Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Back-and-forth shooting method for solving two-point boundary-value problems

Abstract

The paper proposes a special iterative method for a nonlinear TPBVP of the form\(\dot x\)(t)=f(t, x(t),p(t)),\(\dot p\)(t)=g(t, x(t),p(t)), subject toh(x(0),p(0))=0,e(x(T),p(T))=0. Certain stability properties of the above differential equations are taken into consideration in the method, so that the integration directions associated with these equations respectively are opposite to each other, in contrast with the conventional shooting methods. Via an embedding and a Riccati-type transformation, the TPBVP is reduced to consecutive initial-value problems of ordinary differential equations. A preliminary numerical test is given by a simple example originating in an optimal control problem.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Falb, P. L., andDeJong, J. L.,Some Successive Approximation Methods in Control and Oscillation Theory, Academic Press, New York, New York, 1969.

  2. 2.

    Miele, A., Naqvi, S., Levy, A. V., andIyer, R. R.,Numerical Solution of Nonlinear Equations and Nonlinear Two-Point Boundary-Value Problems, Advances in Control Systems, Vol. 8, Edited by C. T. Leondes, Academic Press, New York, New York, 1971.

  3. 3.

    Roberts, S. M., andShipman, J. S.,Two-Point Boundary Value Problems: Shooting Methods, American Elsevier Publishing Company, New York, New York, 1972.

  4. 4.

    Miele, A., andIyer, R. R.,General Technique for Solving Nonlinear, Two-Point Boundary-Value Problems Via the Method of Particular Solutions, Journal of Optimization Theory and Applications, Vol. 5, No. 5, 1970.

  5. 5.

    Keller, H. B.,Shooting and Embedding for Two-Point Boundary Value Problems, Journal of Mathematical Analysis and Applications, Vol. 36, No. 3, 1971.

  6. 6.

    Mataušek, M. R.,Direct Shooting Method for the Solution of Boundary-Value Problems, Journal of Optimization Theory and Applications, Vol. 12, No. 2, 1973.

  7. 7.

    Lastman, G. J.,Obtaining Starting Values for the Shooting Method Solution of a Class of Two-Point Boundary-Value Problems, Journal of Optimization Theory and Applications, Vol. 14, No. 3, 1974.

  8. 8.

    Georganas, N. D., andChatterjee, A.,Invariant Imbedding and the Continuation Method: A Comparison, International Journal of Systems Science, Vol. 6, No. 3, 1975.

  9. 9.

    Meyer, G. H.,Initial Value Methods for Boundary Value Problems, Academic Press, New York, New York, 1973.

  10. 10.

    Mufti, I. H., Chow, C. K., andStock, F. T.,Solution of Ill-Conditioned Linear Two-Point Boundary Value Problems by the Riccati Transformation, SIAM Review, Vol. 11, No. 4, 1969.

Download references

Author information

Additional information

Communicated by S. M. Roberts

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Orava, P.J., Lautala, P.A.J. Back-and-forth shooting method for solving two-point boundary-value problems. J Optim Theory Appl 18, 485–498 (1976). https://doi.org/10.1007/BF00932657

Download citation

Key Words

  • Two-point boundary-value problems
  • shooting methods
  • invariant embedding
  • Riccati transformation
  • iterative numerical methods