A constructive description of the domains of definition of fractional powers of elliptic operators

1976 ◽  
Vol 16 (5) ◽  
pp. 772-778
Author(s):  
L. V. Koledov
Keyword(s):  
Elliptic Operators ◽  
Fractional Powers ◽  
Constructive Description ◽  
Definition Of

A note on fractional powers of strongly positive operators and their applications

2019 ◽  
Vol 22 (2) ◽  
pp. 302-325 ◽  
Author(s):  
Allaberen Ashyralyev ◽  
Ayman Hamad
Keyword(s):  
Banach Space ◽  
Elliptic Operators ◽  
Positive Operators ◽  
Fractional Powers ◽  
Fractional Spaces

Abstract The present paper deals with fractional powers of positive operators in a Banach space. The main theorem concerns the structure of fractional powers of positive operators in fractional spaces. As applications, the structure of fractional powers of elliptic operators is studied.

scholarly journals FRACTIONAL DERIVATIVE AS FRACTIONAL POWER OF DERIVATIVE

2007 ◽  
Vol 18 (03) ◽  
pp. 281-299 ◽  
Author(s):  
VASILY E. TARASOV
Keyword(s):  
Fourier Series ◽  
Fractional Derivative ◽  
Taylor Series ◽  
Fractional Derivatives ◽  
Fractional Power ◽  
Weyl Quantization ◽  
Fractional Powers ◽  
Derivative Operator ◽  
Quantization Procedure ◽  
Definition Of

Definitions of fractional derivatives as fractional powers of derivative operators are suggested. The Taylor series and Fourier series are used to define fractional power of selfadjoint derivative operator. The Fourier integrals and Weyl quantization procedure are applied to derive the definition of fractional derivative operator. Fractional generalization of concept of stability is considered.

scholarly journals High order numerical schemes for solving fractional powers of elliptic operators

2020 ◽  
Vol 372 ◽  
pp. 112627
Author(s):  
Raimondas Čiegis ◽  
Petr N. Vabishchevich
Keyword(s):  
Elliptic Operators ◽  
High Order ◽  
Numerical Schemes ◽  
Fractional Powers

scholarly journals Numerical approximation of fractional powers of elliptic operators

2019 ◽  
Vol 40 (3) ◽  
pp. 1746-1771 ◽  
Author(s):  
Beiping Duan ◽  
Raytcho D Lazarov ◽  
Joseph E Pasciak
Keyword(s):  
Finite Element ◽  
Elliptic Boundary ◽  
Finite Element Approximation ◽  
Elliptic Operators ◽  
Second Order ◽  
Element Approximation ◽  
Fractional Powers ◽  
Time Stepping ◽  
Order Elliptic Operator ◽  
Graded Meshes

Abstract In this paper, we develop and study algorithms for approximately solving linear algebraic systems: ${{\mathcal{A}}}_h^\alpha u_h = f_h$, $ 0< \alpha <1$, for $u_h, f_h \in V_h$ with $V_h$ a finite element approximation space. Such problems arise in finite element or finite difference approximations of the problem $ {{\mathcal{A}}}^\alpha u=f$ with ${{\mathcal{A}}}$, for example, coming from a second-order elliptic operator with homogeneous boundary conditions. The algorithms are motivated by the method of Vabishchevich (2015, Numerically solving an equation for fractional powers of elliptic operators. J. Comput. Phys., 282, 289–302) that relates the algebraic problem to a solution of a time-dependent initial value problem on the interval $[0,1]$. Here we develop and study two time-stepping schemes based on diagonal Padé approximation to $(1+x)^{-\alpha }$. The first one uses geometrically graded meshes in order to compensate for the singular behaviour of the solution for $t$ close to $0$. The second algorithm uses uniform time stepping, but requires smoothness of the data $f_h$ in discrete norms. For both methods, we estimate the error in terms of the number of time steps, with the regularity of $f_h$ playing a major role for the second method. Finally, we present numerical experiments for ${{\mathcal{A}}}_h$ coming from the finite element approximations of second-order elliptic boundary value problems in one and two spatial dimensions.

Gamma-convergence of Gaussian fractional perimeter

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Alessandro Carbotti ◽  
Simone Cito ◽  
Domenico Angelo La Manna ◽  
Diego Pallara
Keyword(s):  
Typical Feature ◽  
Gamma Convergence ◽  
Fractional Powers ◽  
Γ Convergence ◽  
Definition Of

Abstract We prove the Γ-convergence of the renormalised Gaussian fractional s-perimeter to the Gaussian perimeter as s → 1 - {s\to 1^{-}} . Our definition of fractional perimeter comes from that of the fractional powers of Ornstein–Uhlenbeck operator given via Bochner subordination formula. As a typical feature of the Gaussian setting, the constant appearing in front of the Γ-limit does not depend on the dimension.

scholarly journals The approximation of parabolic equations involving fractional powers of elliptic operators

2017 ◽  
Vol 315 ◽  
pp. 32-48 ◽  
Author(s):  
Andrea Bonito ◽  
Wenyu Lei ◽  
Joseph E. Pasciak
Keyword(s):  
Parabolic Equations ◽  
Elliptic Operators ◽  
Fractional Powers

Two-level schemes of Cauchy problem method for solving fractional powers of elliptic operators

2020 ◽  
Vol 80 (2) ◽  
pp. 305-315 ◽  
Author(s):  
Raimondas Čiegis ◽  
Petr N. Vabishchevich
Keyword(s):  
Cauchy Problem ◽  
Elliptic Operators ◽  
Fractional Powers ◽  
Problem Method

Question of the domains of definition of fractional powers of accretive operators

1988 ◽  
Vol 43 (2) ◽  
pp. 129-133
Author(s):  
A. M. Gomilko
Keyword(s):  
Accretive Operators ◽  
Fractional Powers ◽  
Definition Of
Sign in / Sign up

Export Citation Format

Share Document