Neural Network Method For Solving A Nonlinear Problem Of Cross- Diffusion Task With Variable Density
Muhamediyeva Dildora Kabilovna, Madina Eldarovna Shaazizova, Ilxom Tursunbayevich Ismailov, Malika Yuldashovna Doshchanova
Sultanmurat Uali uli Nasirov E-mail: [email protected]
E-mail: [email protected]
Tashkent University of Information Technologies named after Muhammad al-Khwarizmi, Tashkent, Uzbekistan.
Abstract: This paper examines the qualitative properties of solutions for cross-diffusion systems of a
biological population with double nonlinearity and variable density. A self-similar approach is considered. A critical case is investigated.
Keywords: Self-similar equation, nonlinear task, critical case, interdiffusion coefficients, radially
symmetric form.
Introduction
Formalization of the problem in the form of an objective function and a system is an integer nonlinear programming problem.
To solve such problems in general and the distribution problem in particular, various exact and approximate methods of combinatorial optimization are used. In most cases, the method that guarantees finding the optimal solution is a complete enumeration of all possible options. However, the set of options for feasible solutions to such problems grows rapidly with an increase in the dimension of the input data, which makes the use of the exhaustive search method unacceptable in practice.
Statement of the task
It was found that when considering such a problem, a critical case arises and the behavior of the solution in this case changes. In area Q={(t,x): 0< t < , xRN}.
Let’s consider a cross-diffusion system of with double nonlinearity and variable density:
( ) ( )
( ) ( )
1 1
2 2
1 2 1
1 2 1 1 1 1 1
1 2 2
2 1 2 2 2 2 2
( ( ) )
( ) 1 ,
( ( ) )
( ) 1 ,
n m p
n m p
x u div D x u u u x k u u
t
x u div D x u u u x k u u
t
− −
− −
= + −
= + −
(1)
1 t 0 10( )
u = =u x , u2 t=0=u20( )x
. (2) Here: D x u1 n 2m1−1u1 p−2u1,D x u2 n 1m2−1u2 p−2u2
- interdiffusion coefficients,
1, 2, , , 1, 2
m m n p ,D1,D2- positive numeric parameters, (.) (.)
x
−grad , 1, 2 1, ( )x = x−l ,xRN l0; u1=u t x1( , )0, u2 =u t x2( , )0 - solution of a cross-diffusion system of a biological population with double nonlinearity and variable density.
We will study the properties of solutions to problem (1), (2) based on a self-similar analysis of solutions to the system of equations constructed by the method of nonlinear splitting and reference equations and by reducing system (1) to radially symmetric form.
Note that the change in (1)u t x1( , )=e−k t1v1( ( ), ( )) t x , u t x2( , )=e−k t2v2( ( ), ( )) t x will bring it to the form:
( )
( )
1 1 1 2
1 1
2 2 2 1
2 2
2 ( 2) ( 1)
1 1
1
1 2 1 1 1 1
2 ( 2) ( 1)
1 1
2
2 1 2 2 2 2
( ( ) )
( ) ,
( ( ) )
( ) ,
n m p p k m k t
n m p p k m k t
x v div D x v v v x k e v
x v div D x v v v x k e v
− − + − −
− +
− − + + −
− +
= −
= −
(3)
1 t 0 10( )
v = =v x , v2 t=0=v20( )x . (4) Let k p1( −(m1+1))=k p2( −(m2 +1)), then choosing
1 2 1 2 1 2
[( 1) ( 2) ] [( 1) ( 2) ]
1 2 1 2 1 2
( ) ( 1) ( 2) ( 1) ( 2)
m k p k t m k p k t
e e
t m k p k m k p k
= − + − = − + −
− + − − + − , ( )x = x p1 / p1, p1 =(p−(n+l)) / p, we obtain the following system of cross-diffusion equations:
1 1 1
2 2 2
2
1 1
1 1
1 1 1
1 2 1 1
2
1 1
1 1
2 2 2
2 1 2 2
,
.
p
m b
s s
p
m b
s s
w w w
D w a w
w w w
D w a w
−
− +
− −
−
− +
− −
= −
= −
(5)
Here:
( )
11 1 ( 2) 1 ( 1 1) 2 b
a =k p− k + m − k , 1 1 1 1 2
1 1 2
( ( 2)) ( 1)
( 2) ( 1) ,
p k m k
b p k m k
− − − −
= − + −
( )
22 2 ( 2 1) 1 ( 2) 2 b
a =k m − k + p− k , 2 ( 2 ( 2)) 2 ( 2 1) 1,
( 1) ( 2)
p k m k
b m k p k
− − − −
= − + −
( ) ( ), p N l
s p n l
p n l
= − +
− + .
When conditions are met bi =0, and a ti( )=const,i=1,2, cross-diffusion system has the form:
1 1
2 2
2
1 1
1 1
1 1 1
1 2 1 1
2
1 1
1 1
2 2 2
2 1 2 2
,
,
p m
s s
p m
s s
w w w
D w a w
w w w
D w a w
−
− +
− −
−
− +
− −
= −
= −
(6)
Below we describe one of the ways to obtain a self-similar system for the system of equations (5). It consists of the following. Initially, we find a solution to the following system of equations:
1
2
1 1
1 11 2 1
2 2
, . dw a w
d
dw a w
d
+
+
= −
= −
If bi =0, и a ti( )=const,i=1,2, then the solution of the equations has the following form:
1
1 1
1
( ) ( ( )) , 1 w t
= − = , 2 2 2
2
( ) ( ( )) , 1 w t
= − = .
In case bi 0, and a ti( )=const,i=1,2 we find a solution to the following system of equations:
1 1
2 2
1 1
1 11
2 1
2 2
, .
b
b
dw a w
d
dw a w
d
+
+
= −
= −
Solution of the equations is as follows:
1 1 1 1
1
( ) ( ( )) , b 1 w t
− +
= = , 2 2 2 2
2
( ) ( ( )) , b 1 w t
− +
= = .
Using the method of nonlinear splitting, the solution of system (5) is sought in the form:
1 1 1
2 2 2
( , ) ( ) ( ( ), ( )), ( , ) ( ) ( ( ), ( )).
v t x w z t x
v t x w z t x
=
= (7) If 1(p− +2) 2(m1− =1) 2(p− +2) 1(m2 −1), then parameter = ( )t is selected as follows:
1 2 1
1
1 [ ( 2) ( 1)]
1 2 1
1 2 1
( 1) ( 2)
1 1 2 1 2 1
0
1
1 ( ) , 1 [ ( 2) ( 1)] 0,
1 [ ( 2) ( 1)]
( ) ( ) ( ) ln( ), 1 [ ( 2) ( 1)] 0,
( ), 2 1,
p m
m p
T if p m
p m
v t v t dt T if p m
T if p и m
− − + −
−
−
+ − − + −
− − + −
= = + − − + − =
+ = =
Then
for the new variable zi( , ( )), x i=1, 2 we obtain the system of equations:
( )
( )
1 1
2 2
1 2 1
1 1
1
1 2 1 1 1 1 1
1 2 1
1 1
2
2 1 2 2 2 2 2
( z ),
( z ),
m p
s s
m p
s s
z div D z z z z
z div D z z z z
− − +
− −
− − +
− −
= + −
= + −
(8)
where
1 [1( 2 ) 2( 1 1)]
1 2 1
1 2 1
1
( )
1 1 1 2 1
1 , 1 [ ( 2) ( 1) 0,
(1 [ ( 2) ( 1)])
, 1 [ ( 2) ( 1) 0,
p m
if p m
p m
с if p m
− − − + −
− − + −
− − + −
= − − + − =
(9)
2 1 2
2 1 2
2 1 2
2
(1 [ ( 2) ( 1)])
2 1 2 1 2
1 , 1 [ ( 2) ( 1)] 0,
(1 [ ( 2) ( 1)])
, 1 [ ( 2) ( 1)] 0.
p m
if p m
p m
с if p m
− − − + −
− − + −
− − + −
= − − + − =
If 1 [ (− 1 p− +2) 2(m1− =1) 0, self-similar solution of system (9) has the form ( ( ), ) ( ), 1,2, ( ) / [ ( )]1/p
i i
z t = f i= = x t . (10)
Then substituting (10) into (8) with respect to fi( ) we obtain the system of self-similar equations
1 1
2 2
2 1
1 1 1 1 1
2 1 1 1
2 1
1 1 2 2 2
1 2 2 2
( ) (1 ) 0,
( ) (1 ) 0.
p m
s s
p m
s s
d df df df
f f f
d d d p d
d df df df
f f f
d d d p d
−
−
− −
−
− − −
+ + − =
+ + − =
(11)
where 1
1 2 1
1
(1 [ (p 2) (m 1)])
= − − + − and 2
2 1 2
1
(1 [ (p 2) (m 1)])
= − − + − .
System (11) has an approximate solution of the form f1 =A a( −) ,n1 = p/ (p−1),
2
2 ( )n
f =B a− , where А and В constant and
1
1 2
1 2
( 1)( ( 1))
( 2) ( 1)( 1)
p p m
n p m m
− − +
= − − − − , 2 2 2
1 2
( 1)( ( 1))
( 2) ( 1)( 1)
p p m
n p m m
− − +
= − − − − .
In this section, we solve the problem of choosing an initial approximation for the iterative process, which leads to a fast convergence to the solution of the Cauchy problem (1), (2), depending on the values of the numerical parameters and the initial data. For this purpose, the asymptotic representation of the solution found by us was used as an initial approximation.
Construction of the upper solution for cross-diffusion systems of a biological population
Let us start constructing an upper solution for system (11).
Note that the functions f1( ), f2( ) have properties:
( ) ( )
1 2
2 2
2
1 1 1 1 1 1
2 1 1
2
1 2 2 1 1
1 2 2
(0, ),
(0, )
p
m p m p
p
m m p
df df
f A B f C
d d
df df
f A B f C
d d
−
− − − −
−
− − −
= −
= −
and
1 1
2 2
2
1 1 1
1 1 1 1 1 1
2 1 1 1
2
1 1 1
1 1 2 2 1 2
1 2 2 2
,
.
p
m p m
s s p
p
m p m
s s p
d df df df
f A B sf
d d d d
d df df df
f A B sf
d d d d
−
− − −
− − −
−
− − −
− − −
= − +
= − +
Let us choose A and B from the system of nonlinear algebraic equations
1
2
1 1 1
1 1
1 1 1
2 2
1 / , 1 / .
p p m
p m p
A B p
A B p
− − −
− − −
=
=
Then function f1, f2 are a Zeldovich-Kompaneets type solution for system (1) and in
the domain ( )a (p−1)/p they satisfy the system of equations
1
2
2
1 1 1 1 1 1
2 1
2 1
1 1 2 2 2
1 2
0,
0
p
s s m
p m
s s
d df df df s
f f
d d d p d p
d df df df s
f f
d d d p d p
−
−
− −
−
−
− −
+ + =
+ + =
in the classical sense.
Due to the fact that
1
2
2 1
1 1 1
2 1
2 1
1 2 2
1 2
, ,
p m
s s
p m
s s
df df
f f
d d
df df
f f
d d
−
−
−
−
−
−
=
=
function f1( ), f2( ) and the flows have the following smoothness property
( )
1
2 1
1 1 1
1 2 1
0 ( ), 0, ,
p m
s df df s
f f f C
d d
−
−
− =
( )
2
2 1
1 2 2
2 1 2
0 ( ), 0,
p m
s df df s
f f f C
d d
−
−
− = .
Let us choose A and B such that the inequalities
1
2
1 1 1
1 1
1 1 1
2 2
1 / , 1 / .
p p m
p m p
A B p
A B p
− − −
− − −
(12) Then, when
1
2
2 1
1 1 1 1 1
2 1
2 1
1 1 2 2 2
1 2
,
,
p m
s s
p m
s s
d df df df s
f f
d d d p d p
d df df df s
f f
d d d p d p
−
−
− −
−
−
− −
+ = −
+ = −
then due to the fact that
1 0, 2 0 (0, ),
df df
d d при
From (12) we have
1
1
2 1
1 1 1 1 1
2
2 1
1 1 2 2 2
1
0,
0, (0, ).
p m
s s
p m
s s
d df df df
d f d d p d
d df df df
d f d d p d
−
−
− − −
−
−
− −
+
+
Theorem 1. If ui(0, )x ui(0, ),x xR, f1= A a( −) ,n1 = p/ (p−1),
2
2 ( )n
f =B a− , 1 2 1
1 2
( 1)( ( 1))
( 2) ( 1)( 1)
p p m
n p m m
− − +
= − − − − , 2 2 2
1 2
( 1)( ( 1))
( 2) ( 1)( 1)
p p m
n p m m
− − +
= − − − − , then in the
domain Q the solution of problem (1) satisfies the upper bound
1 1
2 2
1 1 1
2 2 2
( , ) ( , ) ( ),
( , ) ( , ) ( ),
k t k t
u t x u t x e f u t x u t x e f
− +
− +
=
=
( ) / [ ( )] .x t 1/p
=
Note that the solution of system (1) for
2
1 2
( 2) ( 1)( 1)
( 1)( ( 1))
i
i
p m m
p p m
= − − − −
− − + has the following
representation for
1 2
1 1 2 2
1 1
/ ( ,1 ) / ( ,1 )
n n
a P B n P B n
= + = +
.
where В(a,b)- Beta Euler function.
It is proved that this representation is the asymptotic behavior of self-similar solutions to systems (1).
1 1
2 2
1
1 1
1
2 2
( ) ,
( ) ,
n
n
a dx P
a dx P
−
+
−
−
+
−
− =
− =
1 1
1 1 1
2 2
2 2 2
1 1 1 1
1 1 1
0
1 1 1 1
2 2 2
0
1 1 1
( ) (1 ) ( ,1 ) ,
1 1 1
( ) (1 ) ( ,1 ) .
n n
n n
n n
n n
a dx a d a B n P
a dx a d a B n P
− −
+
−
− −
+
−
− = − = + =
− = − = + =
Here:
1 2
1 1 2 2
1 1
[ / ( ,1 )]n [ / ( ,1 )] .n
a P B n P B n
= + = +
At n1 0,n2 0, n0 we get the following functions
1 2
1( ) (a ) n, 2( ) (a ) n .
= − + = − +
Here: a0, ( )y + =max
( )
y, 0 , a. For a global solution of the system of equations (1) to exist for the function fi( )
,i=1, 2 must satisfy the following inequality [5]:1 1
1 1
2 1
1 1 1 1 1
2 1 1 1
2 1
1 1 2 2 2
1 1 1 1
(1 ) 0,
(1 ) 0,
(0, ).
p m
s s
p m
s s
d df df df
f f f
d d d p d
d df df df
f f f
d d d p d
−
− − − −
−
−
− −
+ + −
+ + −
Here:
1 1/n2, 2 1/n1
= = .
Development of an algorithm and program for solving cross-diffusion systems Consider in the area D= (0, ), T RN, = −
b x b, =1, 2
two- dimensional reaction problem with diffusion( )
( ) ( ) ( )
1 2
1 1 2 2 1 2
2 2
1 2 1 2
( ) ( ) ( ) 1 ,
, , , ,
u u u
u u u u v t v t k t u u
t x x x x x x
u u x x t x x x
= + − − + −
= = +
(17)
x
with initial and boundary conditions
( )
(0, ) 0 0,
u x =u x (18) ( , )
uГ = x t , t(0, )T , Г – граница (19) which, in the case of degenerating, is equivalent to the Cauchy problem with a finite initial function
( )
2 1/ 1/20
0
, ( ) ; 1; / ; ( ) [ ( )] .
4
t
u t x u t a a x t u d
= − = = =
( , ) : ( )
k t x =k t ; 1 ( ) (1 ) k t = t
+ , 1;
0
1/
( )
( ) 1 ,
1
t
k d
u t
e
−
=
+
where
1) ( ) +, 2) 0
( )
t
k d
q e d
+
.Let 0k t x( , )C(0,+ ) RN. In build a uniform mesh h byx, ( =1, 2) with steps 1 1
1
h b
= n и 2 2
2
h b
= n :
( )
1i, 2j , 1i 1, 2j 2, , 0,1,..., , 1,2
h xij x x x ih x jh i j n
= = = = = = ,
and uniform grid in time =
tk =k , 0, k =0,1,..., ,m m=T
, T 0.The idea is as follows: Enter an intermediate value y = yk+12, где y= yk, yˆ = yk+1 , k - layer number, which can be regarded as the value of y at =t tk+1/2= +tk / 2
( )
12
12
12
12
1 2
1
1 1
1 2
( ), 0.5
0.5 ,
k k
k k k
k k
k k k
y y
y y q y
y y
y y q y
+ +
+
+ + + +
− = + +
−
= + +
(20)
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2
1
1 2 1, 1, , , , 1, 1, ,
1 1
2
2 2 , 1 , 1 , , , , 1 , 1 ,
2 2
1 ,
1 ,
k k k k k k k ij k k
i j i j i j i j i j i j i j i j
ij
k k k k k k k k k
i j i j i j i j i j i j i j i j
y a y y y a y y y y y
h h
y b y y y b y y y y y
h h
+ + + + + + + + +
+ + − +
+ + − +
= − − − + −
= − − − + −
1 2 , ,
( ) ( , i, j) i j(1 i j)
q y =k t x x y −y , xm =
(
x12 +x22)
m.Here the difference coefficients a y
( )
и b y( ) must satisfy the conditions of the second order of approximation and one of the following formulas is used to calculate:а) ,
( )
1, ,2
i j i j
i j
y y
a y K − +
= , ,
( )
, 1 ,2
i j i j
i j
y y
b y K − +
= ,i j, =1,2,...,n −1, =1,2, (21)
б)
( ) (
1,) ( )
,, 2
i j i j
i j
K y K y
a y − +
= , ,
( )
( , 1) ( , )2
i j i j
i j
K y K y
b y − +
= , ,i j=1,2,..,n −1, =1,2, (22) where K(u)=u.
Using formula (22), we have
, 1, ,
( ) 1 ( ) ( )
i j 2 i j i j
a y = y− + y , , 1 , 1 ,
( ) ( ) ( )
i j 2 i j i j
b y = y − + y . We rewrite the initial and boundary conditions as follows:
( )
1 1
2 2
0
, 0
1 1
, 2
, 1
, ,
, при 0 и , , при i 0 и ,
i j h
k k
i j
k k
i j
y u x x
y j j n
y i n
+ +
+ +
=
= = =
= = =
(23)
where 1
(
1)
2(
1)
2 4
k k k k
= + + − + − , which is obtained from system (20), after eliminating the intermediate value yk+12.
We rewrite (20) as:
12
12
1 1 1 1 1
2 2 2 2 2
,
1 , , , 2
1
, 1 k
2 , i,j , 1
, 2 ( ), 0.5
, F 2 ( ),
0.5
k
i j k k k k k k
i j i j i j
k
i j k k k k k
i j i j
y y F F y y q y
y y F y y q y
+
+
+
+ + + + +
+
= + = + +
= + = + +
(24)
we also agree on the following notation: yk = y, yk+12 = y, yk+1= yˆ.
To solve the resulting scheme of nonlinear equations, we use the iterative method.
(
12 12)
1 1 1 1 1 1
,
1, 1, , , , 1, 1, , ,
2
1 1
1 ( ) ( ) 0
0.5
s
s s s s s s s
i j ij k k
i j i j i j i j i j i j i j i j i j
y a y y y a y y y y y F
h h
+
+ + + +
+ +
+ + − +
= − − − + − + =
,
(25)
( )
1 1 1 1 1 2
,
, 1 , 1 , , , , 1 , 1 , ,
2
1 2
ˆ 1 ( )ˆ ˆ ˆ ( )ˆ ˆ ˆ 0
0.5
s
s s s s s s s
i j ij k k
i j i j i j i j i j i j i j i j i j
y b y y y b y y y y y F
h h
+
+ + + +
+ + − +
= − − − + − + =
,
(26) where ai j, ( )y and bi j, ( )y are defined by formula (22).
he iterative process is performed according to the following schemes:
The approximation is performed by the Picard method (simple iteration):
1
1
1 2
1
1
1 2
0.5 ( ),
ˆ ˆ ˆ .
0.5
s
s s
s
s s
y y
y y q y
y y
y y q y
+ +
+ +
− = + +
− = + +
In (25), introducing the notation
1, 1 ,
, 2 , 1, , 2 , ,
1 1 1
, ,
,
0.5 0.5
, ,
1 , 0 1 2 1 2 1, 1, 2,
m m
s s s s s s
i j i j
i j i j i j ij i j i j i j
s s s
i j i j
i j α
x x
A y a y B y a y
h h h
C A B s , , ,... i, j , ,...,n
+
+
= + =
= + + = = − =
difference equation can be written as:
1 1 1
, 1, , , , 1, ,
1
, , при 0, ,
s s s s s s s
i j i j i j i j i j i j i j
A y C y B y F
y i n
+ + +
+ −
− + = −
= =
(27)
, 1,2,..., 1, 1,2.
i j= n − =
Accordingly, (26) can be written as:
1 1 1
, 1, , , , , 1 ,
2
ˆ ˆ ˆ ,
ˆ , at 0, ,
s s s s s s s
i j i j i j i j i j i j i j
A y C y B y F
y i n
+ + +
+ −
− + = −
= =
(28) where
, 1 2 ,
, 2 , 1, , 2 , ,
2 2 1
, ,
,
0.5 0.5
ˆ ˆ , ,
1 , 0 1 2 1 2 1, 1, 2.
m m
s s s s s s
i j i j
i j i j i j ij i j i j i j
s s s
i j i j
i j α
x x
A y b y B y a y
h h h
C A B s , , ,... i, j , ,...,n
+
+
= + =
= + + = = − =
For the numerical solution of problems (27) and (28), the sweep method is used.
System of equations (27) is solved along the lines j=1, 2,...,n2 −1 and is determined y at all grid points h. Then the system of equations (28) is solved along the columns
1,2,..., 1 1
i= n − defining yˆ at all grid points h. When passing from layer k + 1 to layer k + 2, the counting procedure is repeated. The results of the C ++ solution are shown in Fig.1.
Conclusion
The processes of multicomponent cross-diffusion systems of a biological population with double nonlinearity and variable density are simulated on a computer.
Estimates are obtained for solving the Cauchy problem for multicomponent cross- diffusion systems of a biological population with double nonlinearity and variable density.
References
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IOP Conf. Series: Journal of Physics: Conf. Series 1210 (2019) 012101. DOI:10.1088/1742- 6596/1210/1/012101
2. Muhamediyeva D.K. Two-dimensional Model of the Reaction-Diffusion with Nonlocal Interaction // 2019 International Conference on Information Science and Communications Technologies
(ICISCT), Tashkent, Uzbekistan, 2019, pp. 1-5. DOI:
https://doi.org/10.1109/ICISCT47632019.9011854
3. Muhamediyeva D.K., Nurumova A.Yu., Muminov S.Yu. Study Of Multicomponent Cross-Diffusion Systems Of Biological Population With Convective Transfer // European Journal of Molecular &
Clinical Medicine ISSN 2515-8260 Volume 7, Issue 11, 2020, pp. 2934-2944.
4. Bensimon, B. Shraiman, L.P. Kadanoff. Mean Field Theory for a ballistic Model of Aggregation//
Kinetics of Aggregation and Gelation, edited by F. Family, D.P. Landau (Elsevier-North Holland, Amsterdam, 1984), р.75-79.
5. Mittal R. C. and Arora G. Quintic B-spline collocation method for numerical solution of the extended Fisher-Kolmogorov equation. // Int. J. of Appl. Math and Mech. 6 (1): 74-85, 2010.
Solution results
Figure 1. Solution results
