DEFINITION THE FRACTAL MODEL OF DIFFUSION MAIN
PARAMETERS FOR SEMIBOUNDED TWO-LAYER ONEDIMENSIONAL MEDIUM*1
E.S. Mogileva
Penza State University. Penza. Russia
Abstract. Inverse problems without initial conditions for the diffusion equation of
fractional order at the time for semi-bounded two-layered medium are investigated.
The algorithm of identification of the fractal model parameters proposed. Formulas
for calculation of the generalized diffusion coefficient λ, and index of anomalous
diffusion β received. Length values of cross-border layer found.
INTRODUCTION
Inverse coefficient problem for the homogeneous case. Ivashchenko D.S.[1]
considered the solution of the inverse problem for the diffusion equation of fractional
order with constant coefficients in its work. Solution of boundary value problems
without initial conditions or signal problems is realized with the help the Riemann–
Liouville differential operator. The impact of initial conditions practically no effect
on the temperature distribution at the time of observation in the problems of this type.
Boundary condition:
2
t D u ( x, t )
2

u ( x, t ), u (0, t )  Aeiwt .
2
x
2
Solution of boundary value problems without initial conditions is written in the
form u ( x, t )  Ae
(iwt 
(iw) 

x)
.
A real part of the expression is of the form
u ( x, t )  Ae
1
(
w x

e-mail: ElenaSergIvan@yandex.ru
cos

2
)
cos( wt 
w x

sin

2
).
Let`s based on the solution u ( x, t ) and the additional condition u ( x0 , t j )  c j .
We define our values  and  , where u ( x0 , t j ) is experimentally found value u
for the distance x 0 from the source at the moments of time t j . Let u ( x0 , t0 )  c0 ,
u ( x0 , t1 )  c1 are set and condition t1  t0 

fulfilled. Coefficient problems
2w
consist in the search for  ,  and  ,  at the same time.
INVERSE COEFFICIENT PROBLEM FOR THE HOMOGENEOUS CASE
The inverse problem 1 consists in the definition of the generalized diffusion
coefficient  for a given index of anomalous diffusion β to the data of the inverse
problem.
c0  u( x0 , t 0 ) , c1  u( x0 , t1 ) .
c0  Ae

w x

cos

2
cos( wt 0 
w x

sin

2
) , c1  Ae

w x

cos

2
cos( wt1 
w x

sin

2
)
The solution of the inverse problem

2w x0

cos
2
2ln A  ln(c02  c12 )
(1)
The inverse problem 2 consists in the definition index of anomalous diffusion β
for a given of the generalized diffusion coefficient  to the data of the inverse
problem. The solution of the inverse problem w cos(

2
)

2 x0
ln(
A2
)
c02  c12
PROSPECTS
The inverse coefficient problem for the case of ideal contact on the real half-axis.
Statement of the inverse coefficient problem without initial conditions for the
diffusion equation of fractional order at the time with constant coefficients for the
case of one point interface on the real axis. The first inverse coefficient problem - to
define the generalized diffusion coefficient  if  is fixed. Let's calculate
u 20 ( x0 )  u 20 ( x0  2l ) 
2k1
e  ax0  e  a ( x0  2l )
Re Ae iwt
k1  k 2
1  e a 2l

2k1
Re Ae iwt e ax0
k1  k 2

u 21 ( x0 )  u 21 ( x0  2l ) 
2k1
e  ax0  e  a ( x0  2l )
Im Ae iwt
k1  k 2
1  e  a 2l

2k1
Im Ae iwt e ax0
k1  k 2

In the considered formulas from work [1] we will replace c0 , c1 on c0 ,c1 respectively,
let's designate: ci 
k1  k2
k k
u ( x0 , ti )  1 2 u ( x0  2l , ti ), i  0,1.
2k1
2k1
Remark. Ivashchenko D.S. proposes as a solution of the second the inverse problem
equation in the work [1]. This equation does not give explicit algorithm for calculate
the index of anomalous diffusion  . Let's submit an explicit solution to the second of

(c  ic1 ) iwt
the inverse problem. let's define u  c0  ic1 then 0
e
e
A
Passing to logarithms, we have ln
( iw) 

x
.


i w
c0  ic1
 iwt  e 2
x.
A

Let's equate absolute values of the right and left parts:
1
w
2
2
ln( c0  c1 )  ln A  i(  w) 
x0 .
2

Then w  

x0
1
2
2
( ln( c0  c1 )  ln A) 2  (  w) 2 .
2
As a result, we obtain an explicit formula for  :
  c0 2  c12 

ln 
 (  wt )2 
2
x
A
x0
 ,  arg(c  ic ).
  0
0
1
ln w
(2)
The inverse coefficient problem - to define the length l. We calculate the
unknown parameter with the help of available measurements. Let's enter new
designation b – relation of coefficients thermal diffusivity of two substances.
iw

x 
 1  b c  e a1 0 
1

ln 
iw
 1  b  a1 x0

e
 c1 

l
iw
2
a1
(3)
CONCLUSION
Inverse coefficient problem for the homogeneous case, the inverse coefficient
problem for the case of ideal contact on the real half-axis are investigated. Solution
signal problems is realized with the help the Riemann–Liouville differential operator.
Inverse problems without initial conditions for the diffusion equation of fractional
order at the time for semi-bounded two-layered medium are solved. Formulas for
calculation of the generalized diffusion coefficient λ, and index of anomalous
diffusion β received, length values of cross-border layer found. The algorithm of
identification of index of anomalous diffusion β proposed.
REFERENCES
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problems for the diffusion equation of fractional order in time. The Tomsk State
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Prometheus, 2006. - 280 p.
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and Its Applications, Reading, 1984.
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