SIMULATION OF A CLASS OF PARABOLIC SYSTEMS OF QUASILINEAR REACTION-DIFFUSION EQUATIONS WITH DOUBLE N

International Journal of Mechanical and Production Engineering Research and Development (IJMPERD) ISSN (P): 2249–6890; ISSN (E): 2249–8001 Vol. 10, Issue 3, Jun 2020, 3089-3102 © TJPR Pvt. Ltd.

SIMULATION OF A CLASS OF PARABOLIC SYSTEMS OF QUASILINEAR REACTION-DIFFUSION EQUATIONS WITH DOUBLE NONLINEARITY D. K. MUHAMEDIYEVA Tashkent University of Information Technologies named after Muhammad al-Khwarizmi,Amir Temur street, Tashkent, Uzbekistan ABSTRACT In the proposed article, we study the properties of solutions to the initial problem, which is insufficiently studied and therefore interesting to trace the evolution of the reaction process with diffusion. The main results of numerical and analytical studies of nonlinear mathematical models describing population processes in a homogeneous and inhomogeneous medium with nonlinear diffusion by quasilinear parabolic equations are presented. We studied such classes of nonlinear differential equations in which the unknown function and the derivative of this function enter in a power-law manner, since real physical processes are nonlinear, and non-linear mathematical models should be used to describe them adequately.

Received: Jun 03, 2020; Accepted: Jun 23, 2020; Published: Jun 11, 2020; Paper Id.: IJMPERDJUN2020293

1. INTRODUCTION To solve the Cauchy problem of the reaction-diffusion equation, self-similar solutions satisfying the ordinary

Original Article

KEYWORDS:

differential equation are of great interest.

Studies of the self-similar problem carried out in [1-12]. The one-dimensional case well studied. In particular, it proved that there are a finite number of solutions to the self-similar problem on the line.

Two-dimensional problem is less studied. In its study, usually using numerical methods, and there are certain difficulties. The differential problem posed on the entire plane and it is necessary to choose the domain of numerical integration so that the constructed solution is sufficiently small on its boundary. In addition, when using purely implicit difference schemes, the Newton method usually used for iterations over nonlinearity. Therefore, it is necessary to learn how to build good initial approximations, that is, it is necessary to imagine in advance the form of the desired solution.

In [1] considered, the following Cauchy problem in QT  R N   0, T  , N  1 :





N   m1 p2   x u   u Du uxi t i 1 xi ,





u  x,0   u0  x  , x  R N , u0  x   0

suppu0  BR0   x  R0  , u0R N  

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(1)

, ,

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(2)

(4)

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D. K. Muhamediyeva

  1  x  , l  0 l

(5)

.

In [2], the following problem is considered:



ut  div u m1 Du

u m1 Du

p2

p 2



Du  0

u 0 n

в

QT     0, T  ,

  0, T  ,

на

(6)

(7)

u  x,0   u0  x  , x ,

(8)

where   R , N  2 - at m  p  3  0 (6) describes a fast diffusion process.

N

In [3], the Cauchy problem of the following form is considered:



ut  div u Du

m 1



Du  u p ,

where 0  m    1 .

In [4], the existence and non-existence in time of a solution to the following Cauchy problem for initial data slowly tending to zero are established. Considered:



ut  div u Du

 x, t   QT

m 1



Du  u p ,

(9)

 R N   0, T  , T  0, N  1,

u  0, x   u0  x  , x R N

(10)

,

m  1  where m    1, m  0, m    max 0,1   , p  1 . At m    1  0 , equality (9) refers to the equations N   of fast diffusion, the case m  1 arises in plasma physics [3].

In [5] it is shown that if p  p*  1  2 / N , then any non-negative solution to problem (9) - (10) “explodes” in a finite time. In case of exacerbation p  p* .

2. FORMULATION OF THE PROBLEM In Q  {(t, x) : 0  t<, x  R N } consider the cross-diffusion system of a biological population

 

n  u1 k  t   x u1   u2   x n u k 2  t

u1

t 0

 u10 ( x) , u2

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p2

p2

t 0

   k u  u

u2m1  k1  u1  u11  , u

m2 1

1

2

2 2

,

(11)

 u20 ( x) , SCOPUS Indexed Journal

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where m1 , m2 , n, p  0 , 1 ,  2  0 , u1  u1 (t , x)  0 , u2  u2 (t , x)  0 .

In this paper, we study the properties of solutions to problem (11) based on a self-similar approach by the method of nonlinear splitting [12].

Replace in (11) u1 (t , x)  e k1t v1 ( (t ), x), u2 (t , x)  ek2t v2 ( (t ), x) :

 

n  v1 k     x v1   v2   x n v k 2  

v1

t 0

 v10 ( x) , v2

p2

p2

t 0

 a 

v2m1  a1 b1 v11 , v

m2 1

2

(12) b2

2

v2 ,

 v20 ( x) .

Here

 (t ) 

b1 

e[ m1k2 ( p 2) kk1 k1 ]t e[ m2k1 ( p 2) kk2 k2 ]t  , m1k2  ( p  2)kk1  k1 m2 k1  ( p  2)kk2  k2

k   ( p  2)kk1  m2 k1 k11  ( p  2)kk1  m1k2 . , b2  2 2 m2 k1  ( p  2)kk1  k2 m1k2  ( p  2)kk1  k1

Then the solution of system (2) is sought in the form

v2 (t , x)  v2 ( )w2 ( (t ), ( x )),

v1 (t , x)  v1 ( )w1 ( (t ), ( x )),

 1 ( )  (T0   )  ,  2 ( )  (T0   )  , T0  0 , 2

1

Where at b1  0, b2  0 :  1  Then for wi ( , ( x )),

  wk  w1   1 s    s 1 1           w w2k 1 s   s 1   2         where 1



b 1 b 1 1 1 , 2  2 . , and at b1  0, b2  0 :  1  1 , 2  2  1 2 1 1  1 1  1

i  1,2 get the system of equations: w2m1     1 ( w1  w11 ),    p 2 w1m2     2 ( w2  w22 ),   

p 2

(14)

1 1 и 2  .

(1   1[( p  2)k  1]   2 m1 ) 1 (1   2 [( p  2)k  1]   1m2 ) 2

Here at p  n : and at p  n :

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(13)

( x )  x

p1

/ p1 , p1  ( p  n) / p , s  pN / ( p  n) ,

 ( x )  ln( x ).

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Self-similar solution of system (14) has the form

wi ( (t ),  )  fi ( ) ,    ( x ) /  Then substituting (15) in (14) for

  1 s    1 s  

df 1k d s 1 ( d d df d ( s 1 2 d d

where 1



p2

p2

1

p

.

(15)

fi ( ) get a system of self-similar equations

df 2m1  df1 )  1 ( f1  f11 )  0, d 2 d (16)

df  df 2 )  2 ( f 2  f 22 )  0. d 2 d m2 1

1 1 and  2  .

(1   1[( p  2)k  1]   2 m1 ) (1   2 [( p  2)k  1]   1m2 )

System (16) has an approximate solution of the form

f1  A(a   )  n1 , f 2  B(a   )  n2 , where n1



( p  1)[k( p  2)  (m1  1)] ( p  1)[k( p  2)  (m2  1)] , n2  .

2 [k ( p  2)  1]  m1m2 [k ( p  2)  1]2  m1m2

3. PARABOLIC SYSTEM OF TWO QUASILINEAR REACTION-DIFFUSION EQUATIONS Consider in the area Q  {(t, x) : 0  t, x  R}. parabolic system of two quasilinear reaction-diffusion equations: p 2  u u u1  u    1   D1u2m1 1 1   l (t ) 1  k1 (t )u1 1  u21 ,    t  x  x  x x      p 2  u2 u u2  u     l (t ) 2  k2 (t )u2 1  u12 ,   D2u1m2 1 2    t x  x x  x  







u1

(17)



t 0  u10 ( x).

Diffusion coefficients are equal D1u2m1 1

u1 x

p2

, D2u1m2 1

u2 x

p 2

and convective transport with speed l (t ), where

m1 , m2 , p, 1 , 2 - positive real numbers, u1  u1 (t , x)  0, u2  u2 (t , x)  0 - sought solutions.

Next, we investigate the qualitative properties of the problem under consideration by constructing a self-similar system of equations for (17).

We construct the self-similar system of equations by the method of nonlinear splitting [1].

Substitution in (17):

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u1 (t , x)  e

  k1 ( ) d 0

t



v1 ( (t ), ),   x  l ( )d  , u2 (t , x)  e

-3093 t

  k2 ( ) d  0

0

t

v2 ( (t ), ),   x   l ( )d , 0

will lead (1) to the form p 2  v v v1     1  D1v2m1 1 1   k1 (t )e(2 p ) k1  ( 1  m1 1) k2 t v1v21 ,          p 2   v2 v v2    D2 v1m2 1 2   k2 (t )e( 2  m2 1) k1  (2 p ) k2 t v12 v2 ,               

v1

t 0  v10 ( ),

v2

(18)

t 0  v20 ( ).

If (m1  1)k2  ( p  2)k1  (m2  1)k1  ( p  2)k2 , then choosing

 (t ) 

e[(m1 1)k2 ( p 2)k1 ]t e[(m2 1)k1  ( p  2)k2 ]t  , (m1  1)k2  ( p  2)k1 (m2  1)k1  ( p  2)k2

we obtain the following system of equations: p 2  v v v1     1  D1v2m1 1 1   a1 (t ) b1 v1v21 ,          p2   v2 v v2    D2 v1m2 1 2   a2  t  b2 v12 v2 ,               

(19)

where a1  k1  ( p  2)k1  (m1  1)k2  1 , b

b1 

(2  p)k1  ( 1  m1  1)k2 , ( p  2)k1  (m1  1)k2

a2  k2  (m2  1)k1  ( p  2)k2  2 , b

b2 

(  2  m2  1)k1  (2  p)k2 . (m2  1)k1  ( p  2)k2

If bi  0 and ai (t )  const , i  1, 2, then the system takes the form p 2  v v v1     1  D1v2m1 1 1   a1 v1v21 ,               p 2  v2 v2    m2 1 v2    a2 v12 v2 .  D v  2 1             

The Cauchy problem for system (19) in the case when b1  b2  0, studied in [2-6], where the existence of wave www.tjprc.org

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global solutions and blow-up solutions was also proved.

In order to obtain a self-similar system for the system of equations (19), we first find the solution to the system of ordinary differential equations:  d1 1  d  a11 2 ,   d 2  a  2 .

2 1 2  d

In the form

1 ( )  c1 (  T0 )1 ,  2 ( )  c2 (  T0 ) 2 , , T0  0, where

с1  1,  1 

1 1 , с2  1,  2  . 2 1

And then the solution to system (17) is sought as

v1 (t , )  v1 (t ) w1 ( , ), v2 (t , )  v 2 (t ) w2 ( , ), and    (t ) is chosen like this:

1  1[ 1 ( p  2)  2 ( m1 1)] , 1  [ ( p  2)   (m  1)] (T   ) 1 2 1   if 1  [ 1 ( p  2)   2 (m1  1)]  0, 1 ( )   v1( p  2) (t )v2( m1 1) (t )dt   ln(T   ), 0 if 1  [ 1 ( p  2)   2 (m1  1)]  0,  if p  2 и m1  1, (T   ), if 1 ( p  2)   2 (m1  1)   2 ( p  2)  1 (m2  1).

Then for wi ( , x), i  1,2 we get the system of equations: p 2  w w w1     1  D1w2m1 1 1   1 ( w1w21  w1 ),           p  2  w2 w2 w2     D2 w1m2 1   2 ( w2 w12  w2 ),               

(20)

where 1   (1  [ ( p  2)   (m  1)]) , if 1  [ 1 ( p  2)   2 (m1  1)  0, 1   1 2 1  (1[1 ( p2) 2 ( m11)] ) , if 1  [ 1 ( p  2)   2 (m1  1)  0,  1с1

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1   (1  [ ( p  2)   (m  1)]) , if 1  [ 2 ( p  2)   1 (m2  1)]  0, 2   2 1 2  с (1[ 2 ( p  2) 1 ( m2 1)]) , if 1  [ 2 ( p  2)   1 (m2  1)]  0.  2 1

The representation of system (18) in the form of (20) suggests that with    и  i  0 the solution of the last system asymptotically at the front may tend to solve the system p 2  w w w1     1  D1w2m1 1 1 ,           p2  w2 w2    m2 1 w2  .  D w  2 1         

This circumstance is used to find the initial approximation for the construction of the iterative process. If

1  [ 1 ( p  2)   2 (m1  1)  0, then system (20) has the wave solution:

wi ( (t ), )  fi ( ),   c , i  1, 2, where c is the wave velocity and functions wi ( (t ), )  fi ( ) find from the system of self-similar equations:

d  ( f 2m1 1  d  d m2 1  ( f1 d  

i 

df1 d df 2 d

p 2

p2

df1 df )  c 1  1 ( f1  f1 f 21 )  0, d d

(21)

df 2 df )  c 2  2 ( f 2  f 2 f12 )  0, d d

1 , i  1, 2, (1  [ i ( p  2)   3i (mi  1)])

which has a localized solution

f1  A(a   )n1 , f2  B(a   )n2 , n1 

 p  1 p  (m1  1)  n

, n2 

 p  1 p  (m2  1)  n

,

n   p  2    m1  1 m2  1 , 2

if p  2  [ m1  1 m2  1]1/2 , p  (mi  1)  0, i  1, 2, 1  1/ n1 ,  2  1/ n1.

And the coefficients A and B are determined from the solution of a system of nonlinear algebraic equations:

(n1 ) p1 A p1Bm1 1  c, (n2 ) p1 Am2 1B p1  c.

Then taking into account the expressions

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D. K. Muhamediyeva t

u1 (t , x)  e

  k1 ( ) d 0

v1 ( (t ), ),

t

u2 (t , x)  e

  k2 ( ) d  0

v2 ( (t ), ).

We have t

u1 (t , x)  Ae

  k1 ( ) d

(c (t )   ) n1 ,

0

t

u2 (t , x)  Be

  k2 ( ) d 0

(c (t )   ) n2 , c  0.

If t

x  [b (t )   l ( )d ], t  0, 0

then

u1 (t , x)  0, u2 (t , x)  0.

Therefore, the condition for localizing the solutions of system (17) is the conditions е

 l ( y)dy  ,  (t )   for t  0.

(22)

0

Condition (22) is a condition for the appearance of a new effect — the localization of wave solutions (22).

4. SLOW DIFFUSION Case n1  0, n2  0, n  0 (slow diffusion). Using the method [1] to solve equation (17), we obtain the following functions:

1 ( )  (a   )n1 , 2 ( )  (a   )n2 , where a  0 ,

( y)  max  y, 0 ,   a.

It is known [1, 2] that for the global existence of the solution of problem (17) the function f   must satisfy the following inequality: d  ( f 2m1 1  d  d m2 1  ( f1  d

df1 d df 2 d

p 2

p 2

df1 df )  c 1  1 ( f1  f1 f 21 )  0, d d df 2 df )  c 2  2 ( f 2  f 2 f12 )  0, d d

And 1  1/ n2 , 2  1/ n1.

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-3097

Theorem 1. Take functions 1 ( ),  2 ( ) and show that they will be the asymptotics of the finite solutions of system (21)and show that they will be the asymptotics of the finite solutions of system (21).

  a has asymptotics

fi ( )~i   .

Proof. We will seek a solution to equation (21) in the following form:

f i  i   yi ( ), i  1,2,

(23)

where    ln  a    , and    at   a , which allows us to study the asymptotic stability

  .

Substituting (23) into (21), for yi ( ) we get the following equation:

d  mi 1 dyi y n y d  3i d i i

p 2

  e   mi 1 dyi dy ( i  ni yi )     n  ni yi   y3i i   a  e d d   

p 2

(

e  dy  c  i  ni yi   i y ( )(1  e n3i i y3i i )  0,  i d  ae  

 dyi  ni yi )    d 

(24)

where - function defined above.

Note that studying the solution of the last equation is equivalent to studying those solutions of equation (21), each of which in a certain interval

yi    0,

0 ,   satisfies the inequality

dyi  ni yi  0. d

We show first of all that the solutions yi   equations (21) have a finite limit

y0i at    . We introduce the

following notation:

i   

y3mi i1

dyi  ni yi d

p2

(

dyi  ni yi ). d

Then equation (21) takes the form  e

i   

ae



 e  dy   ni  i  c  i  ni yi   i y ( )(1  e ni i y3i i ).  i d  ae   

To analyze the last expression, we introduce a new auxiliary function  e

 ( , )   

ae



 e  dy   ni   c  i  ni yi   i y ( )(1  e ni i y3i i ),  i d  ae   

where  - real number.

Hence it is easy to see that for each value



function

  , 

retains a mark at some interval

1,    0 ,   and for all  1,  one of the inequalities

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i   0, i   0.

And so for the function

i   there is a limit for  1,  . From the expression for i   follows that

lim i   

 

  e



 dy

e





  lim    ni  i  c  i  ni yi   i y ( )(1  e ni i y3i 3i )   0.   i    ae  d     ae Hence, given that

   a   h, lim e  0, lim a  e  a,  

 

i  0.

We obtain the following algebraic equation: 1) i  1, i  1, 2,

 n1 p1 ym2 1y1p1  c,  n 2 p1 y1m 1yp21  c.

1

2

The solution to the latter system gives yi  1 and by virtue (21), f i   ~i   .

2) i  1/ ni , i  1, 2, yi should be a solution to the system

 n1 p1 ym2 1y1p2  y1n y2n ( 1)  c,  n 2 p1 y1m 1y2p2  y1n ( 1) y2n  c.

1

1

1

2

2

2

1

2

Theorem 1 is proved.

5. FAST DIFFUSION Case n1  0, n2  0, n  0 (fast diffusion). For (21) we have

1 ( )  (a   )n1 , 2 ( )  (a   )n2 , where a  0.

Theorem 2. At

   the solution problem (21) disappearing at infinity has the asymptotics

f i ( )~i ( ), i  1, 2.

Proof.To prove the theorem, we use the transformation

fi  i ( ) y(), i  1, 2, where  ln  a    , which leads (22) to the following form.

Substituting (22) into (21), for yi ( ) we get the following equation: Impact Factor (JCC): 8.8746

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d  mi 1 dyi y n y d  3i d i i

p2

(

-3099

  e   mi 1 dyi dyi  ni yi )     n  ni yi   y3i i   a  e  d d   

p 2

(

 dyi  ni yi )    d 

(25)

e  dy  c  i  ni yi   i y ( )(1  e ni i y3i i )  0,  i d  ae   where  - unction defined above.

Note that studying the solution of the last equation is equivalent to studying those solutions of equation (21), each of which in a certain interval 0 ,   satisfies the inequality

yi    0,

dyi  ni yi  0. d

We show first of all that the solutions yi   equations (22) have a finite limit

y0i at   . We introduce the

following notation:

i   

y3mi i1

dyi  ni yi d

p2

(

dyi  ni yi ). d

Then equation (22) takes the form  e

i   

ae



 e  dy   ni  i  c  i  ni yi   i y ( )(1  e ni i y3i i ).  i d  ae   

To analyze the last expression, we introduce a new auxiliary function  e

 ( , )   

ae



 e  dy   ni   c  i  ni yi   i y ( )(1  e ni i y3i i ),  i d  ae   

where  - real number.

Hence it is easy to see that for each value

1,    0 ,  



function

  , 

retains a mark at some interval

и при всех  1,  one of the inequalities

i   0, i   0.

Therefore for function i   there is a limit for

 1,  . From the expression for i   follows that

lim i   

 

  e   e  dyi   n   c  n y   y ( )(1  e  ni i y3i i )   0. i i i i i    i ae  d     a  e 

 lim    

From here, with

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  , i  1, i  1, 2, we obtain the following algebraic equation:

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 n1 p1 ym2 1y1p1  c,  n 2 p1 y1m 1y2p1  c.

1

2

Calculating the last equation gives yi  1 and by virtue (7), f   ~i   .

Theorem 2 is proved.

6. COMPUTATIONAL EXPERIMENT To numerically solve problem (17), we construct a uniform grid and a temporary grid [13-14].

We replace problem (17) with an implicit difference scheme and obtain a difference problem with an error

O  h2  h1  .

As is known, the main problem for the numerical solution in nonlinear problems is the appropriate choice of the initial approximation and the method of linearizing the system (17).

The created program allows you to visually monitor the evolution of the process for various valuesof parameters and data.

Numerical calculations show that in the case of arbitrary values m, p  0,   0 qualitative properties of solutions do not change. Below are the results of numerical experiments for various parameter values (table 1, Table2).

Table 1: Fast Diffusion x1  30 ; x 2  30 ; x1  1 ; x 2  1 ; Parameter Values

x  2

x  30 2

x1  30 ; x 2  40 ; x =50

m1  0.8, m2  0.7, p  2.1 eps  103 1  2  2  5 n0

time1 ( FRAME  1)  time2 ( FRAME  1)

m1  0.4, m2  0.5, p  2.2 eps  103 1  2  2  2

time1 ( FRAME  1)  time2 ( FRAME  1)

time1 ( FRAME  1)  time2 ( FRAME  1)

n0

time1 ( FRAME  1)  time2 ( FRAME  1)

time1 ( FRAME  1)  time2 ( FRAME  1)

time1 ( FRAME  1)  time2 ( FRAME  1)

Table 2: Slow Diffusion Impact Factor (JCC): 8.8746

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Parameter Values

-3101

x1  30 ; x 2  30 ; x  30 2

x1  1 ; x 2  1 ; x  2

x1  30 ; x 2  40 ; x =50

m1  1.9, m2  5, p  2.5 eps  103 1  1.5  2  2 n0

m1  1.5, m2  2, p  2.5

time1 ( FRAME  3)  time2 ( FRAME  3)

time1 ( FRAME  3)  time2 ( FRAME  3)

time1 ( FRAME  3)  time2 ( FRAME  3)

eps  103 1  1.5  2  2 n0

5. CONCLUSIONS

time1 ( FRAME  3)  time2 ( FRAME  3)

time1 ( FRAME  3)  time2 ( FRAME  3)

time1 ( FRAME  3)  time2 ( FRAME  3)

Above properties are established on the basis of the solution comparison theorem, the asymptotics of self-similar solutions are obtained, including for the case of fast diffusion. Based on the solutions found, numerical calculations are performed.

Evolution of solutions developing in an aggravated regime involves several stages. In particular, a quasistationary stage (slow solution growth) and an explosive growth stage are necessarily present. Using the example of finite solutions, it is possible to study the rarely considered stages, when the solution first decreases in amplitude, its carrier increases ("spreading of the solution"), then localization occurs (the carrier ceases to change) and only then the solution begins to grow. Our goal was to find out the relationship between the form of the initial perturbation and the duration of these stages.

REFERENCES 1.

Tedeev AF 2004 Siberian Mathematical Journal Volume 45 1

2.

Boldovskaya OM Tedeev AF 2009 The case of fast diffusion. Report of the Academy of Sciences of Ukraine 6 p 14

3.

Kalashnikov AS 1987 UMN T 42. 2 (254) c 135Andreucci D Tedeev A F A 1999J. Math. Anal. Appl.231p 543

4.

Kaplan S1963Comm. Pure Appl. Math.16p 305

5.

Samarsky AA Galaktionov VA Kurdyumov SP Mikhailov AP 1987 Modes with aggravation in problems for quasilinear parabolic equations (M .: Nauka)

6.

Galaktionov V A Levine H A1998Nonlinear Anal34p 1005

7.

Andreucci D Tedeev A F1998Proc/ Roy/ Soc. Edinburg Sect. A.V 1286p 1163

8.

Liu X Wang M 2001J. Math. Anal. Appl. V. 2571p 170

9.

Mukai K Mochuzuki K Huang Q 200 Nonlinear Anal. V. 39A.1p 33

10. Kamin S Ughi M 1987 J. Math. Anal. Appl.V 128.p 456 11. Aripov M.1998 The method of reference equations for solving nonlinear boundary value problems (Tashkent Fan) 137 p 12. Muxamediyeva D K 2019 IOP Conf.Journal of Physics: Conf. Series 1210

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13. Muxamediyeva D K 2019 International Journal of Mechanical and Production Engineering Research and Development 9Issue 3 p1095

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