# Inhomogeneous Fractional Diffusion Equations

Baeumer, Boris; Kurita, Satoko; Meerschaert, Mark

Fractional Calculus and Applied Analysis (2005)

- Volume: 8, Issue: 4, page 371-386
- ISSN: 1311-0454

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topBaeumer, Boris, Kurita, Satoko, and Meerschaert, Mark. "Inhomogeneous Fractional Diffusion Equations." Fractional Calculus and Applied Analysis 8.4 (2005): 371-386. <http://eudml.org/doc/11246>.

@article{Baeumer2005,

abstract = {2000 Mathematics Subject Classification: Primary 26A33; Secondary 35S10, 86A05Fractional diffusion equations are abstract partial differential equations
that involve fractional derivatives in space and time. They are useful to
model anomalous diffusion, where a plume of particles spreads in a different
manner than the classical diffusion equation predicts. An initial value problem
involving a space-fractional diffusion equation is an abstract Cauchy
problem, whose analytic solution can be written in terms of the semigroup
whose generator gives the space-fractional derivative operator. The corresponding
time-fractional initial value problem is called a fractional Cauchy
problem. Recently, it was shown that the solution of a fractional Cauchy
problem can be expressed as an integral transform of the solution to the
corresponding Cauchy problem. In this paper, we extend that results to
inhomogeneous fractional diffusion equations, in which a forcing function
is included to model sources and sinks. Existence and uniqueness is established
by considering an equivalent (non-local) integral equation. Finally,
we illustrate the practical application of these results with an example from
groundwater hydrology, to show the effect of the fractional time derivative
on plume evolution, and the proper specification of a forcing function in a
time-fractional evolution equation.1. Partially supported by the Marsden Foundation in New Zealand
2. Partially supported by ACES Postdoctoral Fellowship, Nevada NSF EPSCoR RING TRUE II grant
3. Partially supported by NSF grants DMS-0139927 and DMS-0417869, and the Marsden Foundation},

author = {Baeumer, Boris, Kurita, Satoko, Meerschaert, Mark},

journal = {Fractional Calculus and Applied Analysis},

keywords = {26A33; 35S10; 86A05},

language = {eng},

number = {4},

pages = {371-386},

publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},

title = {Inhomogeneous Fractional Diffusion Equations},

url = {http://eudml.org/doc/11246},

volume = {8},

year = {2005},

}

TY - JOUR

AU - Baeumer, Boris

AU - Kurita, Satoko

AU - Meerschaert, Mark

TI - Inhomogeneous Fractional Diffusion Equations

JO - Fractional Calculus and Applied Analysis

PY - 2005

PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences

VL - 8

IS - 4

SP - 371

EP - 386

AB - 2000 Mathematics Subject Classification: Primary 26A33; Secondary 35S10, 86A05Fractional diffusion equations are abstract partial differential equations
that involve fractional derivatives in space and time. They are useful to
model anomalous diffusion, where a plume of particles spreads in a different
manner than the classical diffusion equation predicts. An initial value problem
involving a space-fractional diffusion equation is an abstract Cauchy
problem, whose analytic solution can be written in terms of the semigroup
whose generator gives the space-fractional derivative operator. The corresponding
time-fractional initial value problem is called a fractional Cauchy
problem. Recently, it was shown that the solution of a fractional Cauchy
problem can be expressed as an integral transform of the solution to the
corresponding Cauchy problem. In this paper, we extend that results to
inhomogeneous fractional diffusion equations, in which a forcing function
is included to model sources and sinks. Existence and uniqueness is established
by considering an equivalent (non-local) integral equation. Finally,
we illustrate the practical application of these results with an example from
groundwater hydrology, to show the effect of the fractional time derivative
on plume evolution, and the proper specification of a forcing function in a
time-fractional evolution equation.1. Partially supported by the Marsden Foundation in New Zealand
2. Partially supported by ACES Postdoctoral Fellowship, Nevada NSF EPSCoR RING TRUE II grant
3. Partially supported by NSF grants DMS-0139927 and DMS-0417869, and the Marsden Foundation

LA - eng

KW - 26A33; 35S10; 86A05

UR - http://eudml.org/doc/11246

ER -

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