Abstract
Collocation with piecewise polynomial functions is developed as a method for solving two-point boundary value problems. Convergence is shown for a general class of linear problems and a rather broad class of nonlinear problems. Some computational examples are presented to illustrate the wide applicability and efficiency of the procedure.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahlberg, J. H., Nilson, E. N., Walsh, J. L.: The theory of splines and their applications. New York: Academic Press 1967
Bailey, P. B., Shampine, L. F., Waltman, P. E.: Nonlinear two-point boundary value problems. New York: Academic Press 1968.
Bellman, R. E., Kalaba, R. E.: Quasilinearization and nonlinear boundary-value problems. New York: Elsevier 1965.
Bramble, J. H., Hubbard, B. E.: On a finite difference analogue of an elliptic boundary problem which is neither diagonally dominant nor of nonnegative type. J. Math. and Phys.43, 117–132 (1964).
Carrier, G. F.: Singular perturbation theory and geophysics. SIAM Review12, 175–193 (1970).
Ciarlet, P. G., Schultz, M. H., Varga, R. S.: Numerical methods of high-order accuracy for nonlinear boundary value problems—I. one dimensional problem. Numer. Math.9, 394–430 (1967).
Collatz, L.: Functional analysis and numerical mathematics, trans. by H. J. Oser. New York: Academic Press 1966.
—: The numerical treatment of differential equations, 3rd ed. Berlin-Göttingen-Heidelberg: Springer 1960.
Conte, S. D.: The numerical solution of linear boundary value problems. SIAM Review8, 309–321 (1966).
Davis, H. T.: Introduction to nonlinear differential and integral equations. New York: Dover 1962.
Davis, P. J., Rabinowitz, P.: Numerical integration. Waltham, Mass.: Blaisdell 1967.
De Boor, C.: The method of projections as applied to the numerical solution of two point boundary value problems using cubic splines. Dissertation, University of Michigan, Ann Arbor, Michigan, 1966.
Henrici, P.: Discrete variable methods in ordinary differential equations. New York: Wiley 1964.
Kantorovich, L. V., Akilov, G. P.: Functional analysis in normed spaces. New York: Pergamon 1964.
Keller, H. B.: Accurate difference methods for linear ordinary differential systems subject to linear constraints. SIAM J. Numer. Anal.6, 8–30 (1969).
—: Numerical methods for two-point boundary value problems. Waltham, Mass: Blaisdell 1968.
Krasnosel'skii, M. A.: Topological methods in the theory of nonlinear integral equations. Oxford, England: Pergamon 1964.
Lees, M.: Discrete methods for nonlinear two-point boundary value problems, in Numerical solution of partial differential equations, ed. by, J. H. Bramble, New York: Academic Press 1966.
Luttman, F. W., Rivlin, T. J.: Some numerical experiments in the theory of polynomial interpolation. IBM J. Res. Develop.9, 187–191 (1965).
Naimark, M. A.: Linear differential equations, Part I. English trans. by E. R. Dawson, ed. by W. N. Everitt. New York: Ungar 1967.
Ortega, J. M., Rheinboldt, W. C.: Iterative solution of nonlinear equations in several variables. New York: Academic Press 1970.
Rivlin, T. J.: An introduction to the approximation of functions. Waltham, Mass: Blaisdell 1969.
Roark, A. L., Shampine, L. F.: On the numerical solution of a linear two-point boundary value problem, Sandia Laboratory Technical Memorandum SC-TM-67-588. Albuquerque, New Mexico, 1967.
Shampine, L. F.: Boundary value problems for ordinary differential equations. SIAM J. Numer. Anal.5, 219–242 (1968).
—: Boundary value problems for ordinary differential equations II: Patch bases and monotone methods. SIAM J. Numer. Anal.6, 414–431 (1969).
Shindler, A. A.: Some theorems of the general theory of approximate methods of analysis and their application to the collocation moments and Galerkin methods. Sibirskii Mathematicheskii Zhurnal8, 415–432 (1967).
—: Rate of convergence of the enriched collocation method for ordinary differential equations. Sibirskii Mathematicheskii Zhurnal10, 229–233 (1969).
Timoshenko, S., Woinowsky-Krieger, S.: Theory of plates and shells. New York: McGraw-Hill 1959.
Urabe, M.: Numerical solution of multi-point boundary value problems in Chebyshev series — theory of the method. Numer. Math.9, 341–366 (1967).
— Reiter, A.: Numerical computation of nonlinear forced oscillations by Galerkin's procedure. J. Math. Ann. and Appl.14, 107–140 (1966).
Vainniko, G. M.: On the stability and convergence of the collocation method. Differentsial'nye Uravneniya1, 244–254 (1965).
—: On convergence of the collocation method for nonlinear differential equations. USSR Comp. Math. and Math. Phys.6, 35–42 (1966).
Wilkinson, J. H.: The algebraic eigenvalue problem. London: Oxford University Press 1965.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Russell, R.D., Shampine, L.F. A collocation method for boundary value problems. Numer. Math. 19, 1–28 (1972). https://doi.org/10.1007/BF01395926
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01395926