Feynman averaging of semigroups generated by Schrödinger operators

2018 ◽  
Vol 21 (02) ◽  
pp. 1850010 ◽  
Author(s):  
Leonid A. Borisov ◽  
Yuriy N. Orlov ◽  
Vsevolod Zh. Sakbaev
Keyword(s):  
Diffusion Equation ◽  
Fractional Diffusion ◽  
Linear Operators ◽  
Evolutionary Equation ◽  
Coordinate Space ◽  
Initial Value ◽  
Semigroups Of Linear Operators ◽  
Measurable Map ◽  
Translation Operators ◽  
Complex Valued

The extension of averaging procedure for operator-valued function is defined by means of the integration of measurable map with respect to complex-valued measure or pseudomeasure. The averaging procedure of one-parametric semigroups of linear operators based on Chernoff equivalence for operator-valued functions is constructed. The initial value problem solutions are investigated for fractional diffusion equation and for Schrödinger equation with relativistic Hamiltonian of free motion. It is established that in these examples the solution of evolutionary equation can be obtained by applying the constructed averaging procedure to the random translation operators in classical coordinate space.

Hermite Pseudospectral Method for the Time Fractional Diffusion Equation with Variable Coefficients

2017 ◽  
Vol 18 (5) ◽  
pp. 385-393
Author(s):  
Zeting Liu ◽  
Shujuan Lü
Keyword(s):  
Diffusion Equation ◽  
Pseudospectral Method ◽  
Fractional Diffusion ◽  
Variable Coefficients ◽  
Fractional Diffusion Equation ◽  
Stability And Convergence ◽  
Initial Value ◽  
Fully Discrete ◽  
Time Fractional Diffusion Equation ◽  
Gauss Points

Abstract:We consider the initial value problem of the time fractional diffusion equation on the whole line and the fractional derivative is described in Caputo sense. A fully discrete Hermite pseudospectral approximation scheme is structured basing Hermite-Gauss points in space and finite difference in time. Unconditionally stability and convergence are proved. Numerical experiments are presented and the results conform to our theoretical analysis.

scholarly journals On continuous dependence for the mixed problem of microstretch bodies

2017 ◽  
Vol 25 (1) ◽  
pp. 131-143
Author(s):  
M. Marin ◽  
I. Abbas ◽  
C. Cârstea
Keyword(s):  
Hilbert Space ◽  
Initial Data ◽  
Continuous Dependence ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Linear Operators ◽  
Evolutionary Equation ◽  
Trans Form ◽  
Semigroups Of Linear Operators ◽  
Initial Boundary Value

AbstractWe do a qualitative study on the mixed initial-boundary value problem in the elastodynamic theory of microstretch bodies. After we trans- form this problem in a temporally evolutionary equation on a Hilbert space, we will use some results from the theory of semigroups of linear operators in order to prove the continuous dependence of the solutions upon initial data and supply terms.

scholarly journals Numerical Solution of Nonstationary Problems for a Space-Fractional Diffusion Equation

2016 ◽  
Vol 19 (1) ◽  
Author(s):  
Petr N. Vabishchevich
Keyword(s):  
Diffusion Equation ◽  
Elliptic Operator ◽  
Fractional Power ◽  
Finite Element Approximation ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Evolutionary Equation ◽  
Element Approximation ◽  
First Order ◽  
Space Fractional Diffusion Equation

AbstractAn unsteady problem is considered for a space-fractional diffusion equation in abounded domain. A first-order evolutionary equation containing a fractional power of an elliptic operator of second order is studied for general boundary conditions of Robin type. Finite element approximation in space is employed. To construct approximation in time, regularized two-level schemes are used. The numerical implementation is based on solving the equation with the fractional power of the elliptic operator using an auxiliary Cauchy problem for a pseudo-parabolic equation. The results of numerical experiments are presented for a model two-dimensional problem.

A quasi-boundary regularization method for identifying the initial value of time-fractional diffusion equation on spherically symmetric domain

2019 ◽  
Vol 27 (5) ◽  
pp. 609-621 ◽  
Author(s):  
Fan Yang ◽  
Ni Wang ◽  
Xiao-Xiao Li ◽  
Can-Yun Huang
Keyword(s):  
Diffusion Equation ◽  
Regularization Method ◽  
Regularization Parameter ◽  
A Priori ◽  
Symmetric Domain ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Spherically Symmetric ◽  
Initial Value ◽  
Time Fractional Diffusion Equation

Abstract In this paper, an inverse problem to identify the initial value for high dimension time fractional diffusion equation on spherically symmetric domain is considered. This problem is ill-posed in the sense of Hadamard, so the quasi-boundary regularization method is proposed to solve the problem. The convergence estimates between the regularization solution and the exact solution are presented under the a priori and a posteriori regularization parameter choice rules. Numerical examples are provided to show the effectiveness and stability of the proposed method.

Fractional diffusion equation in half-space with Robin boundary condition

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Jun-Sheng Duan ◽  
Zhong Wang ◽  
Shou-Zhong Fu
Keyword(s):  
Boundary Condition ◽  
Diffusion Equation ◽  
Half Space ◽  
Boundary Value ◽  
Robin Boundary Condition ◽  
Fundamental Solutions ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Initial Value ◽  
Robin Boundary

AbstractThe initial and boundary value problem for the fractional diffusion equation in half-space with the Robin boundary condition is considered. The solution is comprised of two parts: the contribution of the initial value and the contribution of the boundary value, for which the respective fundamental solutions are given. Finally, the solution formula of the considered problem is obtained.

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