scholarly journals Approximate Controllability of Infinite-Dimensional Degenerate Fractional Order Systems in the Sectorial Case

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 735 ◽  
Author(s):  
Dumitru Baleanu ◽  
Vladimir E. Fedorov ◽  
Dmitriy M. Gordievskikh ◽  
Kenan Taş
Keyword(s):  
Control System ◽  
Fractional Order ◽  
Approximate Controllability ◽  
Homogeneous Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Degenerate Equation ◽  
Fractional Order Control ◽  
Infinite Dimensional ◽  
Initial Boundary Value

We consider a class of linear inhomogeneous equations in a Banach space not solvable with respect to the fractional Caputo derivative. Such equations are called degenerate. We study the case of the existence of a resolving operators family for the respective homogeneous equation, which is an analytic in a sector. The existence of a unique solution of the Cauchy problem and of the Showalter—Sidorov problem to the inhomogeneous degenerate equation is proved. We also derive the form of the solution. The approximate controllability of infinite-dimensional control systems, described by the equations of the considered class, is researched. An approximate controllability criterion for the degenerate fractional order control system is obtained. The criterion is illustrated by the application to a system, which is described by an initial-boundary value problem for a partial differential equation, not solvable with respect to the time-fractional derivative. As a corollary of general results, an approximate controllability criterion is obtained for the degenerate fractional order control system with a finite-dimensional input.

scholarly journals Homotopy Analysis Method for a Fractional Order Equation with Dirichlet and Non-Local Integral Conditions

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1167
Author(s):  
Said Mesloub ◽  
Saleem Obaidat
Keyword(s):  
Fractional Order ◽  
Homotopy Analysis Method ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Uniqueness Of The Solution ◽  
Non Local ◽  
Initial Boundary Value

The main purpose of this paper is to obtain some numerical results via the homotopy analysis method for an initial-boundary value problem for a fractional order diffusion equation with a non-local constraint of integral type. Some examples are provided to illustrate the efficiency of the homotopy analysis method (HAM) in solving non-local time-fractional order initial-boundary value problems. We also give some improvements for the proof of the existence and uniqueness of the solution in a fractional Sobolev space.

scholarly journals Non-global solution for visco-elastic dynamical system with nonlinear source term in control problem

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xiaoqiang Dai ◽  
Wenke Li
Keyword(s):  
Dynamical System ◽  
Control System ◽  
Source Term ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Source ◽  
Sharp Condition ◽  
Global Nonexistence ◽  
Existence And Nonexistence ◽  
Initial Boundary Value

<p style='text-indent:20px;'>In this paper, we study the initial boundary value problem of the visco-elastic dynamical system with the nonlinear source term in control system. By variational arguments and an improved convexity method, we prove the global nonexistence of solution, and we also give a sharp condition for global existence and nonexistence.</p>

scholarly journals A grid method for solving the first initial boundary value problem for a loaded differential equation of fractional order convection diffusion

2020 ◽  
pp. 27-40
Author(s):  
Мурат Хамидбиевич Бештоков
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Fractional Order ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Uniform Grid ◽  
Convection Diffusion ◽  
Loaded Differential Equation ◽  
Initial Boundary Value

Рассмотрена первая начально-краевая задача для нагруженного дифференциального уравнения конвекции диффузии дробного порядка. На равномерной сетке построена разностная схема, аппроксимирующая эту задачу. Для решения поставленной задачи в предположении существования регулярного решения получены априорные оценки в дифференциальной и разностной формах. Из этих оценок следуют единственность и непрерывная зависимость решения от входных данных задачи, а также сходимость со скоростью $O(h^2+\\tau^2)$. The first initial boundary value problem for a loaded differential equation of fractional order convection diffusion is considered. A difference scheme approximating this problem is constructed on a uniform grid. To solve the problem, assuming the existence of a regular solution, a priori estimates in differential and difference forms are obtained. From these estimates follow the uniqueness and continuous dependence of the solution on the input data of the problem, as well as the convergence with the rate $O(h^2+\\tau^2)$.

A sixth order degenerate equation with the higher order p-laplacian operator

2010 ◽  
Vol 60 (6) ◽  
Author(s):  
Changchun Liu
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Sixth Order ◽  
Laplacian Operator ◽  
Degenerate Equation ◽  
Discrete Method ◽  
Finite Speed Of Propagation ◽  
Order Degenerate ◽  
Speed Of Propagation ◽  
Initial Boundary Value

AbstractWe consider a initial-boundary value problem for a sixth order degenerate parabolic equation. Under some assumptions on the initial value, we establish the existence of weak solutions by the time-discrete method. The uniqueness, asymptotic behavior and the finite speed of propagation of perturbations of solutions are also discussed.

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