APPROXIMATION BY POSITIVE OPERATORS OF THE C0–SEMIGROUPS ASSOCIATED WITH ONE-DIMENSIONAL DIFFUSION EQUATIONS: PART I

2005 ◽  
Vol 26 (1) ◽  
pp. 1-15 ◽  
Author(s):  
Francesco Altomare ◽  
Rachida Amiar
Keyword(s):  
Positive Operators ◽  
Diffusion Equations ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Approximation By Positive Operators

On the solution of the Fokker–Planck equation for a Feller process

1990 ◽  
Vol 22 (01) ◽  
pp. 101-110
Author(s):  
L. Sacerdote
Keyword(s):  
Diffusion Processes ◽  
Planck Equation ◽  
Hyperbolic Type ◽  
Diffusion Equations ◽  
Time Varying ◽  
Feller Process ◽  
Parameter Group ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Type Boundary

Use of one-parameter group transformations is made to obtain the transition p.d.f. of a Feller process confined between the origin and a hyperbolic-type boundary. Such a procedure, previously used by Bluman and Cole (cf., for instance, [4]), although useful for dealing with one-dimensional diffusion processes restricted between time-varying boundaries, does not appear to have been sufficiently exploited to obtain solutions to the diffusion equations associated to continuous Markov processes.

The Ljapunov equation and an application to stabilisation of one-dimensional diffusion equations

1986 ◽  
Vol 104 (1-2) ◽  
pp. 39-52 ◽  
Author(s):  
Takao Nambu
Keyword(s):  
Elliptic Operator ◽  
Hilbert Spaces ◽  
Linear Operators ◽  
Diffusion Equations ◽  
One Dimensional ◽  
Infinite Dimensional ◽  
Dimensional Diffusion ◽  
General Elliptic ◽  
Geometrical Character ◽  
Specific Control

SynopsisA Ljapunov equation XL − BX = C appears in stabilisation studies of linear systems. Here, the operators L, B, and C are given linear operators working in infinite-dimensional Hilbert spaces, which are derived from a specific control system. We have so far considered the case where L is a general elliptic operator of order 2 in a bounded domain of an Euclidean space. When L is instead a self-adjoint elliptic operator working in an interval of ℝ1, we derive here a stronger geometrical character of the solution X to the Ljapunov equation. The result is applied to stabilisation of one-dimensional diffusion equations.

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