CC BY
14
Chelyabinsk Physical and Mathematical Journal. 2022. Vol. 7, iss. 3. P. 312-325.
DOI: 10.47475/2500-0101-2022-17305
NONLINEAR SYSTEM OF IMPULSIVE
INTEGRO-DIFFERENTIAL EQUATIONS WITH
HILFER FRACTIONAL OPERATOR AND MIXED MAXIMA
T.K. Yuldashev1", T.G. Ergashev2 b, T.A. Abduvahobov3 c
1 Tashkent State University of Economics, Tashkent, Uzbekistan
2 Tashkent Institute of Irrigation and Agricultural Mechanization Engineers, Tashkent, Uzbekistan
3 Tashkent State Agrarian University, Tashkent, Uzbekistan
"[email protected], [email protected], c [email protected]
A nonlocal boundary value problem for a system of ordinary integro-differential equations with impulsive effects, nonlinear mixed maxima and fractional order Hilfer operator is investigated. The nonlinear boundary value condition is given in the nonlinear integral form. The problem is reduced to the nonlinear system of functional integral equations. The system of functional integral equations has terms of nonlinear functions in integral and non-integral forms. The method of successive approximations in a combination with the method of compressing mapping is used in proving the unique solvability of the boundary value problem.
Keywords: impulsive integro-differential equation, Hilfer fractional operator, nonlocal boundary condition, mixed maxima, unique solvability.
1. Introduction
Fractional calculus plays an important role in the mathematical modeling of many problems in scientific and engineering disciplines, continuum and statistical mechanics, in the construction of the Ebola epidemic model, a fractional model for the dynamics of the tuberculosis infection and novel coronavirus (nCoV-2019) and otheres [1-9]. In the applications of fractional derivatives in solving differential equations were obtained interesting results in the many papers (see, for exmample, [10-19]. Many problems in applications are described by fractional order differential equations, the solution of which is a function with first kind discontinuities at some points of the time interval. Fractional differential equations of such type are called fractional differential equations with impulsive effects. Integer order differential equations with impulsive effects are considered, in particular, in [20-27]. On the other hand, in recent years the interest in the studying of differential equations with nonlocal boundary value conditions is increasing (see, for example, [28-37]). Also a lot of publications in studying of differential equations with impulsive effects, describing many natural and practical processes, are appearing [38-45]. In [46-49], the issues of the solution existence for impulsive differential equations with fractional order derivatives are considered.
Differential equations with maxima have singularities in single valued solvability. Moreover, the jumpiness of solutions is a natural thing for differential equations with mixed maxima [50]. In the present paper, we investigate a nonlinear nonlocal boundary
value problem for a system of ordinary Hilfer type fractional order nonlinear integro-differential equations with impulsive effects and nonlinear mixed maxima. The questions of existence and uniqueness of the solution of the nonlinear boundary value problem are investigated. This paper is the further development of the paper [51].
Let (0,T) be an interval, 0 < T < to, a > 0. The Riemann — Liouville a-order fractional integral of a function n(t) is defined as follows:
t
«t) = rO) /(t - s)a-1n(s)ds, a > 0, t e (0,T),
o
where r(a) is the Gamma-function. Let n — 1 <a ^ n e N. The Riemann — Liouville a-order fractional derivative of a function n(t) is defined as
dn
Dotn(t) = ^lont-an(t), t e (0,T). The Hilfer operator will be considered in the form
DT = lY-addt/o1-7, 0 < a ^ 7 ^ 1. Note that this operator can be express through operator V^f from [4, vol.1, p. 55,
a, Y-a
formula (37)] by the equality = V011-a , i. e. 7 = a + ^(1 — a).
2. Problem statement
On the set (0, T) \ {tj}, i =1, 2,... ,p for 0 < a < 7 ^ 1 we consider the following fractional order system of nonlinear integro-differential equations
DoTx(t) = f (t,x(t), Io1t© (t,s,max {x(t)|t e [h 1 : | : h2]})) , (1)
where [hi : | : h^ =
\ [min{hi(t); h2(t)};max{hi(t); h2(t)}] : t e (0,T) \ {tj}, i =
1, 2,... ,p, hj (t) = hj (t,x(t)), j = 1, 2}, f : [0,T] x Rn x Rn ^ Rn, 8 : [0,T]2 x Rn ^ Rn.
We study the unique solvability of the system of nonlinear integro-differential equations (1) with nonlinear nonlocal boundary value condition
A ■ Io1-7 x(0+) + r(7)Io1T [K(t, s) s1-Yx(s)] = B(t, x(t)) (2)
and nonlinear impulsive effect condition
t Y-1
^ 11-7+1 x (t+) — x (t-) = Fj(x (tj)), i = 1, 2,... ,p, (3)
where 0 = to < t1 < ■ ■ ■ < tp < tp+1 = T, A e Rnxn is given matrix, K(t, s) is
T
given (n x n)-dimensional matrix function and det Q(t) = 0, Q(t) = A + JK(t, s)ds,
o
Fj : Rn ^ Rn, B : [0,T] x Rn ^ Rn are given functions; Io1-7 x(0+) = lim Io1-7 x(t),
t^o+
x (t+) = lim x (tj + l), x (t-) = lim x (tj — l) are right-hand sided and left-hand sided limits of function x(t) at the point t = tj, respectively.
2.1. Formulation of problem.
To find the function x(t) e Rn, such that for all t e (0, T) \ {i,}, i = 1, 2,... ,p, 0 < ¿i < t2 < ••• < tp < T it satisfies integro-differential equation (1), nonlinear integral condition (2) and for t = t,, i =1, 2,... ,p nonlinear limit conditions (3) hold for it.
3. Reduction the problem to a functional integral equation
Let a function x(t) e is a solution of nonlocal boundary value problem (1)-(3) on (0,T) \ |tj}, i = 1, 2,...,p. We rewrite fractional differential equation (1) on the interval (0,t1] as
d
^tTdtIo-l'Yx(t) = f (t, ,
where by f (t,x, y) we denote the given nonlinear function
f (t,x(t), /o1T© (t,s, max {x(t)|t G [hi(t,x(t)) : | : h2(t, x(t))]})) . Appling the operator to the both sides of the last differential equation, we obtain
ti
I0\ddtll-7x(t) = f(^/(ti - s)a-1f (s,x,y) ds.
0
Hence, taking into account the formula
d T 1 _Y / S / S t
7-1
10\^I01-7x(i) = x(t-) - ^ lim Io1-7 x(t)
0dt 0^^ r (7) £o+ 0
we obtain
1 ti
17-1 1 ti ,. T1—Y 1
x (t-) = lim IoV x(t) + J (ti - s)a-1f (s, x, y) ds, t G (0, ti].
0
r (7) tio+ u 41 v ' r (a)
Analogously, by integration of the fractional order differential equation (1) on the intervals (t1,t2] , (t2, t3], ..., (tp,tp+1], we obtain:
1 t2 t7-1 1 / x (t-) = ^ lini/11"? x(t) + —- (t2 - s)a-1f (s,X,y) ds, t G (t1,t2] ,
r(7) 12 r(a),'
17-1 1
ti t3
x(t-) = f^T lim+/i12tYx(t) + -— (ta - s)a-1f (s,x,y) ds, t G (t2,ts] r (Y) tit+ r (a).
t Y-1 1
t2 ip+l
x (tp+1) = f"(Y)tlln+/ipiY+i x(t) + F(a)i (tp+1- s)a f (s,x,y) ds, t e (tp,tp+1].
tp
Hence, taking into account the equality x(t-+1) = x(t) on the interval (0,T], we have
t;
t -, If r +Y-1
(t - s)a-1f (s,x,y) ds
r (a) J o
x (t1) - rlY) t1i>m+/0-iY x(t)
+
+
l7-1
x (¿2) - T^T lim 1 htlx(t)
г (y)
t1t+
+ ... +
^7-1
x
(t) — ^ lim I1-+1 x(t)
г (y)
t1t+
^7-1
- r(Y) b1 0-7 x(t)—
t Y-1 ¿2
p / ч lin+ItitY x(t) - X (t1)
i (Y) tit+
t Y-1
¿3
W \ liHlIt2tY x(t) - x (t2)
г (Y) tit+
1 Y-1 ¿p+1
lim 11 tY x(t) — x (tp) Г (y) tit+ tp tp+^ ; KP>
+ x(t).
Due to condition (3), the last equality we rewrite as
x(t)
tY-1 1 Y r(Y) Ä+1 o-Y x(t) +
г (a)
(t — s)a-1/ (s,x,y) ds О , Fi (x (ti)). (4
E
0<ti<t
The last equality has singularity at the point t = 0. We subordinate the function x(t) e Rn in (4) to satisfy boundary value condition (2). Then, multiplying both sides of obtained equality by t1-7, presentation (4) we rewrite as
Г (y) t1-Yx(t) = 10-Y x(0+) +
i(Y) t1-Y Г (a)
(t — s)a-1/ (s,x,y) ds+
+Г (y) t1-Y J] Fi (x (ti)).
0<ti<t
Now we multiple the last equality by K(t, s) and integrate it from 0 to T. Then
T T
Г (y ) / K (t, s) s1-Y x(s) ds = 101-Y x(0+) / K (t, s) ds+
0
T
+ ГЦ/K (t, s) s 1-YJ (s — 0)a-1/(0, x, y) d0 ds+
T
+ Г (y)/ k(t,s) s1-Y ^ Fi (x (ti)) ds. (5
0<ti<t
Substituting presentation (5) into nonlinear condition (2), we obtain:
T
A + J K(t,s) ds 0
I0-Yx(0+)
Г (y)
T
в(t,x(t)) — ^T K(t,s) s1-Y / (s — 0)a-1/(0,x,y) d0 ds— Г (a) J J
T
Г (y) K(t,s) s1-Y ^ Fi (x (ti)) ds. (6
0<ti<t
t
1
t
s
T
By virtue of det Q(t) = det
A + / K(t,s)ds
= 0, the equality (6) one can rewrite as
o
1
/ 1-7 x(0+) = Q-1(t)B(t,x(t))-
T s
Q-1(t) / K(t,s) s1-7 /(s - 0)a-1f (0,x,y) d0ds-
r(a)
o o
T
r(7)Q_1(tW K(t,s) s1-7 J] Fi (x (ti)) ds. (7) 0 0<ti<t
Substituting equality (7) into representation (4), we obtain
t
11 x(t) = t7-1Q-1(t)B(t,x(t)) + — (t - s)a-1f(s, x,y)ds-
o
T s
1 17-1Q-1(t) / K(t,s) s1-7 /(s - 0)a-1f (0,x,y) d0ds-
r(a)
o
T
17-1Q-1(tW K (t,s) s1-7 £ Fi (x (ti)) ds + £ Fi (x (ti)). (8)
0 0<ti<t 0<ti<t Since the following equalities hold
T s
J K(t, s) s1-7 y (s - 0)a-1f (0, x, y) d0ds = 0 0
T T
K(t, 0) 01-7 d0 (t - s)a-1f (s, x, y) ds,
0 s
T T
i K(t, s) s1-7 ^ Fi (x (ti)) ds = ^ i K(t, s) s1-7 ds Fi (x (ti)) ,
0 0<ti<t 0<ti<T^
presentation (8) implies
t
11 x(t) = 17-1Q-1(t)B(t,x(t)) + — (t - s)a-1f (s, x,y)ds-
0
T T
1 7—1^» —/ i /i\ ¿i 1—/i „\a—1
17-1Q-1(t) , , K(t, 0) 01-7d0 (t - s)a-1f (s, x, y) ds-
r(a)
0s
T
17-1Q-1(t) ^ /K (t,s) s1-7 ds Fi (x (ti)) + £ Fi (x (ti)). (9)
0<ti<Tr 0<ti<t
li
After some simplifications in (9) we obtain that the following equalities hold: (t — s)a-1f (s,x,y) ds-
t
1 ' \a—i
r (a) J o
T T
^ t7-1Q-1(t)/J K(t,0) 01-Y d0 (t - s)a-1/(s,x,y) ds
0 s t / s
:Q-1 (t)f(a^/ (a + / K M)d0| (t - s)a-1f (s,x,y) ds-
00
T T
1 17-1Q-1(t) / /K(t,0) 01-7d0 (t - s)a-1f(s,x,y) ds, (10)
r (a)
0s T
E Fi (x (ti)) - 17-1Q-1 (t) ^ /K(t,s) s1-7ds Fi (x (ti))
0<ti<t 0<ti<T 1
ti
Q-1(t) E (a + / K(ti,s)d^Fi (x (ti)) -0<t <t
0<ti<t 0
T
17-1Q-1 (t) ^ f K (t,s) s1-7 ds Fj (x (tj)). (11)
t<ti+i<T /
Taking into account (10), (11) and multiplying both sides by t1-7, from presentation (9) it follows the nonlinear functional integral equation
11-7x(t) = J (t; x) = ^(7) Q-1 (t)B (t,x(t))+ ^ t1-7 G (t,tj) Fj (x (tj)) +
(7) o<ti<t
T T
+ /11-7 G(t,s)(t — s)a-1 f(s,x(s),/ 8 (s,0, max {x(t )|t e [h1 : | : h2]}) d^ds
oo
(12)
for t e (tj, tj+1 ], i = 0,1,... ,p, where hj = hj (0, x(0)), j = 1, 2,
Q-1(t)(A + /K(t,0)d0), 0 ^ s ^ t, G(t,s)=< °t
—17-1Q-1(t) / K(t, 0) 01-7d0, t < s ^ T.
s
4. One value solvability theorem
According to (12), the unknown function x(t) we consider with the weight function t1-7. So, by C ([0,T], Rn) the Banach space of continuous vector functions t1-7x(t), defined on the segment [0,T], with the norm
t1-7 x(t)
\
V max 111-7x7- (t) |,
s 0<-t^T 1
0<t<T j=1 < <
is denoted. By PC ([0,T], Rn) we denote the following linear vector space
PC ([0,T], Rn) = { t1-Y x(t) : [0,T] ^ Rn; x(t) G C ( (ti, ti+1 ] , Rn), i = 1,...,p},
where x (t+) and x (t") (i = 0,1,... ,p) exist and bounded; x (t") = x (ti). Note, that the linear vector space PC ([0,T], Rn) is Banach space with the norm
IIt1-Yx(t) ||PC = max {||t1-Yx(t) ||c, i = 1, 2,... .
For the unique solvability of nonlocal boundary value problem (1)-(3) on the set (0,T) \ {t^, i = 1, 2,... ,p we study the existence of a unique solution of equation (12) in the class PC ([0,T], Rn).
Theorem 1. Suppose the following conditions are fulfilled:
T
1) Mf = max
0<i<T
f t, 0, / 8(t,s, 0) ds
<00 ;
o
1
2) = max | Fi(0) | < Bq = ^ max I Q-1(t)B (t, 0) | <
3) for all t G [0,T], x, y G Rn and i = 1, 2 holds
| f (t, x1, y1) - f (t, x2, y2) | ^ M1(t) t1-Y I x1 - x2 I + M2(t) | y1 - y2 |;
4) for all (t, s) G [0,T]2, x G Rn holds
| ©(t, s, x1) - ©(t, s, x2) | ^ Ms(t, s) s1-Y | x1 - x2 |;
5) for all t G [0,T], x G Rn holds
| B(t, x1) - B(t, x2) | ^ M4(t) t1-Y | x1 - x2 | ;
6) for all t G [0,T], x G Rn holds
| hj(t, x1) - hj(t, x2) | ^ M4+j(t) t1-Y I x1 - x2 I , j = 1, 2;
7) for all x G Rn, i = 0,1,... ,p holds
I Fi(x1) - Fi(x2) I ^ Ni tt1-7 I x1 - x2 I , 0 < Ni;
8) p = S1 + S2 + S3 + S4 < 1, where 0 < Mi(t) G C[0,T], i = 1, 2,... 6,
T
S1 = max i |t1-7G(t, s) | (t - s)a-1M1(s) ds,
T T
\a— 1
I It GU,sj|U - s
0<t<T
S2 = max / |t1-YG(t, s) j (t - s)a-1M2(s) / Ma(s,0) [1 + Mf (M5(0) + Ma(0))] d0ds,
0 0
p
Sa = max J] |t1-YG(t,ti)| ■ ^ S4 = ^0<a<T | Q-1(t) | M4(t).
Then functional integral equation (12) has a unique solution in the class PC ([0, T], Rn). This solution can be found by the following iterative process:
t1-7 xm (t) = J (t; xm-1), t G (t., ti+i), m = 1, 2, 3,... ; t1-Yx0(t) = 0, t G (ti, ti+i), i = 0,1, 2,...,p.
:i3)
Proof. We consider the operator J : PC ([0,T]; Rn) ^ PC ([0,T] x Rn), defined by the right-hand side of equation (12). Taking the first and the second conditions of the theorem, for the first difference of approximations (13) we have the estimates
t1-Y (x 1(t) - x0(t)) || ^ —- max I Q-1(t)B (t,x0(t)) | +
r (7) o^t^T
p
+ maxV|t1-7G(t,ti)||Fi (x0(ti)) | + max I |t1-7G(t,s)| (t - s)a-1
T
x
i=1
x
T
/^s,x°(s),y 8 (s,0,max {x0(r)|r G [h0 : | : h02]}) d^
ds <
^ S0 (Mf + ) + Bq < to, (14
where = hj(0,x0(0)) = hj(0,0), j = 1,2,
T
S0 = max / |t1-7G(t,s)| (t - s)a-1ds ImaxV |t1-7G(t,ti)
0< t<T
0<t<T '
i=1
Mf = max
0<t<T
T
^t,x0(t)^ 8 (t,s,x0(s)) ds^
1
Bq ^^rmax I Q-1 (t)B(t,x0(t)) I , = max I F (x0(t)) I .
Q r(Y ) o<t<T1 v ¿e{i,2,...,p} 1 viv/yi
Then, by the conditions of the theorem, for difference of arbitrary consecutive approximations and for arbitrary t G (ii,ii+1j, we have
t1-Y (xm(t) - xm-1(t)) | ^ — | Q-1 (t) [B (t,xm-1(t)) - B (t,xm-2(t))] | +
T
+ J |t1-YG(t, s)| (t - s)a-1x 0
T
/^s,xm-1(s),y 8 (s,0, max {xm-1(r)|t G [h'-1 : | : h'-1]}) d^ -0
T
^s,xm-2(s),y 8 (s,0, max {xm-2(r)|r G [h''-2 : | : h''-2]}) d^
+ J] 111-7G(t,ti)| | Fi (xm-1(ti^ - Fi (xm-2(ti^ | ^
i=1
^ ^ | Q-1(t) | M4(t) ■ | t1-Y (xm-1(t) - xm-2(t)) | +
T T
+ J |t1-7 G(t, s)| (t - s)a-1 M1(s) ^|s1-7 (x m-1(s) - x m-2(s))| + M2(s)y M3 ( 00 x (01-7 | max {xm-1(r)It G [h^-1 : I : h^-1]} -- max {xm-2(r)It G [h^-1 : I : h^-1]} | + +01-Y | max {xm-2(r)It G h^-1 : I : h^-1} -
- max {xm-2(r)It G h^-2 : I : hT2} |) d0
ds+
p
+ £ |t1-YG(t,ti)| ■ Ni | t1-Y(xm-1(ti) - xm-2(ti)) | . (15)
i=1
Taking into account that
| max {xm-1 (t)It G [h7-1 : I : hm-1]} - max {xm-2(r)Ir G [h^-1 : I : hm-1]} | +
+ | max {xm-2(t)It G [h?f-1 : I : h^-1 ]} - max {xm-2(r)It G [h1-2 : I : h^-2]} | ^ ^ | xm-1(t) - xm-2(t) | + Mf [| h1 (t,xm-1(t)) - h1 (t,xm-2(t)) | + + |h2 (t,xm-1(t)) - h2 (t,xm-2(t)) |] ^ ^ [1 + Mf (M5(t) + M6(t))] | xm-1(t) - xm-2(t) | , from (15) we easily obtain
111-Y (xm(t) - xm-1 (t)) | ^ | Q-1(t) | M4(t) ■ 111-7(xm-1(t) - xm-2(t)) | +
r (y )
+J |t1-7 G(t, s)| (t - s)a-1 M1(s)|s1-7 (x m-1(s) - x m-2(s))| + M2(s^ M3(
T T
m-2
2(s^ M3(s,0)X
0
X [1 + Mf (M5(0) + M6(0))] | 01-7 (xm-1(0) - xm-2(0)) | d0~ p
ds+
xm-1(ti) - xm-2(ti
|t1-7G(t,ti)| ■ Ni ■ | ti1-7(xm-1(ti)
i=1
Hence, by the aid of the introduced norm we get
II t1-Y (xm(t) - xm-1(t)) ||PC ^
^ fL^ | Q-1(t) | M4(t) II t1-Y(xm-1(t) - xm-2(t)) || +
r(Y) 0^t^T
+ max I |t1-7G(t, s)| (t - s)a-1M1(s) ^ s1-7 (xm-1(s) - xm-2(s)) ^ ds+
T
\a—\ || „1—™m—2c
t G(t, s) (t - s
0<t<T ^ 0
T T
1
I 'G(t, s)| (t - s
0<t<T
+ max / |t1-7G(t, s)| (t - s)a-1M2(sW M3(s,0)x
x [1 + Mf (M5(0) + Ma(0))] || 01-7(xm-1 (0) - xm-2(0)) || d0ds+ p
+ ^ |t1-YG(t,ti)| ■ N ■ || t/-Y(xm-1(ti) - xm-2(ti)) || ^
i=1
^ p ■ || t1-Y(xm-1(t) - xm-2(t))|PC , (16)
where p = S1 + S2 + S3 + S4.
According to the last condition of the theorem p < 1. Therefore, from the estimate (16) we have that the operator J on the right-hand side of the equation (12) is contracting. According to fixed point principle, taking into account estimates (14)-(16), we conclude that the operator J has a unique fixed point in the class PC ([0,T], Rn). The theorem is proved. □
Consequently, nonlocal boundary value problem (1)-(3) has a unique solution x(t) G Rn on (0,T) \ {ti}, i = 1, 2,...,p.
5. Conclusion
On the interval (0,T) \ {tj, i = 1,2,...,p for 0 < a < y ^ 1 is considered the fractional order system of nonlinear integro-differential equations (1) with nonlocal integral condition (2) and nonlinear impulsive condition (3). The problem (1)-(3) is reduced on the interval (0,T) \{tj, i = 1, 2,... ,p to the nonlinear system of functional integral equations (12). The system of functional integral equations (12) has terms of nonlinear functions in integral and non-integral forms. The theorem on existence and uniqueness of the solution of the nonlinear boundary value problem (1)-(3) is proved. In proving the theorem on one-valued solvability of the boundary value problem (1)-(3) is used the method of successive approximations in combination it with the method of compressing mapping.
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Article received 10.07.2022.
Corrections received 16.08.2022.
Челябинский физико-математический журнал. 2022. Т. 7, вып. 3. С. 312-325.
УДК 517.911 DOI: 10.47475/2500-0101-2022-17305
НЕЛИНЕЙНЫЕ СИСТЕМЫ ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ С ИМПУЛЬСНЫМИ ВОЗДЕЙСТВИЯМИ, ДРОБНЫМИ ОПЕРАТОРАМИ ХИЛФЕРА И СМЕШАННЫМИ МАКСИМУМАМИ
Т. К. Юлдашев1", Т. Г. Эргашев26, Т. А. Абдувахобов3с
1 Ташкентский государственный экономический университет, Ташкент, Узбекистан
2 Tашкентский институт инженеров ирригации и механизации сельского хозяйства, Ташкент, Узбекистан
3 Ташкентский государственный аграрный университет, Ташкент, Узбекистан а[email protected], [email protected],
Исследуется нелокальная краевая задача для системы обыкновенных интегро-дифференциальных уравнений с импульсными воздействиями, нелинейными смешанными максимумами и дробным оператором Хилфера. Граничное условие задаётся в нелинейной интегральной форме. Задача сводится к нелинейной системе функциональных интегральных уравнений. Система функциональных интегральных уравнений имеет слагаемые в виде нелинейных функций в интегральной и неинтегральной формах. При доказательстве однозначной разрешимости краевой задачи используется метод последовательных приближений в сочетании с методом сжимающих отображений. Доказаны существование и единственность решения нелинейной краевой задачи.
Ключевые слова: интегро-дифференциальное уравнение с импульсными воздействиями, дробный оператор Хилфера, нелокальное граничное условие, смешанные максимумы, однозначная разрешимость.
Поступила в редакцию 10.07.2022. После переработки 16.08.2022.
Сведения об авторах
Юлдашев Турсун Камалдинович, доктор физико-математических наук, доцент, профессор кафедры общих и точных дисциплин, Ташкентский государственный экономический университет, Ташкент, Узбекистан; e-mail: [email protected]. Эргашев Тухтасин Гуламжанович, доктор физико-математических наук, доцент, профессор кафедры высшей математики, Ташкентский институт инженеров ирригации и механизации сельского хозяйства, Ташкент, Узбекистан; e-mail: [email protected].
Абдувахобов Тохиржон Акбарали огли, преподаватель кафедры высшей математики и информационной технологии, Ташкентский государственный аграрный университет, Ташкент, Узбекистан; e-mail: [email protected].
