Modified integral equation solution of viscous flow near sharp corners, Computers and Fluids, Vol.11(4), pp.307–324, 1983.CrossRefMATHGoogle Scholar
- 12.XANTHIS, L.S., BERNAL, M.J.M. and ATKINSON, C., The treatment of singularities in the calculation of stress intensity factors using the boundary integral equation method, Comput. Meth. Appl. Mech. Engrg., Vol.26(3), pp.285–304, 1981.CrossRefMATHMathSciNetGoogle Scholar
- 13.KHADER, M.S., Heat conduction with temperature dependent thermal conductivity, paper 80-HT-4, National Heat Transfer Conf., ASHE, Orlando, Florida, 1980.Google Scholar
- 14.BIALECKI, R. and NOWAK, A.J., Boundary value problems in heat conduction with nonlinear material and nonlinear boundary conditions, Appl. Math. Modelling, pp.417–421, 1981.Google Scholar
- 15.INGHAM, D.B., HEGGS, P.J. and MANZOOR, M., Boundary integral equation solution of nonlinear plane potential problems, IMA J. Num. Anal., Vol.1, pp.416–426, 1981.CrossRefMathSciNetGoogle Scholar
- 16.KHADER, M.S. and HANNA, M.C, An iterative boundary numerical solution for general steady heat conduction problems, Trans. ASME J. Heat. Transfer, Vol.103, pp.26–31, 1981.CrossRefGoogle Scholar
- 17.CARSLAW, H.S. and JAEGER, J.C., Conduction of Heat in Solids, Clarendon Press, Oxford, 1959.Google Scholar
- 18.JOHNSON, L.W. and REISS, R.D., Numerical Analysis, Addison-Wesley, Reading, Mass., USA, 1982.MATHGoogle Scholar
- 19.COULSON, C.A. and BOYD.T.J.M., Electricity, 2nd Edn, Longman, London, 1979.Google Scholar
- 20.KREITH, F., Principles of Heat Transfer, Harper and Row, New York, 1976.Google Scholar
- 21.WHITEMAN, J.R. and PAPAMICHAEL, N., Numerical solution of two dimensional harmonic boundary value problems containing singularities by conformal transformation methods, Brunei University, Dept. of Mathematics Report No. TR/2, June 1971.Google Scholar
- 22.PATTERSON, T.N.L., The optimum addition of points to quadrature formulae, Math. Comp., Vol.22, pp.847–856, 1968.CrossRefMATHGoogle Scholar