# Homogenization of diffusion equation with scalar hysteresis operator

Mathematica Bohemica (2001)

- Volume: 126, Issue: 2, page 363-377
- ISSN: 0862-7959

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topFranců, Jan. "Homogenization of diffusion equation with scalar hysteresis operator." Mathematica Bohemica 126.2 (2001): 363-377. <http://eudml.org/doc/248882>.

@article{Franců2001,

abstract = {The paper deals with a scalar diffusion equation $ c\,u_t = (\{\{F\}\}[u_x])_x + f, $ where $\{F\}$ is a Prandtl-Ishlinskii operator and $c, f$ are given functions. In the diffusion or heat conduction equation the linear constitutive relation is replaced by a scalar Prandtl-Ishlinskii hysteresis spatially dependent operator. We prove existence, uniqueness and regularity of solution to the corresponding initial-boundary value problem. The problem is then homogenized by considering a sequence of equations of the above type with spatially periodic data $c^\varepsilon $ and $\eta ^\varepsilon $ when the spatial period $\varepsilon $ tends to zero. The homogenized characteristics $c^*$ and $\eta ^*$ are identified and the convergence of the corresponding solutions to the solution of the homogenized equation is proved.},

author = {Franců, Jan},

journal = {Mathematica Bohemica},

keywords = {hysteresis; Prandtl-Ishlinskii operator; material with periodic structure; nonlinear diffusion equation; homogenization; initial-boundary value problem; Prandtl-Ishlinskij operator; material with periodic structure; nonlinear diffusion equation; spatially periodic data},

language = {eng},

number = {2},

pages = {363-377},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {Homogenization of diffusion equation with scalar hysteresis operator},

url = {http://eudml.org/doc/248882},

volume = {126},

year = {2001},

}

TY - JOUR

AU - Franců, Jan

TI - Homogenization of diffusion equation with scalar hysteresis operator

JO - Mathematica Bohemica

PY - 2001

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 126

IS - 2

SP - 363

EP - 377

AB - The paper deals with a scalar diffusion equation $ c\,u_t = ({{F}}[u_x])_x + f, $ where ${F}$ is a Prandtl-Ishlinskii operator and $c, f$ are given functions. In the diffusion or heat conduction equation the linear constitutive relation is replaced by a scalar Prandtl-Ishlinskii hysteresis spatially dependent operator. We prove existence, uniqueness and regularity of solution to the corresponding initial-boundary value problem. The problem is then homogenized by considering a sequence of equations of the above type with spatially periodic data $c^\varepsilon $ and $\eta ^\varepsilon $ when the spatial period $\varepsilon $ tends to zero. The homogenized characteristics $c^*$ and $\eta ^*$ are identified and the convergence of the corresponding solutions to the solution of the homogenized equation is proved.

LA - eng

KW - hysteresis; Prandtl-Ishlinskii operator; material with periodic structure; nonlinear diffusion equation; homogenization; initial-boundary value problem; Prandtl-Ishlinskij operator; material with periodic structure; nonlinear diffusion equation; spatially periodic data

UR - http://eudml.org/doc/248882

ER -

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