ON FINDING SOLUTIONS OF A FAMILY OF NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEMS FOR A DIFFERENTIAL EQUATION

This paper investigates a boundary value problem belonging to a family of linear differential equations with nonlinear two-point boundary conditions. For each fixed v alue of t he f amily p arameter, t he origi­nal problem is transformed into a nonlinear two-point boundary value problem for ordinary differential equations. Since such problems are associated with nonlocal boundary value problems, including systems of hyperbolic equations, determining the conditions for their solvability is of considerable interest. The study employs the parameterization method developed by D.S. Dzhumabayev, which is based on reducing the problem to a system of nonlinear algebraic equations with respect to the parameter and solving an initial value problem for a parameter-dependent differential equation. An iterative method is used for the solution process, and the conditions ensuring its convergence are established. The proposed approach combines the numerical solution of Cauchy problems with the solution of a system of nonlinear equations. The developed algorithm is tested on specific benchmark examples, demonstrating its effectiveness. In particular, it is shown that the solution of the corresponding system converges through the iterative pro­cess for a given parameter value. The considered approach provides an efficient method for determining solutions of boundary value problems for families of linear differential equations characterized by nonlinear two-point boundary conditions. The practical efficiency of the proposed method is confirmed by the results of numerical solutions of specific problems.

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