Abstract
The study in this paper is a numerical integration of second-order three-point boundary value problems under two imposed nonlocal boundary conditions at , , and in a general setting, where . We construct a two-stage Lie-group shooting method for finding unknown initial conditions, which are obtained through an iterative solution of derived algebraic equations in terms of a weighting factor . The best is selected by matching the target with a minimal discrepancy. Numerical examples are examined to confirm that the new approach has high efficiency and accuracy with a fast speed of convergence. Even for multiple solutions, the present method is also effective to find them.
Highlights
A Two-Stage LGSM for Three-Point BVPs of Second-Order ODEsThe study in this paper is a numerical integration of second-order three-point boundary value problems under two imposed nonlocal boundary conditions at t t0, t ξ, and t t1 in a general setting, where t0 < ξ < t1
Nonlinear ordinary differential equations ODEs described a majority of engineering problems
We find that the initial slopes A are, respectively, 0.61602 in exact and 0.61611 in numerical, showing that the present TSLGSM can provide very accurate estimation of unknown initial condition of slope
Summary
The study in this paper is a numerical integration of second-order three-point boundary value problems under two imposed nonlocal boundary conditions at t t0, t ξ, and t t1 in a general setting, where t0 < ξ < t1. We construct a two-stage Lie-group shooting method for finding unknown initial conditions, which are obtained through an iterative solution of derived algebraic equations in terms of a weighting factor r ∈ 0, 1. The best r is selected by matching the target with a minimal discrepancy. Numerical examples are examined to confirm that the new approach has high efficiency and accuracy with a fast speed of convergence. Even for multiple solutions, the present method is effective to find them.
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