Abstract
In this paper we use measure theory to solve a wide range of second-order boundary value ordinary differential equations. First, we transform the problem to a first order system of ordinary differential equations (ODE’s) and then define an optimization problem related to it. The new problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures; the optimal measure is then approximated by a finite combination of atomic measures and the problem converted approximatly to a finite-dimensional linear programming problem. The solution to this problem is used to construct the approximate solution of the original problem. Finally we get the error functionalE (we define in this paper) for the approximate solution of the ODE’s problems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Effati and A. V. Kamyad,Solution of Boundary Value Problems for Linear Second Order ODE’s by Using Measure Theory, J. Analysis,6 (1998), 139–149.
G. Anichini and G. Conti,Existence of solutions of a boundary value problem through the solution map of a linearized type problem, Rend. Sem. Mat. Univ. Politec. Torino,48 (1990), 149–159.
L. H. Erbe,Existence of solutions to boundary value problems for ordinary differential equations, Nonlinear Anal.,6 (1982), 1155–1162.
R. E. Gaines and J. Mawhin,Ordinary differential equations with nonlinear boundary conditions, J. Differential Equations,26 (1977), 200–222.
H. Gingold,Uniqueness of solutions of boundary value problems of systems of ordinary differential equations, Pacific J. Math.75 (1987), 107–136.
—,Uniqueness criteria for second order nonlinear boundary value problems, J. Math. Anal. Appl.,73 (1980), 392–410.
A. Granas, R. B. Guenther, and J. W. Lee,On a theorem of S. Bernstein, Pacific J. Math.,74 (1978), 67–82.
—,Some general existence principles in the Caratheodory theory of nonlinear differential systems, J. Math. Pures appl.,70 (1991), 153–196.
D. Hankerson and J. Henderson,Optimality for boundary value problems for Lipschitz equations, J. Differential Equations,77 (1989), 392–404.
G. A. Harris,On multiple solutions of a nonlinear Neumann problem, J. Differential Equations,95 (1952), 75–104.
R. Kannan and J. Locker,On a class of nonlinear boundary value problems, J. Differential Equations,26 (1977), 1–8.
A. C. Lazer and D. E. Leach,On a nonlinear two-point boundary value problem, J. Math. Appl.,26 (1969), 20–27.
J. Mawhin,Topological degree methods in nonlinear boundary value problems, CBMS Regional Conference Series in Mathematics 40, American Mathematical Society, Providence, RI, 1979.
W. V. Petryshyn,Solvability of various boundary value problems for the equation x′’ = f(t,x,x′,x′’) −y, Pacific J. Math.,122 (1986), 169–195.
J. Saranen and S. Seikkala,Solution of a nonlinear two-point boundary value problem with Neumann-type boundary data, J. Math. Anal. Appl.,135 (1988), 691–701.
J. Troch,On the interval of disconjugacy of linear autonomous differential equations, Siam J. Math. Anal.,12 (1981), 78–89.
W. Huaizhong and L. Yong,Neumann boundary value problems for second-order ordinary differential equations across resonance, Siam J. Control and optimization,33 (1995), 1312–1325.
F. Treves,Topological vector spaces, distributions and kernels (New York and London: Academic Press (1967)).
J. E. Rubio,Control and optimization; the linear treatment of nonlinear problems (Manchester, U. K., Manchester University Press (1986)).
G. Choquet,Lectures on analysis, Benjamin, New York (1969).
S. I. Gass,Linear programming, McGraw-Hill, New York (1985).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Effati, S., Kamyad, A.V. & Farahi, M.H. A new method for solving the nonlinear second-order boundary value differential equations. Korean J. Comput. & Appl. Math 7, 183–193 (2000). https://doi.org/10.1007/BF03009936
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF03009936