Integral representation and stabilization of the solution to the Cauchy problem for an equation with two noncommuting operators

1995 ◽  
Vol 58 (1) ◽  
pp. 703-709
Author(s):  
A. V. Glushak
Keyword(s):  
Cauchy Problem ◽  
Integral Representation ◽  
Noncommuting Operators ◽  
The Cauchy Problem

QWN-EULER OPERATOR AND ASSOCIATED CAUCHY PROBLEM

2012 ◽  
Vol 15 (01) ◽  
pp. 1250004 ◽  
Author(s):  
ABDESSATAR BARHOUMI ◽  
HABIB OUERDIANE ◽  
HAFEDH RGUIGUI
Keyword(s):  
Cauchy Problem ◽  
Integral Representation ◽  
White Noise ◽  
Coordinate System ◽  
Functional Integral ◽  
Euler Operator ◽  
The Cauchy Problem ◽  
Sum Formula

In this paper the quantum white noise (QWN)-Euler operator [Formula: see text] is defined as the sum [Formula: see text], where [Formula: see text] and NQ stand for appropriate QWN counterparts of the Gross Laplacian and the conservation operator, respectively. It is shown that [Formula: see text] has an integral representation in terms of the QWN-derivatives [Formula: see text] as a kind of functional integral acting on nuclear algebra of white noise operators. The solution of the Cauchy problem associated to the QWN-Euler operator is worked out in the basis of the QWN coordinate system.

scholarly journals REPRESENTATION OF SOLUTIONS OF KOLMOGOROV TYPE EQUATIONS WITH INCREASING COEFFICIENTS AND DEGENERATIONS ON THE INITIAL HYPERPLANE

2021 ◽  
Vol 9 (1) ◽  
pp. 189-199
Author(s):  
H. Pasichnyk ◽  
S. Ivasyshen
Keyword(s):  
Cauchy Problem ◽  
Fundamental Solution ◽  
Integral Representation ◽  
Initial Condition ◽  
Kolmogorov Type ◽  
Ultraparabolic Equation ◽  
Volume Potential ◽  
Representation Of Solutions ◽  
The Cauchy Problem ◽  
Strong Degeneration

The nonhomogeneous model Kolmogorov type ultraparabolic equation with infinitely increasing coefficients at the lowest derivatives as |x| → ∞ and degenerations for t = 0 is considered in the paper. Theorems on the integral representation of solutions of the equation are proved. The representation is written with the use of Poisson integral and the volume potential generated by the fundamental solution of the Cauchy problem. The considered solutions, as functions of x, could infinitely increase as |x| → ∞, and could behave in a certain way as t → 0, depending on the type of the degeneration of the equation at t = 0. Note that in the case of very strong degeneration, the solutions, as functions of x, are bounded. These results could be used to establish the correct solvability of the considered equation with the classical initial condition in the case of weak degeneration of the equation at t = 0, weight initial condition or without the initial condition if the degeneration is strong.

Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

Fundamental Serrin type regularity criteria for 3D MHD fluid passing through the porous medium

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1287-1293 ◽  
Author(s):  
Zujin Zhang ◽  
Dingxing Zhong ◽  
Shujing Gao ◽  
Shulin Qiu
Keyword(s):  
Cauchy Problem ◽  
Porous Medium ◽  
Regularity Criterion ◽  
Regularity Criteria ◽  
The Cauchy Problem ◽  
Appl Math

In this paper, we consider the Cauchy problem for the 3D MHD fluid passing through the porous medium, and provide some fundamental Serrin type regularity criteria involving the velocity or its gradient, the pressure or its gradient. This extends and improves [S. Rahman, Regularity criterion for 3D MHD fluid passing through the porous medium in terms of gradient pressure, J. Comput. Appl. Math., 270 (2014), 88-99].

scholarly journals A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation

2020 ◽  
Vol 18 (1) ◽  
pp. 1685-1697
Author(s):  
Zhenyu Zhao ◽  
Lei You ◽  
Zehong Meng
Keyword(s):  
Cauchy Problem ◽  
Laplace Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Solution Process ◽  
Tikhonov Regularization Method ◽  
Hermite Expansion ◽  
The Cauchy Problem ◽  
Ill Posed

Abstract In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.

Sign in / Sign up

Export Citation Format

Share Document