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Discrete cubic spline method for second-order boundary value problems. (English) Zbl 1305.65173

A cubic spline method is developed to solve approximatively the boundary value problem \[ y''(x) = f(x)y(x)+g(x),\qquad a\leq x\leq b, \]
\[ y(a)=\alpha, \qquad y(b)=\beta, \] on the uniform mesh \( a=x_{0}<x_{1}<x_{2}< \dots <x_{n}=b \) with the step \(p=x_{i}-x_{i-1}.\)
Using the parameter \(h\in (0,p]\) and the central finite differences of first and second order, the system of cubic splines \( S_{i}(x)\) of degree 3 is obtained. The approximate system for the initial problem has a tridiagonal form and admits the unique solution if \( f(x)\equiv f_{0}>0.\) The error of approximation is of order 4 if \(h=p/\sqrt{2}\) and it is of second order otherwise.

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L12 Finite difference and finite volume methods for ordinary differential equations
65D07 Numerical computation using splines
34B05 Linear boundary value problems for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
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