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 Transforms on white noise functionals with their applications to Cauchy problems

1997 ◽  
Vol 147 ◽  
pp. 1-23 ◽  
Author(s):  
Dong Myung Chung ◽  
Un Cig Ji
Keyword(s):  
Cauchy Problem ◽  
White Noise ◽  
Existence And Uniqueness ◽  
Continuous Linear Operator ◽  
Characterization Theorem ◽  
Cauchy Problems ◽  
Continuous Linear ◽  
The Cauchy Problem ◽  
White Noise Functionals ◽  
Generalized Laplacian

AbstractA generalized Laplacian ΔG(K) is defined as a continuous linear operator acting on the space of test white noise functionals. Operator-parameter - and -transforms on white noise functionals are introduced and then prove a characterization theorem for and -transforms in terms of the coordinate differential operator and the coordinate multiplication. As an application, we investigate the existence and uniqueness of solution of the Cauchy problem for the heat equation associated with ΔG(K)

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.

scholarly journals FUNCTIONAL-INTEGRAL SOLUTION FOR THE SCHRÖDINGER EQUATION WITH POLYNOMIAL POTENTIAL: A WHITE NOISE APPROACH

2011 ◽  
Vol 14 (04) ◽  
pp. 675-688 ◽  
Author(s):  
SONIA MAZZUCCHI
Keyword(s):  
Weak Solution ◽  
Schrödinger Equation ◽  
Integral Representation ◽  
White Noise ◽  
Schrodinger Equation ◽  
Functional Integral ◽  
Integral Solution ◽  
Polynomial Potential ◽  
White Noise Functional ◽  
Functional Integral Representation

A functional integral representation for the weak solution of the Schrödinger equation with polynomially growing potentials is proposed in terms of a white noise functional.

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.

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