MIXED FRACTIONAL INTEGRATION AND DIFFERENTIATION AS RECIPROCAL OPERATIONS Текст научной статьи по специальности «Математика»

ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ

MIXED FRACTIONAL INTEGRATION AND DIFFERENTIATION AS RECIPROCAL

OPERATIONS

Barakaev Dilshod

Bukhara Technological Institute of Engineering, A. Murtazayev 15, Bukhara, Uzbekistan

Abstract. We study the question of the composition of the mixed fractional integral and the mixed fractional derivative in sufficiently broad class of functions. The treatment formula for mixed fractional derivative is obtained.

Аннотация. Исследуется вопрос о композиции смешанного дробного интеграла и смешанной дробной производной в достаточно широком классе функций. Получена формула для смешанной дробной производной.

Keywords: mixed fractional integral, mixed fractional derivative, function of two variables, Riemann-Liouville integrals.

Ключевые слова: смешанный дробный интеграл, смешанная дробная производная, функция двух переменных, интегралы Римана-Лиувилля.

Introduction

Various forms of fractional integrals and derivatives are known. Fractional integrals and Riemann-Liouville derivatives are the most common in the scientific literature [1]. Operators of generalized fractional integro-differentiation with Gauss hypergeometric function.

Direct extension of the Riemann-Liouville fractional integro-differentiation operations to the case of many variables, when these operators are applied for each variable or some of them, gives the so-called partial and mixed fractional integrals and derivatives. They are known [1], as well as [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. Thus, in [2], using the two-dimensional Laplace transform, a solution of the two-dimensional Abel integral equation was obtained.

In this paper, we study the question of the composition of the mixed fractional integral and the mixed fractional derivative in sufficiently broad class of functions. The treatment formula for the mixed fractional derivative is obtained. The results obtained can be applied in the theory of differential equations containing the mixed fractional derivatives.

Lemma 3 on the representability of f(x,y) £ ACn'm(H) function in the form of (6) and Lemma 4 generalized is the previously known Lemmas 1 and 2 for the two-dimensional case. Lemmas 3, 4 permit to prove the theorem (a necessary and sufficient condition for the representability of f(x, y)function as the mixed fractional integral of a summable function) and Theorems 2 and 3 about the composition of a mixed fractional integral and a mixed fractional derivative. Note that Theorems 2 and 3 generalize the results of Theorem 2.4 [1, p. 44] for the two-dimensional case.

Preliminaries

The important role in the theory of fractional integro differentiation is played by absolutely continuous functions.

Let H = {(x,y): a < x < b,c < y < d}, —<x> < a < b < +<», — <x < c < d < +<»

Definition 1 [1, p. 2]. f(x) functionis called absolutely non-discontinuous into segments [a,b], if for any £ > Othere exists 5 > 0 such that for any finite set of pairwise non-intersecting intervals [ak,bk] £ [a,b], k = 1,m, such that Yk=i(^k — ak) < the inequality YJ1^=1\f(bk) — f(ak)l < £ holds. The space of these functions is denote by AC ([a, b]).

Definition 2 [1, p. 2]. Let us denote by ACn([a, b]), where n = 1,2,..., the spaces of functions f(x) which have continuous derivatives up to order n — 1 on [a, b] with f(n-1 (x) £ AC([a,b]).

Definition 3. A function f(x, y) is called absolutely continuous in n, if for any £ > Othere exists 5 > 0 such that for any finite set of pairwise non-intersecting intervals Ak = {(x, y): x1k < x < x2k, y1k < y < y2k}, the sum of the areas of which is less S, the inequality holds

Iik = 1lf(x2k,y2k)—f(X2k,yik)—f(Xlk,y2k)+f(Xlk,yik)l < ^ (1)

and if, moreover, f(a,y) £ AC([c, d])and f(x, c) £ AC([a, b]). The class of all such functions is indicated AC(fl).

Definition 4. ByACn'm(H), where n = 1,2,..., let us denote the class of functions continuously differentiable

Qn+m-2f

on H up to order (n — 1,m — 1), and its mixed partial derivative n-1 m_1 is absolutely continuous in H.

It is known that the class ACn([a, b]) belongs to those and only those functions f(x) that are representable as antiderivatives of Lebesguesummable functions:

f(x) = f^(x)dx + C,^(x) E Li(a,b) .(2)

Lemma 1 [1, p. 39]. The space ACn([a,b]) consists of those and only those functions f(x), which are represented in the form

f(x) =-^X(x - t)n-1$(t)dt + YZ-l Ck(x - a)k, (3)

where <p(x) E L1([a,b]), Ck being arbitrary constants. In the formula (3)

<(t)=fM(t),Ck=f-^. (4)

( \ dnf(x)

The last equality uses the notation f(n)(x) = . A similar property of the functions f(x, y) E AC(D) is as follows.

Lemma 2 [3, p. 238]. The class AC(D)consists of those and only those functions f(x,y) which are represented in the form

f(x,y) = C f? <(t,s)dtds + J*iP(t)dt + J*v(s)ds + C,(5)

where <(x,y) E L1(D),xp(x) E L1([a,b]),T](y) E L1([c,d]), and Cis an arbitrary constant. In order to generalize the last lemma to the case of a class ACn,n(D), we need the following lemma. Lemma 3.Let f(x,y) E AC(D), then

f(xy)=_l_f f f(n'm)(t,s)dids +ffm(a,y)rx-ay +

f(x,y) (n-l)\(m-l).Ja Jc (x-t)i-n(y-s)i-™+L i! (x a) +

+ molf0^(y - c)k - Yn-l Yy-lff(^(x — a)(y — c)k.(6)

In formula (6) the notation used f( (x, y) =

ilk!

dl+kf(x,y)

dxldyli gn+m-2-

Proof. Let be E AC(D). By virtue of Lemma 2, we have

dxn 1dym 1

dn+m~2f— = frX Ly <P(t, s)dtds + (x№dt + fy v(s)ds + C0 (7)

dxn-1 dym-1

Integrating sequentially (7) times n - l by x and times m - l by y, we get

l x y

f(x,y) = (n_1)!(m_1)!J J (x-t)n-1(y-s)m-1<(t,s)dtds +

(y-c)m-1 fx a c (x-a)n-1 fy

+ (n-1)l (m-l). Ja (x-t)n-1*(t)dt+(n-i)!(L-1)!.i (y-s)m-1V(s)ds +

+ m=:01^(y)(x - a) + Yk=dfk(x)(y - c)k,(8)

where ii (y) ( i = 0,n - l),fk (x) (k = 0,m - l) is arbitrary function. When integrating, the well-known for n - multiple integral formula is used [1]

fax dx fax dx^ ft F(x)*C = ^ f^x - t)n-1F(t)dt, (9)

proof, which is easy to implement by mathematical induction. It will be clear from the proof that an arbitrary constant in formula (7) is associated with arbitrary functions of formula (8) by the relation(n - l). in-!-1^^) + (m - l)!f(rni1)(a) = C0. Since f(x,y) E ACnm(fl), then derivatives £+^(0 < i <n,0 < k < m)exist and

are continuous in D. Calculating the derivatives with x respect to the order 0,n- l of the function f(x,y) given by formula (8), and assuming in them x = a, we obtain the equalities

= i! ti(y) + Yk=o f(P(a)(y — c)k,i = 0,n — 2, (10)

= + (n — 1)] f«-i(y) + ^¿^(^(y — C)*. (11)

Similarly, differentiating (8) by y and assuming y = c, we obtain the equality

^g^ =k\fk(x)+ YU f(k)(c)(x — a)i,k = 0^—2 , (12)

" ^j:(^+(™ — 1)!trn-i(y)+Y-o1<m-1\c)(x — ay. (13)

dym-l („-!)! Ja (x-f^-n^ Lrn-1

Expressing from formulas (10) - (13) ify) and fk(x) respectively, we get

n—1 m—1 n—1 ■ / . m—1 \

£ ii(y)(x -ay+Z fk(x)(y - c)k = £ fVM(y - cA +

i=0 k=0 i=0 ' \ k=0 ' m—1 , ,, / , , n—l \ ,

Y <y-cr d"f(x,c) y f(k)(v)(r_n)i\_ <x - a) r V(s)ds

+ L k\ ( dyk L l (y)(X a) ) (n-DKm-Dll (y-s)1—™ k=0 \ i=0 /

■ft_1 ^_1

(y-c)m—1 rx xp(t)dt _s-1(x-a)idif(a,y) ^(y - c)k dkf(x,c)

-(n - 1)\(m - 1)\ Ja (x - t)1—n = /L i\ dx1 + L k\ dyk (x-a)n—1 [y v(s)ds (y-c)m—1 fx xp(t)dt

ry v(s)ds (y-c)m—1 r

(n-1)\(m-l)\Jc (x-t)1—n

- YU Tiï—Kx -aY(y-c)k + (14)

Ql+Kf

Calculating the mixed derivatives dxidykof the function (8) at a point (a, c), we get

1 di+kf(a,c) _ i[k\c) T^(g)

H i ! '

i\k\ dxidyk

Substituting (14), (15) into (8), we get

n—1

1 rx ry <p(t,s)dtds y1dlf(a,y)

>(X,y) (n-1)\(m-1)\Ja J (x-t)1—n(y-s)1—m + Li\ dx1 (X a) +

+ (y-c)k- Y—1 Zk^^-dj^ (x -*Y(y- <)k. (16)

gm+nf(a c)

Equality (6) follows from (16) and from the fact that <p(x,y) = dxndym ■ The lemma is proved.

The following lemma gives a description of the class ACn'm(fl). It generalizes Lemma 1 to the case of two variables and Lemma 2 to the case n + m > 2.

Lemma 4.The space ACn,m(i..) consists of those and only those functions f(x,y), which are represented in the form

m—1

f(X,y) =

1 r rw — V +ftYM , V (y — c)K r xpk(t)dt I (n — 1)! (m — 1)! Ja Jc (y — s)1-m <P(t,s)atas + L(n — 1)! k\ Ja (x — t)1-" +

+ Y-01^-^Jc(y — s)m-1v(s)ds + Y-01 Yk=o Cik(x — aY(y — c)k, (17)

where p(x, y) G L1 (Q), y/k (x) G L1 ([a, b])(k = 0,m—1), rj(y) £ L1([c, d]), (i = 0,n — 1), Cik

being arbitrary constants.

Proof. Necessity. Let f(x,y) £ ACn'm(H). According to the lemma 3

f(x,y) = (n — 1)!(m — 1)!Ja J {x — t)1-n(y — s)1-™dtdS + — a) +

+Ym-1-^^^ - c)k - Yn=i Ym-1-^1^ - a)(y - c)k.(18)

Because f(n-1m-1)(x,y)EAC(ti), then f(n-1m-1)(a,y)EAC([c,d]), consequently, f(n-1l)(a,y) E ACm([c,d]), from here f y )g AC™ ([c, d ]) (i = 0,n- l). Use lemma [1, c.39]

f (a,y) = to-viic + Yk=° —¡¿—W - c) (19)

where r]t(y) E L1([c,d]). Then

n-1

L-i i\ K J ¿-li\(m-l)\Jc (y — s)

i=0 i=0

+ Yn-l Ym-1 (x - a)(y- c)k.(20)

i\k\

Similarly, it is proved that

m-1 m-1

fm)(x,c) (y-c)k fx M) ^

=Lik\(n-l)\i (x — t)1-m + k=0 k=0 a

+ Yn-l Ym-o1-^1^ - a)l(y - c)k. (21)

where ipk(x) E L1([a,b]). Substituting (20), (21) into (18), we obtain the formula (17), in which Cik=^fa'k)(a,c).(22)

q i+kf

Sufficiency. When calculating directly-:-— (0 < i < n,0 < k < m), it is easy to make sure that they

dx1 dyk

are all continuous in D, and

gn+m-2f_ = fax <(t, s)dtds + fax^(t)dt + rv(s)ds + (n - l)\(m - l)\ Cn-1m-1. (23)

dxn-idym-i - Ja fc t ja v^ljul -r jc -r (.ii vu ±j-.Cn-1,m-1

n+ m-2

Obviously E AC(D), from where it follows f(x, y) E ACn,m(D).

The theorem is proven completely. Notice, that

<(x,y) = f(n,m)(x,y);(24) ^k(x) = f(n,k)(x,c),k = 0,m- l;(25) Vi(y) = f(i'm~)(a,y),i = 0,n - l;(26) Cik=-^f(i'k)(a,c).(27)

Definition 5 [1, c. 459]. Let f(x,y) E L1(D). The integral ( Ia'p f)(x v) =_1_fx fy_f(t-s)dtds_(28)

(' a+,c+f )(x,y) r(a)r(p)fa fc (x-t)1-a(y-s)1-P,y28)

where a > 0,p > 0, is called a left-hand sided mixed Riemann-Liouville fractional integral of order (a,p). The fractional integral (28) is obviously defined on functions f(x,y) E L1(D), existing almost everywhere. Using the Fubini theorem, the semigroup property is proved.

Let f(x, y) E L1 (D), a, p, y, 5 be positive numbers, then equality holds almost everywhere in D

a+, + a+, c+ f = a+, + f.(29)

It can be shown that if a > Ofunction f(x, y) is defined in D and f(x, y) E L1 (D), then (Iz+,x)(x,v) E L1([c,d])Vx E (a, b); (lZ+y)(x,y) E L^b^y E (c,d).

In the last equations I£+xf, I%+,yf are partial Riemann - Liouville fractional integrals with respect to the variables x and y, respectively.

Taking these equalities into account, it is directly verified that

(lZ+xcl!+;yf)(x.y) = (lLyIZ+j)(x.y) = (a'+,c+f^) (x, y).(30) Definition 6 [1, c. 460]. For functionf (x, y), given on D, formula

(f)(xy)=_1__gn+^fx fy_f(t,s)dtds_(31)

JVA>yJ r(n-a)r(m-p)gxngymfa fc (x-t)a-n+1(y-s)P-m+1 ( )

where a > 0, p > 0, is called a mixed Riemann-Liouville fractional derivative of order (a, p), n = [a] + l,m = [p] + l.

If the function f(x,y) has a property I^"'™-^ f E ACn,m(D), then the order of taking the derivatives in (31) does not matter, and(D"+^c+f)(x, y) E L1 (D) .

Definition 7 is a two-dimensional analogue of Definition 2.3 [1, p. 43].

Compositions of mixed fractional integral and mixed fractional derivative of the same order

Following [1, p. 44], we define the following classes of functions.

Definition 7. Let I^^?C+(L1) denote the space of function f(x,y), represented by the left-sided mixed fractional integral of order (a,p) of a summable function: f = l"ftC+$, $ £ L1(n).

Definition 8. LetO < a < 1,0 < p < 1. A function f(x,y) £ L1(H) is said to have a summable fractional derivative D^c+f, if £ ACn'm(n).

The following theorem defines the necessary and sufficient condition for the unique solvability of the two-dimensional Abel integral equation.

Theorem 1.In order that f(x, y) £ Ia+C+ (L^_), a > 0, p > 0, itis necessary and sufficient that fn-a,m-p£ACn'm(i}),(32) '

where n = [ a] + 1,m = [p] + 1, and that

fn-lm-p(a,y) = 0,i = 0/n—1(33)

f^°J^)m-p(x,c) = 0,k = 0,m — 1;(34)

fn-a'm—p (a,c) = 0,i = 0,n-1,k = 0,m-1.(35)

Proof. Necessity. Let f = IaftC+ (,( £ L1(H). In view of the semigroup property fn—a,m—p(X,y) = C^cT^f = ¡alc* p,(36)

where p £ L1(H). From here follow feasibility conditions (33) - (35). Feasibility condition (32) follow from Lemma 4. This implies the fulfillment of conditions (33) - (35). The fulfillment of condition (32) follows from Lemma 4.

Sufficiency. Under condition (32), we can present fn—a m—p according to Lemma 3, in the form

f r 1 fV .f. ,

tn—a,m—P(X,y) = (n-1)\(m-1)\.]a Jc (x-t)1—n(y-s)1—mdtdS +

k\

+ L---(х-"У+^-k\-(У -

i=0 k=0 f{ik) (a c)

-YU Yk=o (x - a)l(y - c)k ,(37)

where fn-'™m-ts £ L1(A). Taking into account conditions (33) - (35), the last equality is written in the form

>n-a,m-P c

An,m) , ,

с f \ 1 Гх ГУ 'n-a.m-R(z,i') J.J

fn-a,m-P (x, У) = T^H^ L Jc fr_t-i-nfv_x)i-m dtds(38)

Using the semigroup property (29), we can write

,п-а,т-р с _ ,n,m c(n,m) _ ,п-а,т-р ,a,ji c(n,m) /--зол ' a+,c+ J = 'a+,c+ln-a,m-fS = 'a+,c+ 'a+,c+Jn-a,m-P'( )

From here (f - С^+й-а^п-р) = 0 APP!yingthe integralto this equality we get Jna^c+(f - ia&+fnnamm-P)dxdy=0. (40)

From here f=tf^n-am-ejn-am-ee Li(n).The theorem is p™^.

Note that Theorem 1 is a generalization of Theorem 2.3 [1, p. 43] in the case of two variables. From it, in

а В

particular, it follows that the class of functions having a summable fractional derivative Da'+c+f in the sense of

cc В cc В

Definition 8 is wider than the class of functions Ia+fi+ (L1). Namely, the class 1а'Вс+ (L±) owns only those functions

а , В

that have a summable fractional derivative Da+c+f, for which equalities (33) - (35) hold. Theorem 2.Let a > 0, p > 0. Then equality

DZtXt+f = f(x,y)(4D

performed for any summable function f(x, y). Proof. We have

dn+m

р,а,В та,В с _ d jn-a,m—Bja,B с _

Ua+,c+'a+,c+J = QxnQym 'a+,c+ 'a+,c+J =

_ r-1(B)r-1(n-a) dn+m rx ry dtds rt rs f(u,v)dudv ... = r(a)r(m-B) dxndym 'a 'c (x-t)a(y-s)P 'a 'c (t-u)n-a(s-v)m-P'( )

Changing the order of integration, we get

В та,в с (Г(а)Г(р))-1 dn+m fx Гу DJs+Ia+U = г(п-а)Г(ш-р) dxndym Ja l f(u'v)dudv X

d d

I I —

Ju Jv (X

t)a(y - s)P(t - u)n-a(s - v)m-P

gn+m rx ry i

-I I f(u,v)dudv-

Jn J r *

dxndymJaJc Г(а)Г(п - a)

<-x dt 1 fy ds

x

rx dt 1 Г

Ju (x - t)a(t - u)n-a Г(р)Г(т - p) Jv (y-s)P(s-v)m-P

x

gn+ m

r(n)r(m) gxngym

Q.E.D.

f x f

a c

y

fiuv)

c (x-u)1-n(y-v)1

■dudv = f(x,y) (43)

Theorem 3.For any functionf (x,y) E Ia+,c+(L-i) the equality IaJ,c+DaJ,c+f = f(x,y),(44)

a B

and for any function that has a summable derivative D^+c+f (in the sense of definition 8), the equality

iaut+Dut+f=f(xy

i

=l

(x - a)"

r(a - i)

( n- -1,l) Jn-ao

( a, y)

m-1

-I

(y - c)p-k

- k-1

-f0T---k-1)(x,c) +

+ Yn=o Y

m-1 k=l

k=o F(p-k)

(x-a)a-l-1(y-c)p-k-1 (n-i-1,m-k-1) ( .

'¡n-a,m-p (a, C),(45)

_ jy,S

r(a-i)r(P-k)

where fУ'S(x,V) = Iyt+f.

cc B cc B

Proof. Let f(x,y) E Icl,8c+(L1), thenf(x,y) = Ia'pc+<, <(x,y) E L1(D). Based on Theorem 2, we have

a, 8

l, 8 l, 8 D f =

a +, + a+, c+

Let now Ia-ic'+ 8f E AC(D). According to Lemma 3, the integral fn-aim-p(x,y) = ^

l, 8 l, 8 a+, + Da+, + a+, +

l, 8

< =

a+, c+

< = f(x,y). (46)

n- l, m- 8 a-a+ 8 f can be

represented as

( ) n, m ( n, m) n- l, m- 8 a+, + n- l, m- 8

+

I

f^m-p^y)

(x - a) +

+Y

-(0") (x c)

m-k=l

(y - c)k - Yn=o Ym

k=l

=l

k\ \

(x-a)l(y-c)k .(47)

By the semigroup property, the equality

n, m ( n, m)

a+c+fn-a;m-p ^a+,c+ la+,c+

Further,

n-am-p ja,p f(n,m)

fn(-namm-p.(48)

(x - a) Ai,.

f(-°;m-p(a.y) = il

n-am-p | a+, +

D n

a+, x

-a(x a) Dm-pf(ifi)

c+,y

'fn-¿,m-p(a,v)) +

(x-ay(y-c)m-p-1 m + i\r(m-p) fn-a'1

+ i\r(m-p) fn-a'1(a' C).(49)

From the last equality it follows that

( a, ) = a

n- a, m- p a+, +

I

^-cc'm-p (a'

(x-ay = ina;ac'rp

' n- 1

I

(x - a)-

r(l + i-n + a)

fn%(a,v)) +

(x - a)1

r(l + i- n + a)

f^Aa.y)

+

+ Yn-1 (x-a)\y-c)m P 1 Ai'°) (a c) (50) + Y = V.r(m-p) fn-a'1((1, C),(50)

from where, redesignating the summation index, we get

n-1 f(i'0) (a v)

1 fn-a,m-p (a, y)

I

=l

(x - a) = /,

n- a, m- p a+, +

+ Y = V.r(m-p) fn-a'1(a,C)(51)

Equality is obtained similarly

n-1 f(0'k) (x C) 1 ln-a.m-B v1' L)

I

=l

m-1 ( x- a) n + Y k=l

i I

\

Hy-c)"

(y-c)k = I,

k\r(n-a)

&-p(a,c). (52)

It is not difficult to see that

II

=l =l

n- a, m- p = a+, +

n-1 f( , k) ( a, ) Jn-a'm-B(a,c)

, m- p i\k\

I

=l

(x - a)

a- -1

r(a - i)

-^^(ay

+

n- a, m- p a+, c+

I

k=l

(y-c)p

- k-1

r(p - k)

(l, m- k-1) fl, m- p

(x, c) ) +

(x - a)l(y - c)k =

n-1 m-1

I I

n- a, m- p a+, c+

-p ((x - a)(y - c)h

k=l

i\k\

f( , k) ( a,

>n-a,m-p (a•

C)) =

1

k\

n+ a

=l

=l

n-1

=l

_ jn-am-p I Vn-1 Vm-1(x-a)a 1 1(y-c)P " 1 An-j-1'm-k-1) ( = 'a+,c+ (Yi=° Yk=0 r(a-i)r(p-k) fn-a,m-p (a,c)j. (53)

Taking into account equalities (48), (51) - (53), equality (47) is written in the form

jn-a/m-p c jn-a/m-p ja,p ^afi f , jn-a/m-p IV"1 (x - a) f(n-i-1fl)(n v) | i

1 a+,c+ J 'a+c+ 'a+'c+Da+'c+J +'a+c+ \ I • r(tt~l) j n-afi (a,y) ) +

n-1 m-1

(x-ay(y-c)m-p-1 m -p V (y-c)p-k-1A0'm-k-1)(

1 i.r(m-p) fn-a'1(a,C)+'a+'c+ y1 r(p-k) fo'm-p (

—

=l k=l

Zm-1(x-a)n-a-1(y-c)k (ok)

k\r(n-a) f1m-p (a,c) k=l

_1n-a'm-p (yn-1 rm-1(x-a)a-l-1(y-c)P-k-1 jr(n-i-1'm-k-1) ^ .... 1 a+,c+ (Yi=° Yk=0 r(a-i)r(p-k) fn-a,m-p (a,C)). (54)

By grouping the terms, we get

jn-a,m-p I f _ ja,p j.a.p f ST1 (x - aY ' 1 An-i-1fi) ( a ' a+,c+ \J 'a+,c+Da+,c+l /• F( a — i} f n-afi (a'y)

=l

m-1 k-1

I(y-c)p-k-1f(om-k-1)(xc) + 1 r(P-k) fo'm-p (x,C) +

- a)a-i-1(y - c)p-r(a - i)r(J3 - k)

+ I I1 (x - a)a-i-1(y - c)p-k-1 f(n-i-1'm-k-1)(a c)) =

+ 1 I r(a-i)r(p-k) fn-a'm-p (a,c)) = =l k=l

_ m-1 (x-a)i(y-c)m-P-1 Ail) f„ , rm-1(y-c)k(x-a)n-a-1 ¿-(ok) (

= Yi=° mm-*) fn-a,1(a,c)+Yk=o km-a h,m-p(a,c)(55)

a, p

In the right-hand side of equality (55), under the integral is a summable function. Applying the operator Ia'+c+

to both parts of equality (55), we obtain

rn,m I f _ ja,p na,p f _ \ (x - ^ ' 1 An-i-1fi) ( n _ ' a+'c+\l 'a+,c+Da+,c+l /• F( a — i} f n-afi (a'y)

I F(R-k) fo,m-p (x,C) +

k-=o F(p-k)

+ I I1 (x - a)a-i-1(y - c)p-k-1 fin-i-1,m-k-1)(a c)\ =

+ I I r(a-i)r(p-k) fn-a'm-p (a,c)) = i=0 k=0 /

_ Vn-1 (.x-a)i+aiy-c)m-1 (i'O) (. Vm-1 (.y-c)k+P(x-a)n-1 (ok) ( ,

= Yi=0 r(i + a+1)r(m) fn-a'1(a,C)+Yk=° r(k + 1+p)r(n) f1'm-p(a,C). (56)

Under the integral on the left side of the equality is the summable function, and the right side of the equality

g n+ m

is absolutely continuous. Finding the mixed derivative dxng m of both parts of the equality, we get

f _ ja.p na,p f _ \ (x-a)a 1 1 An-i-1'0), , _ m- (y-c)p k 1 (o,m-k-1) ( n ,

f la+,c+Da+,c+f I r(a-i) fn-al (CI,V) I i(B - k) f0,m-p (x,c) +

=o k=o

i Vn-1 rm-1 (x-a)a-l-1(y-c)P-k-1 jr(n-i-1'm-k-1)r„ n _ n

+ Yi=o Yk=0 r(a-i)rap-k) fn-a,m-p (a, C) = (5 7)

The theorem is proved.

References

1. S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives. Theory and Applications. Gordon&Breach. Sci. Publ., N.York-London, 1993, 1012 pp (book style)

2. S. Vasilache, Asuprauneiecuatii integrale de tip Abel cu doua variable.//Comun.Acad.R.P.Romane, 1953. Vol. 3. P. 109-113 (journal style)

3. V. I. Smirnov, Higher mathematics course. T. 5. M.: OGIZ, 1947. 584 p. (book style)

4. T. Mamatov and S. Samko, "Mixed fractional integration operators in mixed weighted Holder spaces", Fractional Calculus &Applied Analysis (FCAA), vol.13, № 3 (2010),pp. 245-259. (journal style)

5. T. Mamatov, "Mapping Properties Of Mixed Fractional Integro-Differentiation in Holder Spaces", Journal of Concrete and Applicable Mathematics(JCAAM). Volume 12, Num.3-4. 2014. p. 272-290 (journal style)

6. T. Mamatov, Mapping Properties of Mixed Fractional Differentiation Operators in Holder Spaces Defined by Usual Holder Condition, Journal of Computer Science & Computational Mathematics, Volume 9, Issue 2, June 2019. DOI: 10.20967/jcscm.2019.02.003 (journal style)

7. T. Mamatov, D. Rayimov, and M. Elmurodov , Mixed Fractioanl Differentiation Operators in Holder Spaces. Journal of Multidisciplinary Engineering Science and Technology (JMEST), Vol. 6 Issue 4, April - 2019. P. 9855-9857 (journal style)

8. T. Mamatov, F. Homidov, andD. Rayimov, On Isomorphism Implemented by Mixed Fractional Integrals In Holder Spaces, International Journal of Development Research, Vol. 09, Issue, 05 (2019) pp. 2772027730 (journal style)

9. T. Mamatov, Weighted Zygmund estimates for mixed fractional integration. Case Studies Journal ISSN (2305-509X) - Volume 7, Issue 5-May-2018. (journal style)

10. T. Mamatov, Mixed Fractional Integration In Mixed Weighted Generalized Holder Spaces. Case Studies Journal ISSN (2305-509X) - Volume 7, Issue 6-June-2018. (journal style)

11. T. Mamatov, Mixed fractional integration operators in Holder spaces (Russian). «Science and World». Volgograd, № 1 (1). 2013, P. 30-38 (journal style)

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13. T.Mamatov and D.Rahimov, Some properties of mixed fractional integro-differentiation operators in Holder spaces. Journal of Global Research in Mathematical Archives Volume 6, No.11, November 2019. pp.13-22. [Online]. Available: http://www.jgrma.info.(General Internet site)

14. T.Mamatov, R. Sabirova and D.Barakaev. Mixed fractional differentiation operators in Holder spaces defined by usual Holder condition. Scientific journal "Chronos" multidisciplinary collection of scientific publications, "Questions of modern science: problems, trends and prospects" issue 11 (37), Moscow, 2019, pp. 79-82

ПРИНЦИПИАЛЬНЫЕ ОСНОВЫ СОВЕРШЕНСТВОВАНИЯ МЕТОДОВ ОБУЧЕНИЯ

МАТЕМАТИКЕ

Хамраева Зилола Кахрамоновна

Старший преподаватель Бухарский инженерно-технологический институт

г.Бухара, Узбекистан Пулатова Манзура Исхаковна

кандидат мат. физ. наук, доцент Бухарский инженерно-технологический институт

г.Бухара, Узбекистан

FUNDAMENTAL BASIS FOR IMPROVING THE METHODS OF TEACHING MATHEMATICS

Khamraeva Zilola Kakhramonovna

Senior Lecturer Bukhara Engineering Technological Institute Bukhara, Uzbekistan Pulatova Manzura Iskhakovna

Candidate of Mathematical Phys. Sciences, Associate Professor Bukhara Engineering Technological Institute Bukhara, Uzbekistan

Аннотация. Совершенствования методической системы обучения следует уделять внимание каждому элементу её структуры. Совершенствование методической системы обучения должно отправляться от сложившейся ранее системы обучения и воспитания учащихся. Общие принципы совершенствования методической системы и указанные выше конкретные положения реализовались при разработки программ, учебников и методических пособий по новомудля начальной школы учебному предмету - математике.

Abstract. To improve the methodological training system, attention should be paid to each element of its structure. Improvement of the methodological system of teaching should be based on the previously established system of teaching and upbringing of students. The general principles of improving the methodological system and the above specific provisions were implemented in the development of programs, textbooks and teaching aids on a new subject for elementary school - mathematics.

Ключевые слова: методической системы, обучения, метод, структура.

Key words: methodical system, teaching, method, structure.

Методы обучения как важнейший компонент методической системы. Сектор начального обучения при решении поставленной перед ним задачи совершенствования содержания и методов обучения