scholarly journals A Note on the Parabolic Differential and Difference Equations

2007 ◽  
Vol 2007 ◽  
pp. 1-16 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Yasar Sozen ◽  
Pavel E. Sobolevskii
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Difference Scheme ◽  
Positive Operator ◽  
Difference Equations ◽  
Well Posedness ◽  
Holder Space ◽  
Strongly Positive Operator ◽  
Ill Posed ◽  
General Banach Space

The differential equationu'(t)+Au(t)=f(t)(−∞<t<∞)in a general Banach spaceEwith the strongly positive operatorAis ill-posed in the Banach spaceC(E)=C(ℝ,E)with norm‖ϕ‖C(E)=sup−∞<t<∞‖ϕ(t)‖E. In the present paper, the well-posedness of this equation in the Hölder spaceCα(E)=Cα(ℝ,E)with norm‖ϕ‖Cα(E)=sup−∞<t<∞‖ϕ(t)‖E+sup−∞<t<t+s<∞(‖ϕ(t+s)−ϕ(t)‖E/sα),0<α<1, is established. The almost coercivity inequality for solutions of the Rothe difference scheme inC(ℝτ,E)spaces is proved. The well-posedness of this difference scheme inCα(ℝτ,E)spaces is obtained.

scholarly journals Well-posedness of the difference schemes of the high order of accuracy for elliptic equations

2006 ◽  
Vol 2006 ◽  
pp. 1-12 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Pavel E. Sobolevskiĭ
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Continuous Functions ◽  
High Order ◽  
Difference Schemes ◽  
Well Posedness ◽  
Smooth Functions ◽  
Order Of Accuracy ◽  
High Order Of Accuracy ◽  
The Difference

It is well known the differential equation−u″(t)+Au(t)=f(t)(−∞<t<∞)in a general Banach spaceEwith the positive operatorAis ill-posed in the Banach spaceC(E)=C((−∞,∞),E)of the bounded continuous functionsϕ(t)defined on the whole real line with norm‖ϕ‖C(E)=sup⁡−∞<t<∞‖ϕ(t)‖E. In the present paper we consider the high order of accuracy two-step difference schemes generated by an exact difference scheme or by Taylor's decomposition on three points for the approximate solutions of this differential equation. The well-posedness of these difference schemes in the difference analogy of the smooth functions is obtained. The exact almost coercive inequality for solutions inC(τ,E)of these difference schemes is established.

scholarly journals A Note on the Stability of the Integral-Differential Equation of the Parabolic Type in a Banach Space

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Maksat Ashyraliyev
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Difference Scheme ◽  
Unique Solvability ◽  
Parabolic Type ◽  
Stability Estimates ◽  
The Difference ◽  
The Stability ◽  
Integral Differential Equation

The integral-differential equation of the parabolic type in a Banach space is considered. The unique solvability of this equation is established. The stability estimates for the solution of this equation are obtained. The difference scheme approximately solving this equation is presented. The stability estimates for the solution of this difference scheme are obtained.

Asymptotic behavior of evolution systems in arbitrary Banach spaces using general almost periodic splittings

2016 ◽  
Vol 8 (1) ◽  
pp. 1-28
Author(s):  
Josef Kreulich
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Almost Periodicity ◽  
Almost Periodic ◽  
Initial Value ◽  
Dissipative Operators ◽  
Evolution Systems ◽  
General Banach Space

Abstract We present sufficient conditions on the existence of solutions, with various specific almost periodicity properties, in the context of nonlinear, generally multivalued, non-autonomous initial value differential equations, \frac{du}{dt}(t)\in A(t)u(t),\quad t\geq 0,\qquad u(0)=u_{0}, and their whole line analogues, {\frac{du}{dt}(t)\in A(t)u(t)} , {t\in\mathbb{R}} , with a family {\{A(t)\}_{t\in\mathbb{R}}} of ω-dissipative operators {A(t)\subset X\times X} in a general Banach space X. According to the classical DeLeeuw–Glicksberg theory, functions of various generalized almost periodic types uniquely decompose in a “dominating” and a “damping” part. The second main object of the study – in the above context – is to determine the corresponding “dominating” part {[A(\,\cdot\,)]_{a}(t)} of the operators {A(t)} , and the corresponding “dominating” differential equation, \frac{du}{dt}(t)\in[A(\,\cdot\,)]_{a}(t)u(t),\quad t\in\mathbb{R}.

scholarly journals A note on well-posedness of source identification elliptic problem in a Banach space

2020 ◽  
Vol 99 (3) ◽  
pp. 96-104
Author(s):  
A. Ashyralyev ◽  
◽  
C. Ashyralyyev ◽  
V.G. Zvyagin ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Elliptic Problem ◽  
Source Identification ◽  
Elliptic Equations ◽  
Identification Problem ◽  
Stability Estimates ◽  
Well Posedness ◽  
Identification Problems ◽  
Coercive Stability

We study the source identification problem for an elliptic differential equation in a Banach space. The exact estimates for the solution of source identification problem in H¨older norms are obtained. In applications, four elliptic source identification problems are investigated. Stability and coercive stability estimates for solution of source identification problems for elliptic equations are obtained.

scholarly journals A note on the parabolic identification problem with involution and Dirichlet condition

2020 ◽  
Vol 99 (3) ◽  
pp. 130-139
Author(s):  
A. Ashyralyev ◽  
◽  
A.S. Erdogan ◽  
A. Sarsenbi ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Difference Scheme ◽  
Source Identification ◽  
Parabolic Problem ◽  
Identification Problem ◽  
Dirichlet Condition ◽  
Stability Estimates ◽  
Well Posedness ◽  
The Difference

A space source of identification problem for parabolic equation with involution and Dirichlet condition is studied. The well-posedness theorem on the differential equation of the source identification parabolic problem is established. The stable difference scheme for the approximate solution of this problem is presented. Furthermore, stability estimates for the difference scheme of the source identification parabolic problem are presented. Numerical results are given.

Difference Schemes for Nonlinear BVPS on the Semiaxis

2007 ◽  
Vol 7 (1) ◽  
pp. 25-47 ◽  
Author(s):  
I.P. Gavrilyuk ◽  
M. Hermann ◽  
M.V. Kutniv ◽  
V.L. Makarov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Boundary Value Problem ◽  
Difference Scheme ◽  
Adaptive Algorithm ◽  
Order Differential Equation ◽  
Constructive Method ◽  
Numerical Examples ◽  
Exact Difference ◽  
The Given

Abstract The scalar boundary value problem (BVP) for a nonlinear second order differential equation on the semiaxis is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme whose solution coincides with the projection of the exact solution of the given differential equation onto the underlying grid. A constructive method is proposed to derive from the EDS a so-called truncated difference scheme (n-TDS) of rank n, where n is a freely selectable natural number. The n-TDS is the basis for a new adaptive algorithm which has all the advantages known from the modern IVP-solvers. Numerical examples are given which illustrate the theorems presented in the paper and demonstrate the reliability of the new algorithm.

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