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Abdusaidov S. U. Assistant Jizzax Politexnika institute
MURAKKAB TUZILISHDAGI ARALASH MAKSIMUMLI DIFFERENSIAL TENGLAMALAR SISTEMALARI UCHUN
CHEGARAVIY SHART
Annotatsiya: Ushbu maqolada chiziqli bo'lmagan tenglamalar, aralash maksimumli differensial tenglamalar sistemalari uchun chegaraviy shart masalalarini qaraymiz.
Kalit so 'zlar: Differentsial tenglama, chegaraviy shartlar, chegaralangan yopiq to 'plam, Inter jarayoni, segment, boshlang'ich shart.
Abdusaidov S.U. Assistant
Jizzakh Polytechnic Institute.
BOUNDARY CONDITION FOR SYSTEMS OF MIXED MAXIMAL DIFFERENTIAL EQUATIONS OF COMPLEX STRUCTURE
Abstract: In this article, we consider boundary condition problems for systems of nonlinear equations, mixed maximum differential equations. Keywords: Differential equation, boundary conditions, bounded closed set, , Inter process, segment, initial condition.
Ushbu maqolada chiziqli bo'lmagan tenglamalar ko'rib chiqiladi x'(t) = F(t, x(t), max {x(r) | r e [t | j(t, x(t))]}), t e [0; T]
0 <j(t, x) < t T2 =
t; T
t > 0
(1)
(2) va (3)
Chegara sharti bilan x(0) = %
x(T ) = %
Bu yerda x e X œ Rn o'suvchi vektori , X chegaralangan yopiq to'plam, og'ish <j(t, x(t) -qo'shimcha ravishda kerakli funksiyaning o'ziga bo'g'liq x(t).
0 <j(t, x) < T ning T1
0; t
va t <j(t, x) < T ning birinchi qismida
T2
t; T
ning ikkinchi qismida bunday t > 0 shart mavjudligini ko'rib chiqamiz.
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Teorema:
1. o <a(t,x) < t da t e Tl
(4)
2. F (t, x, y) e C (T1 x X x X ) n Bnd ( Mx ) n ( L^y )
(5)
3. a( t, x ) e C (T1 x X )n Lip ( L2x )
(6)
C (Tl x X) sinfda (1) boshlang'ich shartga ko'ra (2) tenglama yagona yechimga ega bo'ladi.
Isbot. Tl segmentda biz ketma-ket yondoshuvlar yordamida (2) shartga ko'ra (1) tenglama yechimini quydagicha tuzamiz
xo(t) = ^o t e Tl
Xm+l (t) = ^o + J F (в, Xm (в), max IXm (r)I r e [а(в, Xm (в)); в]}) de I (V)
o
Jarayonni farqini (V) orqali baholaymiz.
(5) va (7) ga ko'ra, birinchi farq xl (t ) - xo (t ) uchun quydagi baholash o'rinli ||xl(t)-xo(t)|| <Mlt, t eT(l) (S)
(5) ga ko'ra x2 (t ) - xl (t ) farq quydagicha
t
\\x2(t ) - Xl(t )|| < Ll J (I Xl(e) - ^(в)! +
o
max I x (r )I r e [в I а(в; x (в))]}|| -1| max Ix (r )I r e [в I а(в I x (в))]}|| )de
(9)
(9) ning o'ng tomonidagi ikkinchi qismini quydagicha yozamiz max I Xl (r ) I r e [в I а(в; x (в))]} - max I Xo (r ) I r e [в I g(0; Xo(e)) ]} =
max Ix (r)I r e [в I а(в; xl (в))]} - max Ix0 (r)I r e [в I а(в; x0 (в))]} +
+ max Ixo (r ) I r e [в I а(в; Xo (в))]} - max I Xo (r ) I r e [в I а(в; Xo (в)) ]}
(10)
(4) va (S) ni hisobga olgan holda tenglikning birinchi farqi (lo) uchun quydagi baholash o'rinli
|max Ixl(r) I r e [в I сг(в; ^(в))]}- max IXo(r) I r e [в I сг(в; ^(в))]} <
< I|max I(x (r) - X (r)) I r e [в I ст(в; x (в))]} < Mxt, t e T1 (V) ga binoan , tenglikning ikkinchi farqi uchun (lo) olamiz
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l4
max {ф )| r g[01 ¿(0; x (0))]} - max {x0(r )| r g[0 | ¿(0; x0 (0)) ]}|| = 0
U holda (9) tengsizlik quydagicha
t л
|x2 (t) - x (t)|| < l J 2miede = —, t g t 1 (п)
0 2
Endi (6) va (11) ga ko'ra xi(t ) - x0 (t ) farqiga kelsak,
t
HXaCt) - x2(t)|| < Li J(||x2(0) - xi(0)|| +
0
max {x (r)| r g[0 | ¿7(0; x (0))]} - max {x0 (r)| r g[0 | ¿7(0; x0 (0))]} + + max {x (r)| r g[0 | ¿(0; x0 (0))]} - max {x0 (r)| r g[0 | ¿(0; x0 (0))]} d0<
t
< Li J( 2 + L2 Mi )|| x2{0) - xi(0)|| d0<
3
< 2limi [li (2 + l2mi )] l t g t1
Xuddi shunday davom etmoqda ushbu jarayon usuli to'liq xm+i(t) - xm (t) farq uchun matematik induksiyani quydagicha olamiz
>m+i
||xm+i(t) - xm (t)|| < 2LiMi [Li (2 + L2Mi )]m-i J-^ t g T1 (u)
(i2) dan kelib chiqadiki, {xm(t)} ketma-ketlik t bo'yicha T*segmentda tekis yaqinlashadi. Shuning uchun (1) tenglamaning sistemasi (2) boshlang'ich shart bilan Ti segmentda x(t) yechimga yaqinlashadi, bunda x g C1(T1; X) .
Keling , ushbu yechimning yagona ekanligini ko'rsataylik. C1(T1; X )sinfidagi (i) Sistema ham xuddi shunday (2) boshlang'ich shartga ega bo'lgan boshqa z(t) yechimga ega bo'lsin.
z (t ) ni va x0(t ), xi(t ), x2(t ),.....yaqinlashishlarini solishtiramiz.
(5) ga binoan quydagicha baholaymiz
t
\\z(t ) - xl(t )|| < Li J| |z (0) - x(0)|| +
0
+
max {z(r) | r g[0 | ¿(0; z(0))]}- max {x0 (r) | r g[0 | ¿(0; x0 (0))]}
2
2!
t
< 2LM J0d0 = 2ЦЩ —
<
0
(6) va keying farq uchun quydagi tengsizlikni olamiz
0
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i5
t
\\z (t ) - *2(t )|| < L J|| z (0) - x^W +
0
+ ||max {z(r) | r e [01 a(0; z(0))]} - max {x (r)| r e [01 <r{0; x (0))]}|| +
+ max {xi(r) | r e [01 a(0;Xj(0))]}-max {xj(r) | r e [01 a(0;Xj(0))]}d0 <
t
< L J(2 + ¿2M )||z(0) -Xj(0)||d0 <
t
¿M mz(0)-x
0
t
r t
< 2LXMX J(2 + L2Mx)-, t e T1
0
Xuddi shunday
,4
\\z(t ) - X3 (t )|| < 2LjMj [ Lj ( 2 + L2 Mj )]4 -Va hokozo
Ushbu bahodan ko'rishimiz mumkinki m ^œ da ||z(t) -xm(t)|| ^ œ T x segmentda t bo'yicha tekis yaqinlashadi.
Shundan kelib chiqqan holda (1) tenglama yechimi (2) boshlang'ich shart bilan {xm(t)} ketma-ketlikka ko'ra c1 (t X) sinfda yagona yechimga ega
ADABIYOTLAR:
1. Sidorov N., Sidorov D., Dreglea A., Solvability and bifurcation of solutions of nonlinear equations with Fredholm operator. Symmetry. 2020. 12 (6), ID 920, 121.
2. Rojas E.M., Sidorov N.A., Sinitsyn A.V, A boundary value problem for noninsulated magnetic regime in a vacuum diode. Symmetry. 2020. 12 (4), ID 617 14 pp.
3. . Assanova A. T., Bakirova E. A. and Kadirbayeva Z. M. Numerical solution to a control problem for integro-differential equations, Comput. Math. and Math. Phys., 2020. Vol. 60. No. 2. P. 203-221
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