Abstract
An integral representation for the solution of elliptic equations of the form Δnu - P(r2)u=0, developed by Gilbert, is used to construct approximate solutions for problems of this type. Properties of the G-function, needed in the integral representation, are discussed and a numerical scheme for its computation is given. The approximate G-function is used to represent the solution and minimization techniques are used to satisfy the boundary conditions. A discussion of the practical usefulness of methods of this type is given.
This research was supported in part by the Air Force Office of Scientific Research through grant AFOSR-71-2205.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
R. H. Bartels and G. H. Golub, Chebyshev solution to an overdetermined system, Comm. ACM 11(1968), pp. 428–430.
S. Bergman and J. G. Herriot, Application of the method of the kernel function for solving boundary value problems, Num. Math. 3(1961), pp. 209–225.
S. Bergman and J. G. Herriot, Numerical solution of boundary value problems by the method of integral operators, Num. Math. 7(1965), pp. 42–65.
J. R. Cannon, The numerical solution of the Dirichlet problem for Laplace's equation by linear programming, SIAM J. Appl. Math. 12(1964), pp. 233–237.
J. R. Cannon and M. M. Cecchi, The numerical solution of some biharmonic problems by mathematical programming techniques, SIAM J. Numer. Anal. 3(1966), pp. 451–466.
R. P. Gilbert, Integral operator methods for approximating solutions of Dirichlet problems, Proceedings of the Conference on "Numerische Methoden der Approximationstheorie, Oberwolfach, 1969.
M. K. Jain and R. D. Sharma, Cubature method for the solution of the characteristic initial value problem uxy=f(x,y,u, ux,uy), J. Aust. Math. Soc. 8(1968), pp. 355–368.
R. H. Moore, A Runge-Kutta procedure for the Goursat problem in hyperbolic partial differential equations, Arch. Rat. Mech. Anal. 7(1961), pp. 37–63.
M. Protter and H. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, N.J. 1967.
P. Rabinowitz, Applications of linear programming to numerical analysis, SIAM Rev. 10(1968), pp. 121–159.
N. L. Schryer, Constructive approximation of solutions to linear elliptic boundary value problems, SIAM J. Numer. Anal. 9(1972), pp. 546–572.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1974 Springer-Verlag
About this paper
Cite this paper
Gilbert, R.P., Linz, P. (1974). The numerical solution of some elliptic boundary value problems by integral operator methods. In: Colton, D.L., Gilbert, R.P. (eds) Constructive and Computational Methods for Differential and Integral Equations. Lecture Notes in Mathematics, vol 430. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0066271
Download citation
DOI: https://doi.org/10.1007/BFb0066271
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-07021-4
Online ISBN: 978-3-540-37302-5
eBook Packages: Springer Book Archive
