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W. Wasow (1965)
Asymptotic expansions for ordinary differential equations
Differential Equations, Vol. 39, No. 2, 2003, pp. 182–191. Translated from Differentsial'nye Uravneniya, Vol. 39, No. 2, 2003, pp. 171–179. Original Russian Text Copyright c 2003 by Bobochko. ORDINARY DIFFERENTIAL EQUATIONS An Orr{Sommerfeld Equation with a First-Order Di erential Turning Point V. N. Bobochko Kiev National University, Kiev, Ukraine Received December 5, 2001 STATEMENT OF THE PROBLEM Consider the di erential equation 5 (4) 3 000 2 00 L y(x;") " y (x;")+ " a (x)y (x;")+ " a (x)y (x;") " 3 2 (1) + a (x)y (x;")+ a (x)y(x;")= h(x) 1 0 as " ! +0 for x 2 I =[0; 1]. We assume that a (x);h(x) 2 C [I ];i =0; 1; 2; 3;a (x)= xa ~(x); a ~(x) > 0;x 2 I: (2) i 1 Since the factor x corresponding to the turning point x = 0 multiplies the rst derivative, we refer to this point as a rst-order di erential turning point for Eq. (1). The aim of the present paper is to construct a uniform asymptotic expansion of the solution of the nonhomogeneous equation (1) on the entire closed interval [0; 1] including the turning point x =0. The most important results
Differential Equations – Springer Journals
Published: Oct 5, 2004
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