scholarly journals Stability and Convergence of Difference Schemes Approximating a Two-Parameter Nonlocal Boundary Value Problem for Time-Fractional Diffusion Equation

2015 ◽  
Vol 26 (2) ◽  
pp. 252-272 ◽  
Author(s):  
Anatoly A. Alikhanov
Keyword(s):  
Diffusion Equation ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Fractional Diffusion ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Nonlocal Boundary Value Problem ◽  
Time Fractional Diffusion Equation ◽  
Two Parameter

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.

scholarly journals A Difference Method for Solving the Steklov Nonlocal Boundary Value Problem of Second Kind for the Time-Fractional Diffusion Equation

2017 ◽  
Vol 17 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Anatoly A. Alikhanov
Keyword(s):  
Diffusion Equation ◽  
A Priori Estimates ◽  
Nonlocal Boundary ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Difference Schemes ◽  
Stability And Convergence ◽  
Test Problems ◽  
Difference Problem ◽  
Time Fractional Diffusion Equation

AbstractWe consider difference schemes for the time-fractional diffusion equation with variable coefficients and nonlocal boundary conditions containing real parameters α, β and γ. By the method of energy inequalities, for the solution of the difference problem, we obtain a priori estimates, which imply the stability and convergence of these difference schemes. The obtained results are supported by the numerical calculations carried out for some test problems.

scholarly journals Well-Posedness of the Boundary Value Problem for Parabolic Equations in Difference Analogues of Spaces of Smooth Functions

2007 ◽  
Vol 2007 ◽  
pp. 1-16 ◽  
Author(s):  
A. Ashyralyev
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Approximate Solutions ◽  
Difference Schemes ◽  
Stability Estimates ◽  
Well Posedness ◽  
Nonlocal Boundary Value Problem ◽  
Smooth Functions ◽  
Accuracy Difference

The first and second orders of accuracy difference schemes for the approximate solutions of the nonlocal boundary value problemv′(t)+Av(t)=f(t)(0≤t≤1),v(0)=v(λ)+μ,0<λ≤1, for differential equation in an arbitrary Banach spaceEwith the strongly positive operatorAare considered. The well-posedness of these difference schemes in difference analogues of spaces of smooth functions is established. In applications, the coercive stability estimates for the solutions of difference schemes for the approximate solutions of the nonlocal boundary value problem for parabolic equation are obtained.

scholarly journals Modified Crank-Nicolson Difference Schemes for Nonlocal Boundary Value Problem for the Schrödinger Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-15 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Ali Sirma
Keyword(s):  
Schrödinger Equation ◽  
Boundary Value Problem ◽  
Schrodinger Equation ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Approximate Solutions ◽  
Difference Schemes ◽  
Nonlocal Boundary Value Problem ◽  
The Stability ◽  
Crank Nicolson

The nonlocal boundary value problem for Schrödinger equation in a Hilbert space is considered. The second-order of accuracy -modified Crank-Nicolson difference schemes for the approximate solutions of this nonlocal boundary value problem are presented. The stability of these difference schemes is established. A numerical method is proposed for solving a one-dimensional nonlocal boundary value problem for the Schrödinger equation with Dirichlet boundary condition. A procedure of modified Gauss elimination method is used for solving these difference schemes. The method is illustrated by numerical examples.

scholarly journals LOCAL DISPLACEMENT PROBLEM FOR EQUATION OF FRACTIONAL DIFFUSION

2019 ◽  
pp. 28-34
Author(s):  
Ф.М. Лосанова
Keyword(s):  
Boundary Condition ◽  
Diffusion Equation ◽  
Boundary Value Problem ◽  
Linear Combination ◽  
Existence And Uniqueness ◽  
Nonlocal Boundary ◽  
Fractional Diffusion ◽  
Nonlocal Boundary Value Problem ◽  
Displacement Problem ◽  
Uniqueness Of A Solution

Для уравнения дробной диффузии исследуется нелокальная краевая задача первого рода, нелокальность которой проявляется в том, что в граничном условии задается линейная комбинация значений искомой функции. В работе доказана теорема о существовании и единственности решения поставленной задачи. For the fractional diffusion equation, we study a nonlocal boundary value problem of the first kind. The problem nonlocality is manifested in the fact that a linear combination of the values of the desired function is specified in the boundary condition. The theorem on the existence and uniqueness of a solution to the problem is proved.

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