Boundary Value Problem for the Aller — Lykov Moisture Transport Generalized Equation With Concentrated Heat Capacity

Kerefov, Marat Aslanbievich; Nakhusheva, Fatima M.; Геккиева, С Х

doi:10.18287/2541-7525-2018-24-3-23-29

Cited by 3 publications

(2 citation statements)

References 4 publications

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“…In [8] a unique solvability of Dirichlet boundary value problem for the Aller-Lykov equation with constant coefficients was proved. In work [9] there was studied the Neumann boundary value problem.…”

Section: Introductionmentioning

confidence: 99%

Local and nonlocal boundary value problems for generalized Aller - Lykov equation

Gekkieva,

Kerefov,

Nakhusheva

2023

Ufimsk. Mat. Zh.

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In mathematical modelling of solid media with memory there arise equations describing a new type of wave motion, which is between the usual diffusion and classical waves. Here we mean differential equations with fractional derivatives both in time and spatial variables, which are a base for most part of mathematical models in mechanics of liquids, viscoelasticity as well as in processes in media with fractal structure and memory.In the present work we present a qualitatively new moisture transfer equation being a generalization of Aller-Lykov equation. This generalization provides an opportunity to reflect specific features of the studied objects in the nature of the equation such, namely, the structure and physical properties of the going processes, by means of introducing a fractal velocity of moisture varying.The work is devoted to studying local and nonlocal boundary value problems for inhomogeneous Aller-Lykov moisture transfer equation with variable coefficients and Riemann-Liouville fractional time derivative. For a generalized equation of Aller-Lykov type we consider initial boundary value problems with Dirichlet and Robin boundary conditions as well as nonlocal problems involving nonlocality in time in the boundary conditions. Assuming the existence of regular solutions, by the method of energy inequalities, we obtain apriori estimates in terms of Riemann-Liouville fractional derivative, which imply the uniqueness of the solutions to the considered problems as well as their stability in the right hand side and initial data.

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Section: Introductionmentioning

confidence: 99%

Local and nonlocal boundary value problems for generalized Aller - Lykov equation

Gekkieva,

Kerefov,

Nakhusheva

2023

Ufimsk. Mat. Zh.

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“…Численным, в основном разностным, методам исследования нелокальных задач посвящены работы А. В. Гулина, Н. И. Ионкина, В. А. Морозова, В. Л. Макарова, М. Х. Шханукова-Лафишева и др. [5,10]. Разностным методам решения нелокальных краевых задач для уравнений теплопроводности посвящены работы [6][7][8][9].…”

Section: Introductionunclassified

On a Сlass of Non-Local Boundary Value Problems for the Heat Equation

Нахушева,

Керефов,

Геккиева

et al. 2023

Вестник КРАУНЦ. Физико-Математические Науки

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Нелокальные краевые задачи для параболических уравнений, в том числе уравнения теплопроводности, стали объектом исследований достаточно давно. Интерес к такого рода задачам вызван необходимостью дальнейшего развития теории краевых задач со смещением (задач Нахушева), а также в связи с их многочисленными приложениями. Настоящая статья посвящена исследованию вопроса однозначной разрешимости одного класса нелокальных краевых задач для уравнения теплопроводности. Рассмотрена задача отыскания регулярного решения уравнения теплопроводности с дробной производной Римана -Лиувилля в граничных условиях. Рассмотрена задача Коши для уравнения, эквивалентного исходному уравнению, при этом доказано, что рассматриваемая краевая задача редуцируется к первой краевой задаче для уравнения теплопроводности при условии, что задача Коши имеет единственное решение в классе функций, удовлетворяющих условиям А. Н. Тихонова. При этом решение представимо в виде интегрального уравнения, содержащим функцию Барретта в ядре. Также редукцией к системе дифференциальных уравнений с дробной производной Римана – Лиувилля решается вопрос единственности и существования решения поставленной задачи, когда в условии стоят значения решения на другом конце. Полученные в работе результаты послужат основой для дальнейшего исследования нелокальных краевых задач для дифференциальных уравнений параболического типа, лежащих в основе математического моделирования процессов в системах с фрактальной структурой, а также развития теории дифференциальных уравнений дробного порядка. Non-local boundary value problems for parabolic equations, including the equations of thermal conductivity, have been the object of research for a long time. Interest in such problems is caused by the need for further development of the theory of boundary value problems with displacement (Nakhushev’s problems), as well as in connection with their numerous applications. This article is devoted to the study of the question of the unambiguous solvability of one class of nonlocal boundary value problems for the heat equation. The problem of finding a regular solution of the thermal conductivity equation with a fractional Riemann – Liouville derivative under boundary conditions is considered. The Cauchy problem for an equation equivalent to the original equation is considered, and it is proved that the boundary value problem under consideration is reduced to the first boundary value problem for the heat equation, provided that the Cauchy problem has a unique solution in the class of functions satisfying the conditions of A. N. Tikhonov. In this case, the solution is represented as an integral equation containing the Barrett function in the kernel. Also, by reducing to a system of differential equations with a fractional Riemann-Liouville derivative, the question of the uniqueness and existence of a solution to the problem is solved when the values of the solution at the other end are in the condition. The results obtained in this work will serve as a basis for further research of nonlocal boundary value problems for parabolic differential equations underlying mathematical modeling of processes in systems with fractal structure, as well as the development of the theory of fractional differential equations.

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On solution of pseudohyperbolic equation with constant coefficients

Kostikov

Romanenkov

2023

RDLUZ

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Thepaperproposesthemethodofformingtheexactsolutionofthefirstinitialboundaryvalueproblemforone-dimensionallinearpseudohyperbolicequation with constant coefficients. Toobtainthesolutiontype, themodificationofpartitionmethod (Fourier method) is used, when the type of one of the solution functional factors is consideredto be known. Atthesametime, theinitialproblemisreducedto parameterized family of Cauchy problems for ordinary differential equations. Thepaperpresentsexplicitly calculated formulas, whichspecify the solution. The qualitative research of the solution properties has been conducted. Theconditionsforcoefficientsintheformofinequalitieshave been obtained that is indicative ofboundedness and variability of the solutions. Several examples confirming the results obtained have been considered.

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Boundary Value Problem for the Aller — Lykov Moisture Transport Generalized Equation With Concentrated Heat Capacity

Cited by 3 publications

References 4 publications

Local and nonlocal boundary value problems for generalized Aller - Lykov equation

Local and nonlocal boundary value problems for generalized Aller - Lykov equation

On a Сlass of Non-Local Boundary Value Problems for the Heat Equation

On solution of pseudohyperbolic equation with constant coefficients

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