On a uniqueness of the solution of a problem with an analogue of a condition of Frankl on the internal characteristic for the equation of the mixed type

2021 ◽  
Vol 65 (2) ◽  
Keyword(s):  
Mixed Type ◽  
Internal Characteristic ◽  
Uniqueness Of The Solution

A boundary-value problem for the equation of mixed type with generalized operators of fractional differentiation in the boundary conditions

2016 ◽  
Vol 22 (1) ◽  
Author(s):  
Oleg A. Repin ◽  
Svetlana K. Kumykova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Mixed Type ◽  
Original Problem ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Fractional Differentiation ◽  
Uniqueness Of Solutions ◽  
Uniqueness Of The Solution ◽  
Equation Of Mixed Type

AbstractThe paper is devoted to the study of a boundary-value problem for an equation of mixed type with generalized operators of fractional differentiation in boundary conditions. We prove uniqueness of solutions under some restrictions on the known functions and on the different orders of the operators of generalized fractional differentiation appearing in the boundary conditions. Existence of solutions is proved by reduction to a Fredholm equation of the second kind, for which solvability follows from the uniqueness of the solution of our original problem.

scholarly journals THEORETICAL AND COMPUTATIONAL RESULTS FOR MIXED TYPE VOLTERRA–FREDHOLM FRACTIONAL INTEGRAL EQUATIONS

Fractals ◽  
2021 ◽  
Author(s):  
ROHUL AMIN ◽  
HUSSAM ALRABAIAH ◽  
IBRAHIM MAHARIQ ◽  
ANWAR ZEB
Keyword(s):  
Integral Equations ◽  
Mixed Type ◽  
Fractional Integral ◽  
Haar Wavelet ◽  
Algebraic Equations ◽  
Elimination Algorithm ◽  
Collocation Technique ◽  
Uniqueness Of The Solution ◽  
Gauss Elimination ◽  
Mean Square Errors

In this paper, we develop a numerical method for the solutions of mixed type Volterra–Fredholm fractional integral equations (FIEs). The proposed algorithm is based on Haar wavelet collocation technique (HWCT). Under certain conditions, we prove the existence and uniqueness of the solution. Also, some stability results are given of Hyers–Ulam (H–U) type. With the help of the HWCT, the considered problem is transformed into a system of algebraic equations which is then solved for the required results by using Gauss elimination algorithm. Some numerical examples for convergence of the proposed technique are taken from the literature. Maximum absolute and root mean square errors are calculated for different collocation points (CPs). The results show that the HWCT is an effective method for solving FIEs. The convergence rate for different CPS is also calculated, which is nearly equal to 2.

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.

scholarly journals On the inverse problem for a heat-like equation

1987 ◽  
Vol 1 (2) ◽  
pp. 81-97 ◽  
Author(s):  
Igor Malyshev
Keyword(s):  
Inverse Problem ◽  
Boundary Data ◽  
Mixed Type ◽  
Parabolic Type ◽  
Time Dependent ◽  
Nonlinear Volterra Integral Equation ◽  
Parabolic Case ◽  
Special Cases ◽  
Uniqueness Of The Solution ◽  
Explicit Formulae

Using the integral representation of the solution of the boundary value problem for the equation with one time-dependent coefficient at the highest space-derivative three inverse problems are solved. Depending on the property of the coefficient we consider cases when the equation is of the parabolic type and two special cases of the degenerate/mixed type. In the parabolic case the corresponding inverse problem is reduced to the nonlinear Volterra integral equation for which the uniqueness of the solution is proved. For the special cases explicit formulae are derived. Both “minimal” and overspecified boundary data are considered.

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