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# 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.

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## Author information

Communicated by S. M. Roberts

## Rights and permissions

Reprints and Permissions

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