On Gel’fand’s method of chasing for solving multipoint boundary value problems.

*(English)*Zbl 0598.65062
Differential equations and their applications, Equadiff 6, Proc. 6th Int. Conf., Brno/Czech. 1985, Lect. Notes Math. 1192, 267-274 (1986).

[For the entire collection see Zbl 0595.00009.]

We report the formulation of a practical shooting method, namely the method of chasing for an nth order ordinary linear differential equation \((1)\quad x^{(n)}+\sum^{n}_{i=1}p_ i(t)x^{(n-i)}=f(t)\) subject to linearly independent multipoint boundary conditions \((2)\quad \sum^{n-1}_{k=0}c_{ik}x^{(k)}(a_ i)=A_ i,\) \(1\leq i\leq n\) where \(a_ 1\leq a_ 2\leq...\leq a_ n\) \((a_ 1<a_ n)\). This method is originally developed for second order differential equations by Gel’fand and Lokutsiyevskii and first appeared in English literature only recently [I. S. Berezin and N. P. Zhidkov, Computing Methods. II (1965; Zbl 0122.129), Russian original (1959; Zbl 0096.094)]. T. Y. Na [Computational methods in engineering boundary value problems (1979; Zbl 0456.76002)] has briefly described the method and given different formulations for the different particular cases of (1), (2). The general systems derived here include the systems given by Na as special cases. The power of the method is illustrated by solving the known problem studied by J. F. Holt [Commun. Assoc. Comput. Machin. 7, 366-373 (1964; Zbl 0123.118)].

We report the formulation of a practical shooting method, namely the method of chasing for an nth order ordinary linear differential equation \((1)\quad x^{(n)}+\sum^{n}_{i=1}p_ i(t)x^{(n-i)}=f(t)\) subject to linearly independent multipoint boundary conditions \((2)\quad \sum^{n-1}_{k=0}c_{ik}x^{(k)}(a_ i)=A_ i,\) \(1\leq i\leq n\) where \(a_ 1\leq a_ 2\leq...\leq a_ n\) \((a_ 1<a_ n)\). This method is originally developed for second order differential equations by Gel’fand and Lokutsiyevskii and first appeared in English literature only recently [I. S. Berezin and N. P. Zhidkov, Computing Methods. II (1965; Zbl 0122.129), Russian original (1959; Zbl 0096.094)]. T. Y. Na [Computational methods in engineering boundary value problems (1979; Zbl 0456.76002)] has briefly described the method and given different formulations for the different particular cases of (1), (2). The general systems derived here include the systems given by Na as special cases. The power of the method is illustrated by solving the known problem studied by J. F. Holt [Commun. Assoc. Comput. Machin. 7, 366-373 (1964; Zbl 0123.118)].