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 , gG f F f F gG
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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IJRET: International Journal of Research in Engineering and Technology
[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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IJRET: International Journal of Research in Engineering and Technology
eISSN: 2319-1163 | pISSN: 2321-7308
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 , tm1 , 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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IJRET: International Journal of Research in Engineering and Technology
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
ut ( 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 zi1 i 1, j zi1 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 i1 4 2hz 2hz
w uˆ 2 k zi1 4hz 2hz
1 n 2 i 1, j
n Fij ,
(11)
1 i I 1, 1 j J 1,
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n ij
where F
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n k y j 1 uˆin, j 1 (k y j1 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
ut ( 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 11
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 11
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 zi1 ui 1, j (k zi1 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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IJRET: International Journal of Research in Engineering and Technology
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 j1 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 11
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 j1 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,j11
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 j1 uˆ I , j 1 uˆ Ij 2 4q I
n 1 k y j1 uˆ I , j 1 4q I
(k
y j 1
k y j )uˆ Ijn 1
h
k
yj
uˆ In,j11
2 y
Theorem 1 Suppose that for the system
b0 uˆ0 c0 uˆ1 f 0 , ai uˆi1 b0 uˆi ci uˆi1 f i , an uˆ n1 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 xi1 , i 1,..., I 1 , y0 f0 /b0 , yi fi ai y y 1 / bi ai xi 1 , i 1,..., I 1 , un yn , ui xiui1 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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