scholarly journals A note on the well-posedness of the nonlocal boundary value problem for elliptic-parabolic equations

2011 ◽  
pp. 1-16 ◽  
Author(s):  
A. Ashyralyev
Keyword(s):  
Boundary Value Problem ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Well Posedness ◽  
Nonlocal Boundary Value Problem

scholarly journals Well-Posedness of Nonlocal Parabolic Differential Problems with Dependent Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Asker Hanalyev
Keyword(s):  
Differential Equation ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Continuous Functions ◽  
Parabolic Differential Equation ◽  
Well Posedness ◽  
Nonlocal Boundary Value Problem ◽  
Hölder Condition ◽  
Linear Positive Operator

The nonlocal boundary value problem for the parabolic differential equationv'(t)+A(t)v(t)=f(t)  (0≤t≤T),  v(0)=v(λ)+φ,  0<λ≤Tin an arbitrary Banach spaceEwith the dependent linear positive operatorA(t)is investigated. The well-posedness of this problem is established in Banach spacesC0β,γ(Eα-β)of allEα-β-valued continuous functionsφ(t)on[0,T]satisfying a Hölder condition with a weight(t+τ)γ. New Schauder type exact estimates in Hölder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are established.

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 Coercive solvability of the nonlocal boundary value problem for parabolic differential equations

2001 ◽  
Vol 6 (1) ◽  
pp. 53-61 ◽  
Author(s):  
A. Ashyralyev ◽  
A. Hanalyev ◽  
P. E. Sobolevskii
Keyword(s):  
Banach Space ◽  
Boundary Value Problem ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Stability Estimates ◽  
Nonlocal Boundary Value Problem ◽  
Parabolic Differential Equations ◽  
Coercive Solvability ◽  
Coercive Stability

The nonlocal boundary value problem,v′(t)+Av(t)=f(t)(0≤t≤1),v(0)=v(λ)+μ(0<λ≤1), in an arbitrary Banach spaceEwith the strongly positive operatorA, is considered. The coercive stability estimates in Hölder norms for the solution of this problem are proved. The exact Schauder's estimates in Hölder norms of solutions of the boundary value problem on the range{0≤t≤1,xℝ n}for2m-order multidimensional parabolic equations are obtaine.

scholarly journals A remark on elliptic differential equations on manifold

2020 ◽  
Vol 99 (3) ◽  
pp. 75-85
Author(s):  
A. Ashyralyev ◽  
◽  
Y. Sozen ◽  
F. Hezenci ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Elliptic Type ◽  
Well Posedness ◽  
Nonlocal Boundary Value Problem ◽  
Elliptic Boundary Value

For elliptic boundary value problems of nonlocal type in Euclidean space, the well posedness has been studied by several authors and it has been well understood. On the other hand, such kind of problems on manifolds have not been studied yet. Present article considers differential equations on smooth closed manifolds. It establishes the well posedness of nonlocal boundary value problems of elliptic type, namely Neumann-Bitsadze-Samarskii type nonlocal boundary value problem on manifolds and also DirichletBitsadze-Samarskii type nonlocal boundary value problem on manifolds, in H¨older spaces. In addition, in various H¨older norms, it establishes new coercivity inequalities for solutions of such elliptic nonlocal type boundary value problems on smooth manifolds.

scholarly journals On well-posedness of the nonlocal boundary value problem for parabolic difference equations

2004 ◽  
Vol 2004 (2) ◽  
pp. 273-286 ◽  
Author(s):  
A. Ashyralyev ◽  
I. Karatay ◽  
P. E. Sobolevskii
Keyword(s):  
Boundary Value Problem ◽  
Difference Equations ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Stability Estimates ◽  
Well Posedness ◽  
Nonlocal Boundary Value Problem ◽  
The Difference ◽  
Type Boundary ◽  
The Stability

We consider the nonlocal boundary value problem for difference equations(uk−uk−1)/τ+Auk=φk,1≤k≤N,Nτ=1, andu0=u[λ/τ]+φ,0<λ≤1, in an arbitrary Banach spaceEwith the strongly positive operatorA. The well-posedness of this nonlocal boundary value problem for difference equations in various Banach spaces is studied. In applications, the stability and coercive stability estimates in Hölder norms for the solutions of the difference scheme of the mixed-type boundary value problems for the parabolic equations are obtained. Some results of numerical experiments are given.

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