Problem of linear conjugation on a circumference

1987 ◽  
Vol 41 (3) ◽  
pp. 195-198
Author(s):  
I. T. Khabibullin
Keyword(s):  
Linear Conjugation ◽  
Problem Of Linear Conjugation

A Problem of Linear Conjugation for Analytic Functions with Boundary Values from the Zygmund Class

2002 ◽  
Vol 9 (2) ◽  
pp. 309-324
Author(s):  
V. Kokilashvili ◽  
V. Paatashvili
Keyword(s):  
Explicit Form ◽  
Analytic Functions ◽  
Boundary Values ◽  
Linear Conjugation ◽  
Zygmund Class ◽  
Solvability Conditions ◽  
Piecewise Continuous ◽  
Problem Of Linear Conjugation

Abstract The solvability conditions are established for a problem of linear conjugation for analytic functions with boundary values from the Zygmund class 𝐿(ln+ 𝐿) α when the conjugation coefficient is piecewise-continuous in the Hölder sense. Solutions of the problem are constructed in explicit form.

scholarly journals Solution of one hypersingular integro-differential equation defined by determinants

2021 ◽  
pp. 17-28
Author(s):  
Andrei P. Shilin
Keyword(s):  
Differential Equation ◽  
Analytic Functions ◽  
Arbitrary Order ◽  
Exact Analytical Solution ◽  
Linear Equations ◽  
Linear Conjugation ◽  
Traditional Classification ◽  
Linear Differential ◽  
Problem Of Linear Conjugation

The paper provides an exact analytical solution to a hypersingular inregro-differential equation of arbitrary order. The equation is defined on a closed curve in the complex plane. A characteristic feature of the equation is that if is written using determinants. From the view of the traditional classification of the equations, it should be classified as linear equations with vatiable coefficients of a special form. The method of analytical continuation id applied. The equation is reduced to a boundary value problem of linear conjugation for analytic functions with some additional conditions. If this problem is solvable, if is required to solve two more linear differential equations in the class of analytic functions. The conditions of solvability are indicated explicitly. When these conditions are met, the solution can also be written explicitly. An example is given.

On A Problem of Linear Conjugation in the Case of Nonsmooth Lines and Some Measurable Coefficients

2002 ◽  
Vol 9 (3) ◽  
pp. 507-524
Author(s):  
E. Gordadze
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Linear Conjugation ◽  
Measurable Coefficients ◽  
Problem Of Linear Conjugation

Abstract A boundary value problem of linear conjugation is considered for more general curves than those studied previously. A condition on the coefficient is found, under which the classical results are valid for these curves.

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