Approaches to the numerical solving of fuzzy

IJRET: International Journal of Research in Engineering and Technology

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APPROACHES TO THE NUMERICAL SOLVING OF FUZZY DIFFERENTIAL EQUATIONS D.T.Muhamediyeva1 1

Leading researcher of the Centre for development of software products and hardware-software complexes, Tashkent, Uzbekistan

Abstract One of the main problems of the theory of numerical methods is the search for cost-effective computational algorithms that require minimal time machine to obtain an approximate solution with any given accuracy. In article considered fuzzy analog of the alternating direction, combining the best qualities of explicit and implicit scheme is unconditionally stable (as implicit scheme) and requiring for the transition from layer to layer a small number of actions (as explicit scheme).

Keywords: fuzzy set, quasi-differential equation, the scheme of variable directions, finite-difference methods.

--------------------------------------------------------------------***-----------------------------------------------------------------1. INTRODUCTION The concept of fuzzy differential equations was introduced

by O.Kaleva in 1987. In [1] he proved the theorem of existence and uniqueness of solutions of such equations.

Later in [2-6] were been obtained properties of fuzzy differential equations and their solutions. To determine fuzzy derivative O.Kaleva used M.L.Pur and D.A.Rulesku’s approach [7] to the differentiability of fuzzy mappings, which, in turn, is based on the M.Hukuhara’s idea [8] about differentiability of multivalued mappings. In this regard, the O.Kaleva’s approach adopted all the shortcomings typical differential equations with the Hukuhara’s derivative.

In 1990 J.P.Aubin [9] and V.A.Baidosov [10,11] introduced into consideration fuzzy differential inclusion. Their approach to solving such equations is based on the latest information for ordinary differential inclusions. In the future, fuzzy inclusion were considered in works [12-15].

In [17], in the same way as was done in the theory of differential equations with multivalued right-hand side [16], introduced the concept of quasi-differential equation. This allows on the one hand to avoid the difficulties that arise in the solution of fuzzy differential equations and inclusions, and with other - to get some of their properties by available methods.

2. STATEMENT OF THE TASK n

Let Conv( R ) - the space of nonempty convex compact subsets

R n with Hausdorf metric

  h( F , G )  max sup inf f  g , sup inf f  g  , gG f F  f F gG 

Now we introduce the space

 : R n  [0,1] ,

which

E n of the mapping

satisfying

the

following

conditions:

 - semi-continuous from above, i.e. for any x' R n and for any   0 there is  ( x' ,  )  0 such that for all x  x'   runs condition  ( x)   ( x' )   ; 1.

2.



that 3.

- normally, i.e. there exists a vector

x0  R n such

 ( x0 )  1 ;



- fuzzy convex, i.e. for any x' , x' ' R

n

and any

  (0,1] true inequality  (x'(1   ) x' ' )  min{ ( x' ),  ( x' ' )} ; n 4. The closure of the set x  R  ( x)  0 is compact.

Zero

in

the

En

space

are

elements

1, x  0

 ( x)  

n 0, x  R \ 0

Definition 1



0   1

let's call a set



- cutting [  ] of the mapping

cutting of the map

x  R

n

  En

 ( x)  0.

  E n at x  R n  (x)   . Zero

let's call the closure of the set

E n metric D : E n  E n  [0, ) ,   fancy D(  , )  sup h([ ] , [ ] ) .

Let’s define in space

0 1

f : [0, T ]  E n called weakly continuous at the point t 0  (0, T ) , if for any fixed Definition 2 Mapping

Where, under

 refers Euclidean norm in space R n .

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  [0,1] and arbitrary   0 there is  ( ,  )  0 such that h( f  (t ), f  (t 0 ))   for all t 0  [0, T ] such, that t  t 0   ( ,  ) . n

I

is

f ' (t 0 ) call fuzzy derivative f (t ) in the point t0 .

Mapping f : I  E

n

called differentiable on I , if it is

differentiable at each point

Definition 3 Integral of the mapping f : I  E over the interval

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the

such

that

[ g ]   f (t )dt for all   (0,1] . n

called differentiable at

  [0,1] multivalued mapping f (t ) differentiable by Hukuhara [198] in the point t0 , its DH f (t 0 ) and the derivative is equal to {DH f (t 0 ) :  [0,1]} set family defines an element. n

n

is called uniformly

continuous on G  E , if for each   0 there is  ( )  0 such that for all x, y  G , satisfying the

D ( x, y )   D( f ( x), f ( y ))   .

the point t 0  I , if fr all

Definition 5 For u , v  E , w  E

n

inequality

I

Definition 4 Mapping f : I  E

Definition 8 Mapping f : I  E n

g  En

element

t I .

called u and

Definition 9 Fuzzy number L  R type, if

assessment

is called a fuzzy number of

u  uL ( )  ,   L (u )  1  uL  u ( )     (u )  1  uR ( )  u ,  R uR

v , if

u  v  w , and it is written as w  u Hv .

u~

fair

u - clear value of the number u~ , i.e. u  u L (1)  u R (1) ; u L and u R respectively the left and ~ ; u ( ) and u ( ) right stretching of fuzzy number u L R ~ respectively the left and right values of the fuzzy number u of definition  . where

Definition 6 Function F : [a, b]  E

n

differentiable in

t 0  (a, b) , if there is a F ' (t 0 )  E such that there are F (t 0  h) H F (t 0 ) lim limits and h 0  h F (t 0 ) H F (t 0  h) lim and they are equal to F ' (t 0 ) . h 0  h n

If F differentiable in t 0  (a, b) , hen for all



From the definition it follows that if

u~( )  {u, uL ( ), uR ( )} , then

slices

u L ( )  u  (1   )u L ; uR ( )  u  (1   )uR .

F (t )  [ F (t )] there is a Hukuhara differential in t0 and

[ F ' (t0 )]  DF (t ) , where DF is called Hukuhara differential F . f : [a, b]  E n Seikkala SDf (t ) in a such way

Definition 7 For function introduced

the

concept

[ SDf (t )]  [ f1' (t , ), f 2' (t , )], 0    1 .

For

t [a, b], [ SDf (t )]

is

fuzzy.

If

Consider the algebraic action on fuzzy Addition:

L  R type:

u~  v~  {u  v  (1   )(u L  vL );u  v  (1   )(u R  vR )} .

Subtraction:

u~  v~  {u  v  (1   )(uL  vL );u  v  (1   )(uR  vR )} mapping

.

f : I  E n differentiable at the point t 0  I , then Multiplication: 1) for u  0; v  0

u~  v~  {u  v; (1   )(uvL  vuL )  (1   )uL vL ; (1   )(uvR  vuR )  (1   )uR vL } ; 2) for u  0; v  0

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u~  v~  {u  v; (1   )(uvR  vuL )  (1   )uL vL ; (1   )(uvL  vuR )  (1   )u R vL } ; 3) for u  0; v  0

u~  v~  {u  v; (1   )(uvR  vuL )  (1   )uR vR ; (1   )(uvL  vuL )  (1   )uL vL } .

Division:

In this paper discusses some of the issues of building fuzzy analogs of finite difference methods.

u~ ~ 1 u~. v~ v Definition

10

We

say

Let’s consider the initial-boundary task that

the

mapping

 : [0,  ]  [0, ]  [0, T ]  E n  E n

specifies the local quasi movement, if satisfied following conditions: 1) initial conditions axiom  (0,0, t , y )  y ; 2) quasiprimitive axiom:

u  Lu  f ( x, t ), ( x, t )  QT , t

(1)

u ( x,0)  u 0 ( x),

(2)

x  D,

u ( x, t )   ( x, t ), x  , t  [0, T ] ,

D( (h , ,0, y0 ), (hN , m , tm1 , y N 1 ))  0(h) ,

 2u  2u , L u  , 2 x12 x 22

Lu  u  L1u  L2 u, L1u  Where

3)

h

N

m

 hn

   ts ;

n 1

D  0  x  l ;   1,2,

s 0

axiom

of

(3)

continuity:

QT  D  (0, T ], x  ( x1 , x2 ) .

mapping

 (h, , y ( x, t )) is - weakly continuously.

Approximation equation

Suppose that u 0 ( x), ( x, t ), f ( x, t ) - are fuzzy functions.

D( y ( x  h, t   ), (h, , t , y ( x, t )))  0(h, )

u 0 ( x) 

0

( x)] ,

[ 0 ,1]

,

 ( x, t ) 

y( x,0)  u0 ( x) , x  D ,

y ( x, t )   ( x, t ),

  [U 

 [ ( x, t )] ,  

(4)

[ 0 ,1]

f ( x, t ) 

x  , t  [0, T ]

 [ F ( x, t )] ,   [0,1]  

(5)

[ 0 ,1]

we will call fuzzy quasi-differential equation.

Definition 11 Continuous map y : QT  E , satisfying the approximation equation will be called a solution of fuzzy quasi-differential equation. n

We introduce two-dimensional spatial grid and temporary one-dimensional grid:





w h  w h1 ,h2  wh   h  ( x n1 , x n2 )  D; 0  n  N  ;   1,2 , t t

s

1 2

 t1  0,5 ; y  y

s

1 2

In this case, the construction of the discrete solution of the task (1)-(5) is reduced to the definition in the grid of fuzzy s 

numbers [ y ] , such, that for for exact solutions u ( x, t )

,  s  [ f ( x n , t s )]

of

the

u 0 ( x) 

task

(1)-(

  [U

 [ 0 ,1]

3)

with 

0

( x)]

any

initial

condition and

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IJRET: International Journal of Research in Engineering and Technology

 ( x, t ) 

 [ ( x, t )] ,  

fair

inclusions

[ 0 ,1]

u( xs , t ) 

 

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[ y s 1 ]  [ y 0,5

s 

[y ] .

[ 0 ,1]

s

1 2 

]

 1[ y

s

1 2 

]   2 [ y s ]  [ s ] , (7)

Let the function f ( x, t ) 



  [ F ( x, t )]

,   [0,1]

 [ 0 ,1]

defined for points ( x, t )  QT and F ( x, t ) satisfies the following conditions:

[ y( x,0)]  [U 0 ( x)] , x  wh ,

(8)

[ y s 1 ]  [ ] , n2  0, n2  N ,

(9)



( x, t )  QT .

2) [ F ( x, t )] - monotonously to inclusion of h0 , i.e. from ensue X 1  X 2 , T1  T2   [ F ( X 1 , T1 )]  [ F ( X 2 , T2 )] .

3) There is a real constant l  0 , such that when true inequality ( x, t )  QT     ([F ( x, t )] )  l ( ([ x] )   ([t ] )) .

1) [ F ( x, t )] - defined and continuous for all 

[y

s

1 2 

]  [] , n1  0, n2  N .

(10)

Equation (6) is implicit in the first direction and clear for the second, and equation (7) is explicit in the first direction, and implicit in the second.

From (6) and (7) we get

Describe the fuzzy variant of the finite-difference methods for tasks (1)-(5).

2  2 [ y ]   1 [ y ]  [ F ] ; [ F ]  [ y ]  [ y ]  [] ,  

3.

2  2 [ yˆ ]   2 [ yˆ ]  [ F ] ; [ F ]  [ y ]  1 [ y ]  [] ,  

FUZZY

VARIANT

OF

THE

FINITE-

DIFFERENCE METHODS. Replace differential operators with finite-difference:

Lu  y  1 y   2 y,  y  yx , x ,   1,2 . 

 1 1 1 1 [ y n1 1 ]  2 2  [ y n1 ]  2 [ y n1 1 ]  [ Fn1 ] , n1  1,2,..., N1  1 2 h h1  h1   ,

One of the main problems of the theory of numerical methods is the search for cost-effective computational algorithms that require minimal time machine to obtain an approximate solution with any given accuracy   0 .

As is known, the explicit scheme requires a lot of activity, but its stability is at a sufficiently small time step, the implicit scheme is unconditionally stable, but it requires a large number of arithmetic operations.

We can build a system that combines the best of explicit and implicit scheme is unconditionally stable (as implicit scheme) and requiring for the transition from layer to layer a small number of actions (as explicit scheme).

One of the first cost-effective schemes is the scheme of variable directions, built in 1955 by Pismen and Recordon.

Pismen and Reckford scheme makes the transition from layer S on a layer S  1 in two steps, using intermediate (fractional) layer.

[y

1 s 2 





[ y n1 ]  [n1 ] ; n1  0, n1  N1 .

 1 1 1 1 [ yˆ ]  2 2  [ yˆ n2 ]  2 [ yˆ n2 1 ]  [ F n2 ] , n2  1,2,..., N 2  1 2 n2 1 h h2  h2   , 



[ y n2 ]  [n2 ] ; n2  0, n2  N 2 ,

xn  (n1h1 , n2 h2 ), [ F ]  [ Fn1n2 ] , [ y]  [ yn1n2 ] .

Thus, the construction of the discrete solution of problem (1)-(5) is reduced to the definition in the grid of fuzzy s 

numbers [ y ] , such that for exact solutions u ( x, t ) of the task (1)-(3) with any initial condition

u 0 ( x) 

  [U 

0

( x)]

and

[ 0 ,1]

 ( x, t ) 

 [ ( x, t )] ,  

fair

inclusions

[ 0 ,1]

s ]  [ y s ]   1 [ y 2 ]   2 [ y s ]  [ s ] , 0,5 1

u( xs , t ) 

[ y 

s 

] .

[ 0 ,1]

(6)

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4. NUMERICAL EXPERIMENT Let’s consider the idea of the alternating direction method of class cheap schemes, applied to the solution of the first 

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initial-boundary value task for two-dimensional heat equation.







u  u   u    u    [q ( z )]     (k z )   [ w]     (k y )   [au ] , z  y   t   z  z   y 

( z , y, t )  (0, L)  ( M , M )  (0, T ) , 



[u ( z, y,0)]  [ ( z, y)] ,

differential equation is approximated by two differential equations, one of which connects layers t  t

( z, y)  [0, L]  (M , M )

- initial conditions,

[u ( z, y, t )]  [ ( z, y, t )] ,

( z, y)  , 0  t  T -

t t

step

[ ( z, y )] ,[ ( z, y, t )] -  - sections of the set of fuzzy functions. L, M , T - given numbers. 



, and the second layers t  t

1 n 2

and t  t

First phase. Transition to more smart n 

boundary conditions где   (0, L)  (  M , M ),  is -  domain border;

1 n 2

n

n 1

and .

1 - th layer with 2

 . Time derivative is approximated by the formula 2 n

ut ( zi , y j , t n ) 

1

uˆij 2 H uˆijn



,

2 Initial equation is approximated by the combination of two difference schemes, each of which corresponds to only one spatial direction. Each element of the sum approximated by explicit and implicit structures.

Derivative by z approximated on n 

To do this, along with a layer t  t , alculation of which is carried out at this stage, the inclusion of an additional layer

Included in the right part of the initial equation fuzzy

derivative by y on

1 - th layer, 2

n - th layer.

n

t t

n

1 2

. Then for the transition from layer t  t

layer t  t

n 1

by using the layer t  t 

 n  12  n uˆij   uˆij qi         2 

k

  (k

 n y j 1 i , j 1

uˆ

 

y j 1

1 n 2

n

to the



function [ f ( z, y, t , q)] is replaced by its grid view.

Corresponding difference scheme has the form:

the original 







1 1 1 n  n  n     2 2 2 ˆ ˆ ˆ k u  ( k  k ) u  k u  zi1 i 1, j   zi1   zi i 1, j  zi ij       w  2  hz

 

 k y j )uˆijn

h 

  k 

yj

uˆin, j 1

2  y









 n  12   n  12  uˆi 1, j   uˆi 1, j        2hz 



 a n 1    uˆij 2  , 2 

1  i  I  1, 1  j  J  1,



  n 1  1   n Where uˆij 2   uˆ ( zi , yi , t   )  or in traditional recording 2     







    n  12      a   n  12  ˆ k  w  u  k  k  q  z i  2 zi   i 1, j   2 zi   uˆij   4hz   2hz2 i1 4    2hz  2hz   

     w  uˆ  2 k zi1  4hz    2hz

1 n 2 i 1, j



 n    Fij , 

 

(11)

1  i  I  1, 1  j  J  1,

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IJRET: International Journal of Research in Engineering and Technology

 

n  ij

where F

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

 n  k y j 1 uˆin, j 1  (k y j1  k y j )uˆijn  k y j uˆin, j 1   qi uˆij   .

2 hy2  

For each j  1,..., J  1 need to solve three-diagonal system of fuzzy linear algebraic equations, as each equation contains three unknown values 



Second phase. The transition to ( n  1) - th temporary layer from intermediate n 



 n  12   n  12   n  12  uˆi 1, j  , uˆij  , uˆi 1, j  , rest of the values are taken       from n - th layer. In other words, under this approach, the

1  - th with step . Time 2 2

derivative is approximated by the formula n

ut ( zi , y j , t n ) 

scheme is implicit in the direction z and explicit in direction y. The desired value in the intermediate layer are calculated by the fuzzy sweep method by direction z , i.e. by the longitudinal direction.

uˆijn 1 H uˆij

1 2

,



2 derivative by z approximated on the n  derivative by y on the ( n  1) - th layer.

1 - th layer, 2

Corresponding difference scheme has the form:

uˆ 

n 1  ij

qi 



k

uˆ in, j 11



y j 1

 n 1   uˆ ij 2   

  2  







y j 1



 



  (k



1 1 1 n  n  n     2 2 2 ˆ ˆ ˆ k u  ( k  k ) u  k u  zi 1 i 1, j   zi 1   zi i 1, j  zi ij       w  2  hz

 k y j )uˆ ijn 1

h 

  k 

yj

uˆ in, j 11





2  y





 n  12   n  12  ˆ u   i 1, j  uˆ i 1, j        2hz 



a    uˆ ijn 1  , 2 

1  i  I  1, 1  j  J  1,

or in the traditional record: 

   n 1  2 k y j   uˆ i , j 1   2h y







   n 1  2 k y j 1   uˆ i , j 1 2 h  y 





   a  n 1   2 k y j  2 k y j 1  qi    uˆ ij 4  2h y  2h y







  Q 

1 n 2 ij

 







  , 

(12)

1  i  I  1, 1  j  J  1,

Where 

 Q 

1 n 2 ij



    qi uˆ  

1 n 2 ij

1 1 1 1 1 n n n  n n    2 2 2 2 2 ˆ ˆ ˆ   k zi1 ui 1, j  (k zi1  k zi )uij  k zi ui 1, j   w uˆi 1, j  uˆi 1, j  .      2  2 hy2 2hz     



For each i  1,..., I  1 need to solve three-diagonal system of fuzzy linear algebraic equations, as each equation contains three unknown values

uˆ  , uˆ  , uˆ  n 1  i , j 1

n 1  ij

n 1  i , j 1

in the intermediate layer are calculated by the sweep method in the direction y, i.e. the transverse direction. Diagonal elements prevail.

,

1 the rest of the values are taken from ( n  ) - th layer. In 2 other words, under this approach, the scheme is implicit in the direction z and a clear by direction y. The desired value

Under implementation is the lack of dividing the interval containing zero on each  cutting, and by sustainability limited impact of an error in the calculation at some stage of the final result.

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uˆ     ; uˆ     , n  i ,0

The advantages of the method, achieved through the introduction of intermediate layer, should include the splitting of the initial task into two simpler, solved with the help of sweep algorithm.

0  ij



ij

uˆ     ; uˆ     ,

0  j  J; 0  n  N,

n



n  I, j

n  I, j

n  i,J

0  j  J; 0  n  N,



n

n

i

i, j





j



ij

i

j

 n  12  For the intermediate layer is required values uˆ ij   

OZ

0, j

n  i,J

    ( x , y , t ) ,     ( x , y ) .

0  i  I; 0  j  J ,

on planes perpendicular to the axis n  0, j

n 

i ,o

Where

The initial and boundary conditions is presented in the form:initial layer

uˆ     ,

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

on

the sides of the settlement of the region defined by the equations x  0 and x  L . Subtracting (12) from (11), we obtain

on planes perpendicular to the axis OY 







  (k 

n  uˆijn  uˆijn 1   n  12     k y j1 uˆi , j 1   uˆij     2   4qi    





n 1    k y j 1 uˆi , j 1    4qi 

  (k 

y j 1

 k y j )uˆijn 1

h 

  k 

yj

uˆin, j 11

y j 1





2  y

 k y j )uˆijn

h 

  k 

yj

uˆin, j 1







2  y





1 n   a       uˆijn 1  uˆij 2  .  8qi   

Hence when i  0 ( x  0) we have 







  (k 

n  uˆ0n j  uˆ0n j 1   n  12     k y j1 uˆ0, j 1   uˆ0 j     2   4 q0    





n 1    k y j 1 uˆ0, j 1    4 q0 

  (k 

y j 1

 k y j )uˆ0n j 1

h 

  k 

yj

uˆ0n,j11

y j 1

 k y j )uˆ0n j

h 

  k 

yj

uˆ0n, j 1







2  y





2  y





1 n   a   n 1 2 ˆ ˆ   u  u 0j  .   0j  8q0   

Similarly, we get the relation at i  I ( x  L) 







n  uˆ Ijn  uˆ Ijn 1   n  12     k y j1 uˆ I , j 1   uˆ Ij     2   4q I    





n 1    k y j1 uˆ I , j 1    4q I 

  (k 

y j 1

 k y j )uˆ Ijn 1

h 



  k 

yj

uˆ In,j11

2  y

Theorem 1 Suppose that for the system

b0  uˆ0   c0  uˆ1    f 0  , ai  uˆi1   b0  uˆi   ci  uˆi1    f i  , an  uˆ n1   bn  uˆ n    f n  , by the formulas

  (k

y j 1





 k y j )uˆ Ijn

h 

2  y



  k 

yj

uˆ In, j 1





 

1 n   a   n 1 2 ˆ ˆ   u  u  .

Ij   Ij  8q I   

x0   c0  /b0  ,     xi    ci  / bi  ai xi1  , i  1,..., I 1 , y0    f0  /b0  ,     yi    fi  ai y y 1  / bi  ai xi 1  , i  1,..., I  1 , un   yn  ,    ui    xiui1    yi  , i  I 1, I  2,..., 0

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IJRET: International Journal of Research in Engineering and Technology

defined

ui 

Then we have

uˆ   u     i

i

Where 

ai 

     2 kyj  ,   2h y

bi 

   a    2 k y j  2 k y j 1  qi   , 2h y 4   2hy

ci 

     2 k y j 1  ,  2hy 

ai 

      2 k zi  w , 4hz   2hz

bi 

   a    2 k zi  2 k zi 1  qi   , 2hz 4  2hz

ci 

      2 k zi 1  w , 2 h 4 k z  z 

















 fi 



 n 1   Qij 2   



в (2) и







 f i 

 

 Fijn



.

5. CONCLUSIONS Introduced analogues of numerical solutions of fuzzy differential equations by the method of variable directions generalize earlier reviewed fuzzy differential equations and inclusions and provide an opportunity to examine their properties.

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