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An existence theorem for fuzzy partial differential equation Efendiyeva H.1, Rustamova L.2 (Republic of Azerbaijan) Теорема о существовании нечеткого дифференциального уравнения
в частных производных Эфендиева Х. Д.1, Рустамова Л. А.2 (Азербайджанская Республика)
'ЭфендиеваХеджер Джавид /Efendiyeva Hedzher - кандидат физико-математических наук, преподаватель; 2Рустамова Ламия Аладдин /RustamovaLamiya - кандидат физико-математических наук, преподаватель,
кафедра математической экономики, Бакинский государственный университет, г. Баку, Азербайджанская Республика
Abstract: in this paper, first a space of fuzzy numbers is constructed and a scalar product is introduced. The derivative offuzzy function in this space is defined. Further, Poisson's equation with first boundary condition for fuzzy functions is considered. It is shown that if problems data (right hand site and boundary function) are fuzzy then solution of this problem also is fuzzy function.
Аннотация: в данной статье вначале было построено пространство нечетких чисел и представлено скалярное произведение. Была определена производная функции принадлежности нечеткого множества в данном пространстве. Далее рассматривается уравнение Пуассона с граничными условиями первого рода для функций принадлежности нечеткого множества. Показано, что если данные задачи (правая часть и граничная функция) содержит нечеткие числа, то и решение этой задачи - это функция принадлежности нечеткого множества.
Keywords: fuzzy partial differential equation, fuzzy function, poisson's equation.
Ключевые слова: нечеткие дифференциальные уравнения в частных производных, функция принадлежности нечеткого множества, уравнение Пуассона.
AMS Subject Classification: 3'A30, 34K36, 35R13, 60E'0
1. Introduction
The complexity of the world makes the events we face uncertain in furious forms. Besides, randomness, fuzziness and is important uncertainty, which plays an essential role in the real world. Fuzzy set theory has been developed very fast since it was introduced by scientist on cybernetics Zadeh [1] in 1965. A fuzzy set characterized with its membership function by Zadeh. For the purpose of measuring fuzzy events, Zadeh [2] presented the concept of possibility measure and the term of fuzzy variable in 1978.
To investigate fuzzy differential equations at first one has to introduce the definition of the derivative of fuzzy function. This definition must allow one to investigate ordinary and partial differential equation.
The concept of fuzzy derivative was first introduced by Chang and Zadeh [3], and it was followed up by Dubois and Prade [4], who used the extension principle in their approach.
Other methods have been discussed by Puri and Ralescu [5]. A thorough theoretical research of fuzzy Cauchy problems was given by Kaleva [6], Seikkala [7]. Fuzzy partial differential equations were formulated by Buckley [8], and T. Allahviranloo [9] used a numerical method to solve the (FPDE).
In the present work, introducing the space of fuzzy numbers, the derivative of the fuzzy function is determined. Using this approach, the method is proposed to investigate fuzzy partial differential equations.
2. The space of the pairs of fuzzy numbers
A fuzzy set A is characterized by a generalized characteristic function ЦАО, called membership function, defined on a universe X , which assumes values in [0,1]. For any a e [0,1] denote by
Aa = {x e X : цА (x) > a} the a - cut of A . Let JUA(.) is an upper semicontinuous function and
supp(A) = {x e X: /лА (x) > a} is bounded set of X . A fuzzy set is a fuzzy number if X ^ R and for any a e [0,1], the a -cut
Aa is convex and the height of A , that is, supцА(x) has to be equal to one. This fuzzy number usually
xeX
is called convex normal fuzzy number.
Let's define by F the class of convex normal fuzzy numbers. Then for any a e F the set of d -cut of fuzzy number a the interval ad = [La (d), Ra (d)] , d e [0,1] , is defined ([7]). Let a e F ,
b e F and ad = [La (d), Ra (d)], bd = [Lb (d), Rb (d)]. Then d -cut of fuzzy number
a + b and ka, k > 0, define as ad + bd = [La (d) + Lb (d), Ra (d) + Rb (d)] and
kct = [kLa (d), kRa (d)] , respectively.
Note that F is not a linear space (the operation of subtraction is not defined in F ). We consider the set of pairs (a, b) e F x F and define the operation of addition, multiplication and equivalency as
a a2) + b b2 ) = (ai + a2 + b2 )
k • (a, b) = (ka, kb), k > 0,
(-1) • (a, b) = (b, a), (1)
(a, a) = (b, b2) ^ a + b = a + b
As zero element of this space is taken the pair (0,0) , i. e. the set of elements (a, a), a e F . From last relation (1) we get (a, a) = (0,0) . For any X = (a, b) , — X = (b, a) . It is clear that
x + (—x) = (a, b) + (b, a) = = (a + b, a + b) = (0,0).
The set of all pairs (a, b) e F x F forms a structure of a linear space. Let
x = (a, a ) e F x F, y = (b, b2 ) e F x F.
Then
ad = L (d), Rai (d)] , bd = [Lbi (d), Rb (d)], d e [0,1]
For any x, y e F x F define the scalar product as
X ° y =1 {[(La, (d) — L (d))(Lb (d) — L^ (d)) +
2 o 1 21 2 (2)
+ (Rai (d) — Rai (d))(Rb (d) — R^ (d))]dd
It may be shown that this definition satisfies all requirements of the scalar product. We denote this space by LF . Norm in this space is defined as
H2 = 1 {[(La (d) — La2 (d))2 +(Rat (d) — Ra2 (d))2]dd (3)
2 0
We define distance between two fuzzy numbers a e F and be F as
b) = | |X — y|| (4)
where X = (a,0), y = (b,0) .
3. Derivative of the fuzzy function
Now, let's consider fuzzy function f (t) e F for each t e [t0, ^ ]and define a derivative of the function f (t) .
For any de[0,1],
fd (t) = [Lf (t) (d), Rf (t) (d)], d e [0,1] (5)
is called d -cut of the function f (t) .
Definition. Let there exists such <(t) e F, j(t) e F , t e [t0, t1 ], that
lim (f(/ + — (f(t)0) = (<(/)j(/)). (6)
A/ ^0 A/
Then the pair (<(t), j(t)) e F x F is called a derivative of the function f (t) at the point t e[t0, tx] . This definition may be written in the following form
lim (fd(/ + A/^^ — (fd(/),0) = (<(/j/)), (7)
A/^0 A/
where (<d (t), jd (t)) are d -cut for the functions <(t) , lj/(t) .
It is shown that, if Lf (t )(d), Rf (t )(d) is continuous differentiable relatively t, then f (t) is
differentiable. Each function f (t)may be considered as an element (f (t),0) from F x F . Then
(fl(/) ± f2(/))' = /(/) ± №). (8)
Now, let f(t) be a pair of fuzzy functions, i.e.
f (/) = (fi(/),/2(/)), V/ e (/0,0.
From relation
f (t) = 01(t ),0) + (0, f2(t)) = (f(t ),0) — (f2(t ),0)
we see, that the derivative of the function f (t) also is a pair from F x F .
For any T/ = ^l(t) e F x F , which Tj (t) e F x F , consider the scalar product
f'(t) oj(t) defined by the formula (2). It can be shown that T T
{ f'(r) o j(t)dr = f (r) o t(t)\T — { f (t) o t'(t)dt, V/, T e (/0, /1).
t t
(9)
One may show that this derivative satisfies the "necessary natural" conditions. Example 1. Let f (t) be fuzzy function whose d -cut is defined as follows:
fd (t) = [t2 — (1 — d)t, t2 + (1 — d)t], t > 0.Then it is not difficult to show that
d) = (<d(t ),jd(t)),
where
<d (t) = [2t — (1 — d),2t + (1 — d)], jd (t) = 0. Example 2. Let
S* / ^ r 9 1 d O 1 d —m „
fd(t) = [t2 —t2 + —], t > 0.
In this case
Pa(t) = 0, ¥a(t) = [2t-2t +
Analogically we can define partial differential on y\, ^2v-, yn .
Example 3. Let a £ F , a2 £ F be fuzzy number and u(y, y2) = yi aj + y2 a2, y{ £ R be fuzzy function. It is no difficult to show that, for
y ^ 0, i = i,2,
du
fyi In obverse
du
= (2 ya ,0).
= (0,2 yfa ,0).
dyi
Also it is clear that
d 2u
= (2ai ,0).
dy2
4. Fuzzy elliptic equation
Let D £ R" be a given bounded domain with smooth boundary S and fuzzy function
u = u(y) £ F x F depends on the parameter y = (yl,y2,---, yn) £ D , i.e.
U = U(y), y £ D . We'll write U £ C(D) , if the function ||u(y)| continues on y in D .
Analogically we can define U £ C (D) . Consider the boundary problem
Au = -f (y), y £ D , (10) U({) = g(£), ^ £ S. (11)
Let f (y) = (f (y), f2 (y)) £ F X F, y £ D,
g(4) = (g (4), g2 (4)) £ F X F, 4 £ D.
In the difference of traditional problems, here solution of the problem (6), (7) is fuzzy function u = u(y) £ F or pair of the fuzzy function u(y) = (ul(y), u2(y)) £ F X F. For the of
simplicity, this type functions we'll call fuzzy function. Equation (10) and boundary condition (11) we understand as equality pair of the domains.
Theorem 1. Let f £ C (D) ^ C(D ) and g £ C(S), i = 1,2. Then there exists unequal
solution u(y) = (ul(y),u2(y)) £ F X F,y £ D of the problem (10), (11).
It is interesting to investigate problem (10), (11), when f (y), g(4) are fuzzy functions, e.i. f (y) £ F, g(4) £ F, y £ D,4 £ S. There is, that in this case solution of the problem (10), (11) also
is fuzzy function from F .
Theorem 2. Let for any y £ D and 4 £ S be fuzzy function and
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f £ Cl(D) H C(D), g £ C(S). Then there exists unequal fuzzy function u = u(y) £ F solution of the problem (10), (11).
References
1. Zadeh L. A. Fuzzy sets. Information and control. 1965, v. 8, p. 338-353.
2. Zadeh L. A. Fuzzy sets as a basis for theory of possibility. Fuzzy set and systems, 1978, v. 1, p. 3-28.
3. Chang S. I., Zadeh L. A. On fuzzy mapping and control. IEEE. Trans. Systems Man Cybernet. 1972; 2: 30-34.
4. Dubois D., Prade H. Towards fuzzy differential calculus. III. Differentiation, Fuzzy Sets and Systems, 1982; 8 (3): 225-233.
5. PuriM. L., Ralescu D. A. Differentials for fuzzy function. J. Math. Anal. Appl. 1978; 64: 369-380.
6. Kaleva O. Fuzzy differential equations, fuzzy Sets and Systems 24 (1987) 301-317.
7. Seikkala S. On the fuzzy initial value problem, Fuzzy Sets and Systems, 1987; 24 (3): 319-330.
8. Buckley J. J., Feuring T. Fuzzy differential equations, Fuzzy Sets and Systems, 2000, vol. 110, pp. 43-54.
9. Allahviranloo T. Difference methods for fuzzy partial differential equations, Computational methods in appliead mathematics, 2 (2002), №. 3, PP. 233-242, 169 (2005) 16-33.
To the question about of physical essence in process of dilation temporal in special and general theory's of relativity. Part 1 Zlobin I. (Republic of Finland) К вопросу о физической сущности процесса замедления времени в специальной и общей теории относительности. Часть 1 Злобин И. В. (Финляндская Республика)
Злобин Игорь Владимирович / Zlobin Igor — ведущий специалист, член Финляндской астрономической ассоциации, Отдел технической и программной поддержки компьютерного центра, Высшая техническая школа «SETMO», г. Хельсинки, Финляндская Республика
Аннотация: показано, что процесс замедления Времени в специальной и общей теории относительности - это физическое явление, имеющее унитарный характер. Сформулированы базовые элементы методики, необходимые для установления факта интегрирования двух метаморфизмов в один. Используются понятия фазового угла Времени Wz и темпоральных токов Времени ji. Найдена функция, обеспечивающая корреляцию между Wz и действительными темпоральными процессами.
Abstract: it is shown that the process of Time dilation in special and general theory of relativity is a physical phenomenon having a unitary character. Presented basic elements techniques necessary for ascertaining the fact integration two processes into one. Uses the concept ofphase angel Time Wz and Time's temporal current ji. Found function provides the correlation between Wz and the actual temporal processes.
Ключевые слова: Эйнштейн, специальная и общая теория относительности, фазовый угол времени, токи времени.
Keywords: special and general theories of relativity, time's currents, phase angel time, Einstein.
УДК530.12:53'.'8; УДК530.12:53 '.5' PACS number(s): 0330. + p; 04.20. — q; 95.30. Sf
1. Введение
А. Эйнштейну удалось в 1905 г. в работе «К электродинамике движущихся тел» [6] сформулировать основные принципы специальной теории относительности (СТО). Позднее, в 1916 г. им же, но уже в работе «Основы общей теории относительности» [6] в окончательном виде излагается общая теория относительности (ОТО), включая и гравитацию.
Решающим аргументом в пользу справедливости построенных теорий явились предсказанные СТО и ОТО специфицированные эффекты. Данные астрономических наблюдений, а также большое число физических экспериментов подтвердили правильность ожидаемых процессов, что способствовало позитивному укреплению новых представлений в физике.
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