Abstract
In the studies of acoustic waveguides in ocean, buckling of columns with variable cross sections in applied elasticity, transverse vibrations in nonhomogeneous strings, etc., we encounter a new class of problems of the type defined on an interval [d1, d2] and defined on the interval [d2, d3] satisfying certain matching conditions at the interface point x = d2.Earlier, in Part I, we constructed a fundamental system for (L1, L2) and derived certain estimates for the same.Here, in Part II, we consider four types of boundary value problems associated with (L1, L2) and study the corresponding spectra.
Highlights
With the same notation as in Part I [4], consider the pair of Sturm-Liouville equationsP"C" ́ C"ww ;"ÐBÑC" œ -C"ß ! Ÿ B Ÿ 2ß Ð"ÑP#C# ́ C#ww ;#ÐBÑC# œ -C#ß 2 Ÿ B Ÿ "ßÐ#Ñ together with the matching conditions at the interface B œ 2 given byC"Ð2Ñ œ C#Ð2Ñ, A"C"w Ð2Ñ œ A#C#w Ð2ÑÐ$Ñ where - is a complex constant, ! 2 " and Ð;"ß ;#Ñ − P#G Ò!ß 2Ó ‚ PG# Ò2ß "ÓÞ consider the following set of boundary conditions at the end points B œ ! and B œ ".Dirichlet boundary conditions: C"Ð!Ñ œ C#Ð"Ñ œ ! Ð%ÑNeumann boundary condition: C"w Ð!Ñ œ C#w Ð"Ñ œ ! Ð&Ñ
In Part II, we consider four types of boundary value problems associated with ÐP"ß P#Ñ and study the corresponding spectra
Proof: We prove the theorem for boundary value problem (BVP) (I)
Summary
Sri Sathya Sai Institute of Higher Learning Department of Mathematics and Computer Science. Prasanthinilayam 515134 Andhra Pradesh, India (Received February, 1995; Revised February, 2000). In Part I, we constructed a fundamental system for ÐP"ß P#Ñ and derived certain estimates for the same. In Part II, we consider four types of boundary value problems associated with ÐP"ß P#Ñ and study the corresponding spectra. AMS subject classifications: 34XX, 34B05, 34B10, 34B25
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