scholarly journals Finite Rank Solution for Conformable Degenerate First-Order Abstract Cauchy Problem in Hilbert Spaces

2021 ◽  
Vol 14 (2) ◽  
pp. 493-505
Author(s):  
Fakhruddeen Seddiki ◽  
Mohammad Al-horani ◽  
Roshdi Rashid Khalil
Keyword(s):  
Cauchy Problem ◽  
Tensor Product ◽  
Fractional Derivative ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Finite Rank ◽  
Main Idea ◽  
Abstract Cauchy Problem ◽  
Conformable Derivative ◽  
First Order

In this paper, we find a solution of finite rank form of fractional Abstract Cauchy Problem. The fractional derivative used is the Conformable derivative. The main idea of the proofs are based on theory of tensor product of Banach spaces.

scholarly journals An Application Of The Theory Of Scale Of Banach Spaces

2015 ◽  
Vol 29 (1) ◽  
pp. 51-59
Author(s):  
Łukasz Dawidowski
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Banach Spaces ◽  
Abstract Cauchy Problem ◽  
Initial Boundary ◽  
Scale Of Banach Spaces ◽  
The Cauchy Problem ◽  
Selection Of

AbstractThe abstract Cauchy problem on scales of Banach space was considered by many authors. The goal of this paper is to show that the choice of the space on scale is significant. We prove a theorem that the selection of the spaces in which the Cauchy problem ut − Δu = u|u|s with initial–boundary conditions is considered has an influence on the selection of index s. For the Cauchy problem connected with the heat equation we will study how the change of the base space influents the regularity of the solutions.

scholarly journals Fractional-hyperbolic systems

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Anatoly Kochubei
Keyword(s):  
Cauchy Problem ◽  
Fundamental Solution ◽  
Fractional Derivative ◽  
Hyperbolic Systems ◽  
Time Variable ◽  
Spatial Variables ◽  
First Order ◽  
Linear Partial Differential Equations ◽  
The Cauchy Problem ◽  
Evolution Systems

AbstractWe describe a class of evolution systems of linear partial differential equations with the Caputo-Dzhrbashyan fractional derivative of order α ∈ (0, 1) in the time variable t and the first order derivatives in spatial variables x = (x 1, …, x n), which can be considered as a fractional analogue of the class of hyperbolic systems. For such systems, we construct a fundamental solution of the Cauchy problem having exponential decay outside the fractional light cone {(t,x) : |t -α| ≤ 1}.

Fractional fractals

2020 ◽  
Vol 23 (5) ◽  
pp. 1329-1348
Author(s):  
J.A. Tenreiro Machado ◽  
Daniel Cao Labora
Keyword(s):  
Fractional Derivative ◽  
Iterated Function System ◽  
Main Idea ◽  
Higher Order ◽  
Numerical Calculations ◽  
Function System ◽  
Limit Sets ◽  
First Order ◽  
Iterated Function

Abstract This paper introduces the notion of “fractional fractals”. The main idea is to establish a connection between the classical iterated function system and the first order truncation of the Gründwald-Letnikov fractional derivative. This allows us to consider higher order truncations, and also to study the limit sets for these higher order systems. We prove several results involving the existence and dimension of such limit sets, that will be called “fractional fractals”. Some numerical calculations and representations illustrate relevant examples.

scholarly journals On Semilinear Integro-Differential Equations with Nonlocal Conditions in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-26
Author(s):  
Tran Dinh Ke ◽  
Valeri Obukhovskii ◽  
Ngai-Ching Wong ◽  
Jen-Chih Yao
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Banach Spaces ◽  
Topological Structure ◽  
Continuous Dependence ◽  
Nonlocal Conditions ◽  
Abstract Cauchy Problem ◽  
Nonlinear Perturbations ◽  
Continuous Dependence Results

We study the abstract Cauchy problem for a class of integrodifferential equations in a Banach space with nonlinear perturbations and nonlocal conditions. By using MNC estimates, the existence and continuous dependence results are proved. Under some additional assumptions, we study the topological structure of the solution set.

C-Semigroups, subordination principle and the Lévy α-stable distribution on discrete time

2020 ◽  
pp. 2050063
Author(s):  
Edgardo Alvarez ◽  
Stiven Díaz ◽  
Carlos Lizama
Keyword(s):  
Cauchy Problem ◽  
Discrete Time ◽  
Explicit Solution ◽  
Stable Distribution ◽  
Difference Operator ◽  
Abstract Cauchy Problem ◽  
Fractional Difference ◽  
First Order ◽  
The Subject ◽  
Discrete Setting

In this paper, we introduce the notion of Lévy [Formula: see text]-stable distribution within the discrete setting. Using this notion, a subordination principle is proved, which relates a sequence of solution operators — given by a discrete [Formula: see text]-semigroup — for the abstract Cauchy problem of first order in discrete-time, with a sequence of solution operators for the abstract Cauchy problem of fractional order [Formula: see text] in discrete-time. As an application, we establish the explicit solution of the abstract Cauchy problem in discrete-time that involves the Hilfer fractional difference operator and prove that, in some cases, such solution converges to zero. Our findings give new insights on the theory, provide original concepts and extend as well as improve recent results of relevant references on the subject.

scholarly journals Normal periodic solutions for the fractional abstract Cauchy problem

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jennifer Bravo ◽  
Carlos Lizama
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Periodic Solution ◽  
Periodic Boundary ◽  
Abstract Cauchy Problem ◽  
Periodic Boundary Conditions ◽  
Sectorial Operator ◽  
Closed Linear Operator ◽  
Resolvent Set ◽  
Spectral Angle

AbstractWe show that if A is a closed linear operator defined in a Banach space X and there exist $t_{0} \geq 0$ t 0 ≥ 0 and $M>0$ M > 0 such that $\{(im)^{\alpha }\}_{|m|> t_{0}} \subset \rho (A)$ { ( i m ) α } | m | > t 0 ⊂ ρ ( A ) , the resolvent set of A, and $$ \bigl\Vert (im)^{\alpha }\bigl(A+(im)^{\alpha }I \bigr)^{-1} \bigr\Vert \leq M \quad \text{ for all } \vert m \vert > t_{0}, m \in \mathbb{Z}, $$ ∥ ( i m ) α ( A + ( i m ) α I ) − 1 ∥ ≤ M  for all  | m | > t 0 , m ∈ Z , then, for each $\frac{1}{p}<\alpha \leq \frac{2}{p}$ 1 p < α ≤ 2 p and $1< p < 2$ 1 < p < 2 , the abstract Cauchy problem with periodic boundary conditions $$ \textstyle\begin{cases} _{GL}D^{\alpha }_{t} u(t) + Au(t) = f(t), & t \in (0,2\pi ); \\ u(0)=u(2\pi ), \end{cases} $$ { D t α G L u ( t ) + A u ( t ) = f ( t ) , t ∈ ( 0 , 2 π ) ; u ( 0 ) = u ( 2 π ) , where $_{GL}D^{\alpha }$ D α G L denotes the Grünwald–Letnikov derivative, admits a normal 2π-periodic solution for each $f\in L^{p}_{2\pi }(\mathbb{R}, X)$ f ∈ L 2 π p ( R , X ) that satisfies appropriate conditions. In particular, this happens if A is a sectorial operator with spectral angle $\phi _{A} \in (0, \alpha \pi /2)$ ϕ A ∈ ( 0 , α π / 2 ) and $\int _{0}^{2\pi } f(t)\,dt \in \operatorname{Ran}(A)$ ∫ 0 2 π f ( t ) d t ∈ Ran ( A ) .

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