scholarly journals Boundary Value Problems with Integral Gluing Conditions for Fractional-Order Mixed-Type Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
A. S. Berdyshev ◽  
E. T. Karimov ◽  
N. Akhtaeva
Keyword(s):  
Mixed Type ◽  
Differential Operators ◽  
Type Equation ◽  
Telegraph Equation ◽  
Fractional Diffusion ◽  
Tricomi Problem ◽  
Green Functions ◽  
Mixed Type Equation ◽  
Energy Integrals ◽  
Gellerstedt Problem

Analogs of the Tricomi and the Gellerstedt problems with integral gluing conditions for mixed parabolic-hyperbolic equation with parameter have been considered. The considered mixed-type equation consists of fractional diffusion and telegraph equation. The Tricomi problem is equivalently reduced to the second-kind Volterra integral equation, which is uniquely solvable. The uniqueness of the Gellerstedt problem is proven by energy integrals' method and the existence by reducing it to the ordinary differential equations. The method of Green functions and properties of integral-differential operators have been used.

scholarly journals A BOUNDARY VALUE PROBLEM WITH AN OFFSET FOR A MODEL EQUATION OF MIXED TYPE IN AN UNBOUNDED DOMAIN

2020 ◽  
pp. 31-41
Author(s):  
Р.Т. Зуннунов ◽  
Ж.А. Толибжонов
Keyword(s):  
Mixed Type ◽  
Model Equation ◽  
Type Equation ◽  
Green Functions ◽  
Horizontal Strip ◽  
Existence Of A Solution ◽  
Method Of Integral Equations ◽  
Energy Integrals ◽  
Uniqueness Of The Solution ◽  
Equation Of Mixed Type

В данной работе для уравнения смешанного типа в неограниченной области эллиптическая часть которой является горизонтальной полосой исследуется задача со смещением на характеристиках разных семейств. Единственность решения задачи доказывается методом интегралов энергии, а существование решения задачи методом функций Грина и методом интегральных уравнений. In this paper, for a mixed type equation in an unbounded region, the elliptical part of which is a horizontal strip, we study the problem with a shift on the characteristics of different families. The uniqueness of the solution of the problem is proved by the method of energy integrals, and the existence of a solution of the problem by the method of Green functions and the method of integral equations.

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