scholarly journals Cauchy problem and integral representation associated to the power of the QWN-Euler operator

2012 ◽  
Vol 6 (4) ◽  
Author(s):  
Aymen Ettaieb ◽  
Habib Ouerdiane ◽  
Hafedh Rguigui
Keyword(s):  
Cauchy Problem ◽  
Integral Representation ◽  
Euler Operator

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.

The Resolvent Operator

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Laplace Transform ◽  
Heat Kernel ◽  
Resolvent Operator ◽  
Contour Integration ◽  
Resolvent Kernel ◽  
The Laplace Transform ◽  
Elliptic Estimates ◽  
The Resolvent Operator

This chapter describes the construction of a resolvent operator using the Laplace transform of a parametrix for the heat kernel and a perturbative argument. In the equation (μ‎-L) R(μ‎) f = f, R(μ‎) is a right inverse for (μ‎-L). In Hölder spaces, these are the natural elliptic estimates for generalized Kimura diffusions. The chapter first constructs the resolvent kernel using an induction over the maximal codimension of bP, and proves various estimates on it, along with corresponding estimates for the solution operator for the homogeneous Cauchy problem. It then considers holomorphic semi-groups and uses contour integration to construct the solution to the heat equation, concluding with a discussion of Kimura diffusions where all coefficients have the same leading homogeneity.

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