Constructive method for solving a boundary value problem for ordinary differential equations

Abstract

We suggest a method for solving a boundary value problem for ordinary differential equations with boundary conditions in the presence of state and integral constraints. The method is based on the embedding principle, which permits one to reduce the original boundary value problem to a special optimal control problem with the use of the general solution of a Fredholm integral equation of the first kind. STATEMENT OF THE PROBLEMConsider the boundary value probleṁwith the boundary conditionsthe phase constraintsand the integral constraintsHere A(t) and B(t) are given n × n and n × m matrices, respectively, with piecewise continuous entries, μ(t), t ∈ I, is a given n-dimensional vector function with piecewise continuous components, f (x, t) is an m-dimensional vector function that is jointly continuous with respect to the variables (x, t) ∈ R n × I and satisfies the conditionsand S is a given convex closed set. The function F (x, t) = (F 1 (x, t), . . . , F r (x, t)), t ∈ I, is an r-dimensional jointly continuous vector function, and γ(t) = (γ 1 (t), . . . , γ r (t)) and δ(t) = (δ 1 (t), . . . , δ r (t)), t ∈ I, are given continuous functions.149

To the solution of a boundary value problem with a parameter for an ordinary differential equations

“…This is the principal difference of the proposed method in comparison with known methods. This work is a continuation of the studies presented in [6][7][8][9][10][11][12].…”

“…The proof of Theorems 1, 2 was given in works [6,7]. Application of Theorems 1, 2 to control problems was presented in [8][9][10], while the boundary value problems for ordinary differential equations were discussed in [11,12].…”

Similar constructing method for solving nonhomogeneous mixed boundary value problem of n‐interval composite ode and its application

“…, Y i (x) is the solution of the ith interval of the boundary value problem (1), then solutions of each interval are expressed, respectively, as follows:…”

“…Some authors have studied the analytical solutions or semianalytical solutions of some boundary value problems for differential equation. Aisagaliev and Kalimoldaev put forward a method for solving boundary value problems for ODEs with boundary conditions under state constraints and integral constraints. Li and Qiang built the mathematical model of round constant pressure and round closed homogeneous radial composite reservoir that considers the impact of well‐bore storage and skin factor.…”