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Global existence and uniqueness for the Kummer transformation problem subject to non-Cauchy data. (English) Zbl 0743.34043

This paper is a sequel of the authors’ previous two papers [Asymptotic and computational analysis. Conference in honor of Frank W. J. Olver’s 65th birthday, Proc. Int. Symp., Winnipeg/Can. 1989, Lect. Notes Pure Appl. Math. 124, 707-722 (1990; Zbl 0704.34006); Math Comput. 55, No. 192, 591-612 (1990; Zbl 0676.65041)] on evaluation zeros of solutions of second order differential equations. In this paper, the authors establish the global existence and uniqueness for the non-Cauchy problem associated with Kummer third order nonlinear differential in 1834, namely (1) \(Q(x)x'{}^ 2=q(t)-{1\over 2}(x,t)\), where \[ (x,t)={x'''\over x'}-{3\over 3}{x''{}^ 2\over x'{}^ 2}, (2) \] which plays a central role in the theory of transformations between any two linear homogeneous ordinary differential equations like (3) \(d^ 2y/dt^ 2+q(t)y=0\), \(t\in j\equiv(a,b)\), (3’) \(d^ 2Y/dT^ 2+Q(T)Y=0\), \(t\in J\equiv (A,B)\), where \(q\in C^ 0(j)\), \(Q\in C^ 0(J)\), \(-\infty\leq a<b\leq\infty\), \(- \infty\leq A<B\leq+\infty\). The authors concentrate on the special case of (1) obtained when \(Q=1\), i.e. (4) \(\alpha'{}^ 2=q(t)-{1\over 2}(\alpha,t)\), whose solution is the function \(\alpha(t)\) three times differentiable and with \(\alpha'\neq 0\) on \(j\). The authors also demonstrate the connections between the Kummer transformation problem and the Liouville-Green approximation in the oscillatory case while they illustrate applications to Bessel functions, central dispersions and numerical analysis at the end of the paper.
Reviewer: F.M.Ragab (Cairo)

MSC:

34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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