scholarly journals Scaling limits of solutions of linear evolution equations with random initial conditions

2017 ◽  
Vol 17 (03) ◽  
pp. 1750019 ◽  
Author(s):  
Miłosz Krupski
Keyword(s):  
Evolution Equations ◽  
Initial Conditions ◽  
Linear Operators ◽  
Scaling Limits ◽  
Initial Value ◽  
Linear Evolution ◽  
Semigroup Of Linear Operators ◽  
Random Initial Conditions ◽  
Stationary Random Field ◽  
Linear Evolution Equations

We consider a linear equation [Formula: see text], where [Formula: see text] is a generator of a semigroup of linear operators on a certain Hilbert space related to an initial condition [Formula: see text] being a generalised stationary random field on [Formula: see text]. We show the existence and uniqueness of generalised solutions to such initial value problems. Then we investigate their scaling limits.

Linear evolution equations in two Banach spaces

1982 ◽  
Vol 91 (3-4) ◽  
pp. 287-303 ◽  
Author(s):  
Niko Sauer
Keyword(s):  
Banach Space ◽  
Infinitesimal Generator ◽  
Evolution Equations ◽  
Linear Operators ◽  
Initial Condition ◽  
Semi Group ◽  
Initial State ◽  
Linear Evolution ◽  
The Family ◽  
Linear Evolution Equations

SynopsisWe consider the ordinary differential equation [Bu(t)]′ = Au(t) with A and B linear operators with domains in a Banach space X and ranges in a Banach space Y. The initial condition is that the limit as (t → 0 of Bu(t) is prescribed in Y. We study the properties of the “solution operator” S(t) which maps the initial state in Y to the solution u(t) at time t. The notion of infinitesimal generator A of S(t) is introduced and the relationships between S(t), an associated semi-group E(t), the operators A and B and some other operators are studied. In particular a pair of operators Ao and Bo, derived from A and B, determine the family S(t) of operators. These so-called “generating pairs” are characterized. The operators A and B and Ao and Bo need not be closed, but form so-called closed pairs which is a weaker condition. We also discuss two applications of the theory.

scholarly journals Linear evolution equations on the half-line with dynamic boundary conditions

2021 ◽  
pp. 1-33
Author(s):  
D. A. SMITH ◽  
W. Y. TOH
Keyword(s):  
Boundary Conditions ◽  
Heat Equation ◽  
Evolution Equations ◽  
Half Line ◽  
Robin Problem ◽  
Dynamic Boundary Conditions ◽  
Transform Method ◽  
Linear Evolution ◽  
Robin Condition ◽  
Linear Evolution Equations

The classical half-line Robin problem for the heat equation may be solved via a spatial Fourier transform method. In this work, we study the problem in which the static Robin condition $$bq(0,t) + {q_x}(0,t) = 0$$ is replaced with a dynamic Robin condition; $$b = b(t)$$ is allowed to vary in time. Applications include convective heating by a corrosive liquid. We present a solution representation and justify its validity, via an extension of the Fokas transform method. We show how to reduce the problem to a variable coefficient fractional linear ordinary differential equation for the Dirichlet boundary value. We implement the fractional Frobenius method to solve this equation and justify that the error in the approximate solution of the original problem converges appropriately. We also demonstrate an argument for existence and unicity of solutions to the original dynamic Robin problem for the heat equation. Finally, we extend these results to linear evolution equations of arbitrary spatial order on the half-line, with arbitrary linear dynamic boundary conditions.

scholarly journals Lumpability of linear evolution Equations in Banach spaces

2017 ◽  
Vol 6 (1) ◽  
pp. 15-34 ◽  
Author(s):  
Fatihcan M. Atay ◽  
◽  
Lavinia Roncoroni ◽  
Keyword(s):  
Banach Spaces ◽  
Evolution Equations ◽  
Linear Evolution ◽  
Linear Evolution Equations
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