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A new method for solving the nonlinear second-order boundary value differential equations

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.

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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

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AMS Mathematics Subject Classification

  • 34B15
  • 49J15

Key Word and Phrases

  • ODE’s
  • measure theory
  • optimal control
  • linear programming