scholarly journals Inner boundary value problem with an integral condition for fractional diffusion equation

2021 ◽  
pp. 24-29
Author(s):  
Ф.М. Лосанова
Keyword(s):  
Diffusion Equation ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Integral Condition ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Integral Conditions ◽  
Interior Boundary ◽  
Uniqueness Of The Solution ◽  
Fractional Differentiation Operator

В данной работе рассматривается нелокальная внутреннекраевая задача для уравнения дробной диффузии с оператором дробного дифференцирования в смысле Римана-Лиувилля с интегральными условиями. Исследуемая задача эквиваленто сведена к системе двух интегральных уравнений Вольтерра второго рода. Доказана теорема существования и единственности решения поставленной задачи. In this paper, we consider a nonlocal interior boundary value problem for the fractional diffusion equation with a fractional differentiation operator in the sense of Riemann-Liouville with integral conditions. The problem under study is equivalently reduced to a system of two Volterra integral equations of the second kind. The theorem of existence and uniqueness of the solution of the posed problem is proved.

Inner boundary value problem for fractional diffusion equation

2020 ◽  
Vol 20 (3) ◽  
pp. 14-18
Author(s):  
F.M. Losanova ◽  
Keyword(s):  
Diffusion Equation ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Existence And Uniqueness Theorem ◽  
Nonlocal Boundary Value Problem ◽  
Linear Combinations

In this paper, we prove the existence and uniqueness theorem for a nonlocal boundary value problem for the fractional diffusion equation with boundary conditions presented in the form of linear combinations.

On the equivalence of two representations of Green's function of first boundary value problem for fractional-order diffusion equation

2020 ◽  
Vol 20 (2) ◽  
pp. 12-15
Author(s):  
F.G. Khushtova ◽  
Keyword(s):  
Diffusion Equation ◽  
Boundary Value Problem ◽  
Green’S Function ◽  
Fractional Order ◽  
Green's Function ◽  
Boundary Value ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation

In this paper, the equivalence of two forms of representations of the Green's function of the first boundary value problem for the fractional diffusion equation is proved.

scholarly journals Locally one dimensional scheme of the Dirichlet boundary value problem for fractional diffusion equation with space caputo fractional derivative

2016 ◽  
Author(s):  
А.К. Баззаев
Keyword(s):  
Diffusion Equation ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Dirichlet Boundary Value Problem ◽  
One Dimensional

В работе рассматриваются локально-одномерные схемы для уравнения диффузии дробного порядка с дробной производной в младших членах с граничными условиями первого рода. С помощью принципа максимума получена априорная оценка для решения разностной задачи в равномерной метрике. Доказаны устойчивость и сходимость построенных локально-одномерных схем.

scholarly journals Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation

2012 ◽  
Vol 15 (1) ◽  
Author(s):  
Yuri Luchko
Keyword(s):  
Diffusion Equation ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Initial Boundary Value Problems ◽  
The Maximum Principle ◽  
Time Fractional Diffusion Equation ◽  
Initial Boundary Value

AbstractIn this paper, some initial-boundary-value problems for the time-fractional diffusion equation are first considered in open bounded n-dimensional domains. In particular, the maximum principle well-known for the PDEs of elliptic and parabolic types is extended for the time-fractional diffusion equation. In its turn, the maximum principle is used to show the uniqueness of solution to the initial-boundary-value problems for the time-fractional diffusion equation. The generalized solution in the sense of Vladimirov is then constructed in form of a Fourier series with respect to the eigenfunctions of a certain Sturm-Liouville eigenvalue problem. For the onedimensional time-fractional diffusion equation $$(D_t^\alpha u)(t) = \frac{\partial } {{\partial x}}\left( {p(x)\frac{{\partial u}} {{\partial x}}} \right) - q(x)u + F(x,t), x \in (0,l), t \in (0,T)$$ the generalized solution to the initial-boundary-value problem with Dirichlet boundary conditions is shown to be a solution in the classical sense. Properties of this solution are investigated including its smoothness and asymptotics for some special cases of the source function.

scholarly journals Maximum principle and its application to multi-index Hadamard fractional diffusion equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Xueyan Ren ◽  
Guotao Wang ◽  
Zhanbing Bai ◽  
A. A. El-Deeb
Keyword(s):  
Maximum Principle ◽  
Diffusion Equation ◽  
Classical Solution ◽  
Boundary Value ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Multi Index ◽  
Initial Boundary Value

AbstractThis study establishes some new maximum principle which will help to investigate an IBVP for multi-index Hadamard fractional diffusion equation. With the help of the new maximum principle, this paper ensures that the focused multi-index Hadamard fractional diffusion equation possesses at most one classical solution and that the solution depends continuously on its initial boundary value conditions.

scholarly journals Fractional diffusion equation described by the Atangana-Baleanu fractional derivative and its approximate solution

2021 ◽  
Vol 2 (1) ◽  
pp. 60-75
Author(s):  
Ndolane Sene
Keyword(s):  
Approximate Solution ◽  
Diffusion Equation ◽  
Fractional Derivative ◽  
Integral Method ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Double Integral ◽  
Uniqueness Of The Solution ◽  
The Impact ◽  
The Heat Balance

In this paper, we propose the approximate solution of the fractional diffusion equation described by a non-singular fractional derivative. We use the Atangana-Baleanu-Caputo fractional derivative in our studies. The integral balance methods as the heat balance integral method introduced by Goodman and the double integral method developed by Hristov have been used for getting the approximate solution. In this paper, the existence and uniqueness of the solution of the fractional diffusion equation have been provided. We analyze the impact of the fractional operator in the diffusion process. We represent graphically the approximate solution of the fractional diffusion equation.

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