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 A novel series method for fractional diffusion equation within Caputo fractional derivative

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 695-699 ◽  
Author(s):  
Sheng-Ping Yan ◽  
Wei-Ping Zhong ◽  
Xiao-Jun Yang
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Series Solution ◽  
Expansion Method ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Series Method ◽  
Time Fractional Diffusion Equation ◽  
Series Expansion Method

In this paper, we suggest the series expansion method for finding the series solution for the time-fractional diffusion equation involving Caputo fractional derivative.

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.

Sensitivity of the pressure distribution to the fractional order α in the fractional diffusion equation

2015 ◽  
Vol 93 (1) ◽  
pp. 18-36 ◽  
Author(s):  
Nadeem A. Malik ◽  
R.A. Ghanam ◽  
S. Al-Homidan
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Pressure Distribution ◽  
Fractional Diffusion ◽  
Linear Increase ◽  
Fractional Diffusion Equation ◽  
One Dimensional ◽  
Rock Formations ◽  
Rate Of Increase ◽  
Porous Reservoir

In reservoir engineering, an oil reservoir is commonly modeled using Darcy’s diffusion equation for a porous medium. In this work we propose a fractional diffusion equation to model the pressure distribution, p(x, t), of fluid in a horizontal one-dimensional homogeneous porous reservoir of finite length, L, and uniform thickness. A chief concern in this work is to examine the sensitivity of the pressure distribution, p(x, t), to different forms of pseudo-diffusivity, K, including cases when it depends upon the order of the fractional derivative (α), 0 ≤ α < 1 (e.g., K ∝ (1 – α)), which may be more realistic for some types of rock formations. In all cases the systems show a near-linear increase in the pressure difference P(x, t) = (p(x, t) – pi)/pi in the reservoir for large times, where pi = p(x, t = 0). For x/L < 0.4, the rate of increase of P with time increases with α, but there is a crossover at x/L = 0.4 and this trend reverses for x/L > 0.4. When K = 10k (k is the conventional permeability when α = 0), the solutions are almost independent of α, and when K = 0.1k the rate of increase in P depends upon α. This effect is enhanced when K = (1 – α)k; furthermore, in this case towards the closed end of the reservoir the pressure distribution remains practically undisturbed as α → 1. These results show that the pressure distribution in a porous reservoir is very sensitive to the dependence of the pseudo-diffusivity on the order of the fractional derivative, α.

scholarly journals Existence of a unique weak solution to a non-autonomous time-fractional diffusion equation with space-dependent variable order

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Karel Van Bockstal
Keyword(s):  
Weak Solution ◽  
Diffusion Equation ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Differential Operators ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Variable Order ◽  
Unique Weak Solution ◽  
Bounded Lipschitz Domain

AbstractIn this contribution, we investigate an initial-boundary value problem for a fractional diffusion equation with Caputo fractional derivative of space-dependent variable order where the coefficients are dependent on spatial and time variables. We consider a bounded Lipschitz domain and a homogeneous Dirichlet boundary condition. Variable-order fractional differential operators originate in anomalous diffusion modelling. Using the strongly positive definiteness of the governing kernel, we establish the existence of a unique weak solution in $u\in \operatorname{L}^{\infty } ((0,T),\operatorname{H}^{1}_{0}( \Omega ) )$ u ∈ L ∞ ( ( 0 , T ) , H 0 1 ( Ω ) ) to the problem if the initial data belongs to $\operatorname{H}^{1}_{0}(\Omega )$ H 0 1 ( Ω ) . We show that the solution belongs to $\operatorname{C} ([0,T],{\operatorname{H}^{1}_{0}(\Omega )}^{*} )$ C ( [ 0 , T ] , H 0 1 ( Ω ) ∗ ) in the case of a Caputo fractional derivative of constant order. We generalise a fundamental identity for integro-differential operators of the form $\frac{\mathrm{d}}{\mathrm{d}t} (k\ast v)(t)$ d d t ( k ∗ v ) ( t ) to a convolution kernel that is also space-dependent and employ this result when searching for more regular solutions. We also discuss the situation that the domain consists of separated subdomains.

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.

scholarly journals Local exact controllability of the diffusion equation in one dimension

2003 ◽  
Vol 2003 (14) ◽  
pp. 793-811 ◽  
Author(s):  
Marius Beceanu
Keyword(s):  
Diffusion Equation ◽  
Boundary Value Problem ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Exact Controllability ◽  
Null Controllability ◽  
One Dimension ◽  
Dirichlet Boundary Value Problem ◽  
Distributed Controls ◽  
Exact Null Controllability

This paper establishes the local exact null controllability of the diffusion equation in one dimension using distributed controls in the case of the Dirichlet boundary value problem. Most of the techniques used in the course of the proof are borrowed from Barbu (2002).

Sign in / Sign up

Export Citation Format

Share Document