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

  • Derek B. Ingham
  • Mark A. Kelmanson
Part of the Lecture Notes in Engineering book series (LNENG, volume 7)


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.


Boundary Integral Equation Straight Line Segment Solution Domain Nonlinear Elliptic Equation Integral Equation Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    JASWON, M.A. and SYMM, G.T., Integral Equation Methods in Potential Theory and Elastostatics, Academic Press, London, 1977.zbMATHGoogle Scholar
  2. 2.
    BREBBIA, C.A. and WALKER, S., Boundary Element Techniques in Engineering, Butterworth, London, 1980.zbMATHGoogle Scholar
  3. 3.
    BREBBIA, C.A. (Editor), Proc. 4th Int. Conf. on Boundary Element Methods in Engineering, Springer Verlag, 1981.Google Scholar
  4. 4.
    SPIEGEL, M.R., Vector Analysis, McGraw-Hill, London, 1974.Google Scholar
  5. 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. 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.CrossRefzbMATHMathSciNetGoogle Scholar
  7. 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.CrossRefzbMATHGoogle Scholar
  8. 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. 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.CrossRefzbMATHMathSciNetGoogle Scholar
  10. 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. 11.
    KELMANSON, M.A., Modified integral equation solution of viscous flow near sharp corners, Computers and Fluids, Vol.11(4), pp.307–324, 1983.CrossRefzbMATHGoogle Scholar
  12. 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.CrossRefzbMATHMathSciNetGoogle Scholar
  13. 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. 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. 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. 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. 17.
    CARSLAW, H.S. and JAEGER, J.C., Conduction of Heat in Solids, Clarendon Press, Oxford, 1959.Google Scholar
  18. 18.
    JOHNSON, L.W. and REISS, R.D., Numerical Analysis, Addison-Wesley, Reading, Mass., USA, 1982.zbMATHGoogle Scholar
  19. 19.
    COULSON, C.A. and BOYD.T.J.M., Electricity, 2nd Edn, Longman, London, 1979.Google Scholar
  20. 20.
    KREITH, F., Principles of Heat Transfer, Harper and Row, New York, 1976.Google Scholar
  21. 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. 22.
    PATTERSON, T.N.L., The optimum addition of points to quadrature formulae, Math. Comp., Vol.22, pp.847–856, 1968.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin, Heidelberg 1984

Authors and Affiliations

  • Derek B. Ingham
    • 1
  • Mark A. Kelmanson
    • 1
  1. 1.Department of Applied Mathematical StudiesUniversity of LeedsLeedsEngland

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