Formal adjoint operators and asymptotic formula for solutions of autonomous linear integral equations

We study a certain conjugation problem for a pair of elliptic pseudo-differential equations with homogeneous symbols inside and outside of a plane sector. The solution is sought in corresponding Sobolev–Slobodetskii spaces. Using the wave factorization concept for elliptic symbols, we derive a general solution of the conjugation problem. Adding some complementary conditions, we obtain a system of linear integral equations. If the symbols are homogeneous, then we can apply the Mellin transform to such a system to reduce it to a system of linear algebraic equations with respect to unknown functions.

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On Pseudo-Resolvents of Linear Integral Equations in General Analysis

Annals of Mathematics ◽

10.2307/2007221 ◽

1920 ◽

Vol 21 (4) ◽

pp. 323

Author(s):

T. H. Hildebrandt

Keyword(s):

Integral Equations ◽

General Analysis ◽

Linear Integral ◽

Linear Integral Equations

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On the location of the Weyl circles

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500020163 ◽

1981 ◽

Vol 88 (3-4) ◽

pp. 345-356 ◽

Cited By ~ 36

Author(s):

F. V. Atkinson

Keyword(s):

Integral Equations ◽

Complex Plane ◽

Riccati Equations ◽

Sturm Liouville ◽

Linear Integral ◽

Linear Integral Equations

SynopsisThe paper deals with explicit estimates concerning certain circles in the complex plane which were associated with Sturm–Liouville problems by H. Weyl. By the use of Riccati equations instead of linear integral equations, improvements are obtained for results of Everitt and Halvorsen concerning the behaviour of the Titchmarsh–Weyl m-coefficient.

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Integral equations and relations for Lamé functions and ellipsoidal wave functions

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100042638 ◽

1968 ◽

Vol 64 (1) ◽

pp. 113-126 ◽

Cited By ~ 7

Author(s):

B. D. Sleeman

Keyword(s):

Wave Function ◽

Green’S Function ◽

Integral Equations ◽

Green's Function ◽

Function Theory ◽

Natural Generalization ◽

Wave Functions ◽

Linear Integral ◽

First Time ◽

Linear Integral Equations

AbstractNon-linear integral equations and relations, whose nuclei in all cases is the ‘potential’ Green's function, satisfied by Lamé polynomials and Lamé functions of the second kind are discussed. For these functions certain techniques of analysis are described and these find their natural generalization in ellipsoidal wave-function theory. Here similar integral equations are constructed for ellipsoidal wave functions of the first and third kinds, the nucleus in each case now being the ‘free space’ Green's function. The presence of ellipsoidal wave functions of the second kind is noted for the first time. Certain possible generalizations of the techniques and ideas involved in this paper are also discussed.

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