pde in fractal domains
Jerome Ellepola
PLEASE REPLY TO
michele.c...@eng.ox.ac.uk
Dear friend,
we are working in materials science, and we used fractal models to interpret some
strange features of fracture behaviour of disordered composites. Despite
many efforts, the attempt to extend classical continuum mechanics fields
(eg, strains, displacements, energy flux) to fractal domains was
unsuccessful. Only qualitative scaling laws could be obtained by means of
renormalization group, that is, strictly exploiting the self-similar
properties of the sets.
There are several ideas that we are thinking about, and we are not able
to have a unifiyed vision, and to see the link between them.
One problem we can look at [1] is the closest to classical, in the sense
that considering a continuum with a continuous distribution of
singularities, in the form of dislocation, one end up with a domain that
is not Riemanian any more. Then, the computation of deformation field,
under some boundary conditions (like the loads or displacements
distributed along boundaries) reduces to the solution of generalized
Laplace equation (diffusion process) or Navier equation (vector problem).
The fractal properties of the resulting material can be understood on
solving these equations. Now, what do you know about the techniques to do
this (analitycally, for simple problems (Green's functions)), or
numerically???
Vice-versa (?), to model stress and strains (elasticity framework) in
fractal (sponge-like) domains,
one can use the classical iterated function systems (IFS) approach, i.e.
computing the classical integer dimension fields, at a particular stage
of the generation of the fractal domain, and then consider the asymptotic
properties of this process. However, is it possible to formulate the
problem directly in term of the renormalized physical quantities, that
are necessarily non-integer any more??? And is the result equivalent to
the study with IFS??? Maybe the answer is in understanding whether a
field, defined on a general step of the IFS only on in part of the domain
, is equivalent in the limit to a generalized function, defined on the
entire domain, but of fractional dimension, i.e. defined from fractional
operators. Do you have any idea about that???
The majority of the attempt to use fractional calculus in elasticity has
been in the direction of relaxation fenomena, or materials with memory,
where the only fractal discretization is in the time domain. An
interesting review of these problems is in [2, 3] by Prof.Mainardi, with
whom we are in touch, but he is not well at the moment.
Our impression is that, using the same ideas, we can get Green's
functions of Laplace and Navier equations in fractal domain, using
Fourier Transform, after defining the appropriate fractional Laplace and
Navier operators. The idea is to generalize a classical book like SNeddon
"Fourier Transform", especially the part regarding elasticity in 2D. What
do you think about this??. What kind of equation are we likely to obtain
in the transform domain (a guess: ordinary but fractional differential
equation)???
As your answer to these question are likely to be important for the
procecution of our efforts, and for the direction to take, could you pay
some attention in giving us any:-
a) suggestion
b) general or unifoed view
c) source of reference
d) other people working on similar subjects
in the most possible close to engineering formalism?? By the way, I saw
some of your paper regarding, if I understand well, fractional analytic
functions, therefore some partial answer of my problems maybe is already
there, but a non-formal answer from you is necessary for me, to get rid
of the clouds that my poor mathematical backgrounds imply.
Also, attached you will find some reference to paper that I think maybe
of some importance, but again, your comments will guide me in the choice.
I will not take as an insult, moreover, if you say that the mathematics
that we need is very advanced even for a mathematician, therefore
requiring some time for engineering purposes, as this will at least move
me to less sophisticated work, where it is more likely to have success in
the close future. (This is because I have to think on convincing people
from industries to give me money, not because I'm not fascinated by these
subjects!)
Sorry for the long list of questions, and hope we can have seldom
discussions on the subject.
Thank you.
Yours,
Michele Ciavarella
Department of Engineering Science
University of Oxford
P.S. Here are the original question of my friend Dino Chiaia, from Turin,
that are closely related to mine, and maybe are more coincise.
1) are you aware of any attempt to extend the use of fractional calculus to
fractal spatial domains ?
2) Mandelbrot explained fractional brownian motion by means of fractional
derivatives. Tricot et al. tried to define the upper order of fractional
derivation for self-affine graphs like the Weierstrass function. These are
the only attempts, to my knowledge.
3) Our need is to find fractional operators to model stress and strains
(elasticity framework) in fractal (sponge-like) domains. The key
should be extension of Gauss-Green formulation and of the Laplacian
operator at least in 2 dims.
4) We do not want to model fractals as the limit of iterated function
systems.
Our major aim is to change dimensionality of the physical quantities and
therefore we need proper new operators. Could you give us some hints in
this direction ???
Dino Chiaia
Politecnico Di Torino
Italy
REFERENCES:
[1]
TI- NON-RIEMANNIAN AND FRACTAL GEOMETRIES OF FRACTURING IN GEOMATERIALS
AU- NAGAHAMA, H
NA- TOHOKU UNIV,FAC SCI,INST GEOL & PALAEONTOL,SENDAI,MIYAGI 980,JAPAN
JN- GEOLOGISCHE RUNDSCHAU
PY- 1996
VO- 85
NO- 1
PG- 96-102
IS- 0016-7835
AB- The mechanism of earthquakes is presented by use of the elastic
dislocation theory. With consideration of the continuous dislocation
field, the general problem of medium deformation requires analysis
based on non-Riemannian geometry with the concept of the continuum
with a discontinuity (''no-more continuum''). Here we derive the
equilibrium equation (Navier equation) for the continuous dislocation
field by introducing the relation between the concepts of the
continuous dislocation theory and non-Riemannian geometry. This
equation is a generalization of the Laplace equation, which can
describe fractal processes like diffusion limited aggregation (DLA)
and dielectric breakdown (DB). Moreover, the kinematic compatibility
equations derived from Navier equation are the Laplace equations and
the solution of Navier equation can be put in terms of functions
which satisfy the biharmonic equation, suggesting a close formal
connection with fractal processes. Therefore, the relationship
between the non-Riemannian geometry and the fractal geometry of
fracturing (damage) in geomaterials as earthquakes can be understood
by using the Navier equation. Moreover, the continuous dislocation
theory can be applied to the problem of the earthquake formation with
active folding related with faulting (active flexural-slip folding
related to the continuous dislocation field).
[2]
TI- THE FUNDAMENTAL-SOLUTIONS FOR THE FRACTIONAL DIFFUSION-WAVE EQUATION
AU- MAINARDI, F
NA- UNIV BOLOGNA,DEPT PHYS,VIA IRNERIO 46,I-40126 BOLOGNA,ITALY
JN- APPLIED MATHEMATICS LETTERS
PY- 1996
VO- 9
NO- 6
PG- 23-28
IS- 0893-9659
AB- The time fractional diffusion-wave equation is obtained from the
classical diffusion or wave equation by replacing the first- or
second-order time derivative by a fractional derivative of order 2
beta with 0 < beta less than or equal to 1/2 or 1/2 < beta less than
or equal to 1, respectively. Using the method of the Laplace
transform, it is shown that the fundamental solutions of the basic
Cauchy and Signalling problems can be expressed in terms of an
auxiliary function M(z; beta), where z = \x\/t(beta) is the
similarity variable. Such function is proved to be an entire function
of Wright type.
[3]
TI- FRACTIONAL RELAXATION-OSCILLATION AND FRACTIONAL DIFFUSION-WAVE
PHENOMENA
AU- MAINARDI, F
NA- UNIV BOLOGNA,DEPT PHYS,I-40126 BOLOGNA,ITALY
JN- CHAOS SOLITONS & FRACTALS
PY- 1996
VO- 7
NO- 9
PG- 1461-1477
IS- 0960-0779
AB- The processes involving the basic phenomena of relaxation, diffusion,
oscillations and wave propagation are of great relevance in physics;
from a mathematical point of view they are known to be governed by
simple differential equations of order 1 and 2 in time. The
introduction of fractional derivatives of order alpha in time, with 0
< alpha < 1 or 1 < alpha < 2, leads to processes that, in
mathematical physics, we may refer to as fractional phenomena. The
objective of this paper is to provide a general description of such
phenomena adopting a mathematical approach to the fractional calculus
that is as simple as possible. The analysis carried out by the
Laplace transform leads to certain special functions in one variable,
which generalize in a straightforward way the characteristic
functions of the basic phenomena, namely the exponential and the
gaussian. Copyright (C) 1996 Elsevier Science Ltd.
OTHER REFERENCES:
TI- OVERVIEW OF ELECTRICAL PROCESSES IN FRACTAL GEOMETRY - FROM
ELECTRODYNAMIC RELAXATION TO SUPERCONDUCTIVITY
AU- LEMEHAUTE, A;HELIODORE, F;DIONNET, V
NA- ALCATEL ALSTHOM RECH,F-91460 MARCOUSSIS,FRANCE
JN- PROCEEDINGS OF THE IEEE
PY- 1993
VO- 81
NO- 10
PG- 1500-1510
IS- 0018-9219
AB- This paper is devoted to a general analysis of the consequences of
the parametrization of the fractal set on the electrodynamics of this
set. The relevance of scaling properties to electrochemical,
dielectric, and magnetic relaxations is considered with a special
emphasis on the use of noninteger derivative operators in
electromagnetism and superconductivity.
In electromagnetism, the above analysis gives a brief overview of the
main results already obtained, focusing especially on the
introduction of dissipative terms in the equation of propagation and
on the generalized form of the uncertainty principle in fractal
media. The new Laplacian and d'Alembertian operators are evoked as
well as the scale relativity on which this new analysis is founded.
For superconductivity, the analysis introduces a geometrical
interpretation founded on frustration acting not only on topology but
on the metric of the space-time in a particular type of fractal
geometry. Although this point of view may appear as a break-through
in the theory of superconductors, the model offers some relations
with the theory of fractional statistics and the theory of Anyons.
PA- 8510325 FR;CASSOUX_P
165340041 FR;LEMEHAUTE_A
CR- ALEXANDER_S, 1983 Vol.44 p.13, J PHYS LETT
ALEXANDER_S, 1983 p.805, J PHYS-PARIS
BROIDE_ML, 1986 Vol.2 p.65, FRACTAL ASPECTS MATE
TI- THE EXACT SOLUTION OF CERTAIN DIFFERENTIAL-EQUATIONS OF
FRACTIONAL ORDER BY USING OPERATIONAL CALCULUS
AU- LUCHKO, YF;SRIVASTAVA, HM
NA- BELARUSSIAN STATE UNIV,DEPT MECH & MATH,MINSK 220050,BYELARUS
UNIV VICTORIA,DEPT MATH & STAT,VICTORIA,BC V8W 3P4,CANADA
JN- COMPUTERS & MATHEMATICS WITH APPLICATIONS
PY- 1995
VO- 29
NO- 8
PG- 73-85
IS- 0898-1221
AB- In the present paper, the authors first develop an operational
calculus for the familiar Riemann-Liouville fractional differential
operator. This operational calculus is then used here to solve a
Cauchy boundary-value problem for a certain linear equation involving
the Riemann-Liouville fractional derivatives. Relevant connections
are also indicated with the special cases of the equation, which were
solved earlier by using other methods.
TI- FRACTIONAL GREEN-FUNCTIONS
AU- MILLER, KS;ROSS, B
NA- PROMETHENS INC,NEWPORT,RI,02840
UNIV NEW HAVEN,DEPT MATH,W HAVEN,CT,06516
JN- INDIAN JOURNAL OF PURE & APPLIED MATHEMATICS
PY- 1991
VO- 22
NO- 9
PG- 763-767
AB- In this brief paper we shall show how the problem of finding
solutions to a wide class of fractional differential systems may be
reduced to a problem in ordinary differential equations. With this
method the only way the fractional calculus enters into the picture
is through the computation of fractional derivatives of known
functions. We achieve this goal by introducing the concept of
fractional Green's function.
Dr Huang
http://server.drhuang.com/input/?guess=dsolve%28ds%28y%2Ct%2C0.5%29%2B2ds%28y%2Cx%2C2%29%2B4ds%28y%2Cx%2C1%29%2Bexp%28x%29*t%29&inp=ds%28y%2Ct%2C0.5%29%2B2ds%28y%2Cx%2C2%29%2B4ds%28y%2Cx%2C1%29%2Bexp%28x%29*t&lang=null
Dr Huang
button and method to solve partial differential eq are changed to pdsolve( ) or PDE( ) from dsolve().
more info
Analytical Solution of Fractional Differential Equation by MathHand.com
http://drhuang.com/science/mathematics/fractional_calculus/fractional_differential_equation.htm