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Список литературы /References
1. Баранова Е.К., Бабаш А.В. Информационная безопасность и защита информации. М.: РИОР ИНФРА-М, 2019. 335 с.
2. Данко П.Е., Попов А.Г. Высшая математика в упражнениях и задачах. Часть 1. М.: Высшая школа, 1974. 415 c.
3. Грешилов А.А. Математические методы принятия решений. М.: МГТУ им. Баумана, 2006. 583 с.
4. Бодров В.И., Лазарева Т.Я., Мартемьянов Ю.Ф. Математические методы принятия решений. Тамбов, изд-во ТГТУ, 2004. 124 с.
ON THE SOLVABILITY OF BOUNDARY VALUE PROBLEM WITH AN INTEGRAL CONDITION Tankeyeva A.K. Email: Tankeyeva690@scientifictext.ru
Tankeyeva Aigerim Kieyvna - Master of Science, DEPARTMENT OF NATURAL SCIENCES, REPUBLIC STATE INSTITUTION MILITARY INSTITUTE OF AIR DEFENSE FORCES NAMED AFTER TWICE HERO OF THE SOVIET UNION T. YA. BEGELDINOV, AKTOBE, REPUBLIC OF KAZAKHSTAN
Abstract: there is investigated boundary value problem with an integral condition for the system of partial differential equation. For solution considered problem we use a method of parameterization. The signs of the unique and correct solvability of a nonlocal boundary value problem for a system of hyperbolic equations with a mixed derivative are established in terms of a special matrix compiled from the original matrices of the equation and the matrices of the boundary condition. The hyperbolic system of equations reduces to the linear problem of a family of ordinary differential equations. Coefficient sufficient conditions for the existence of a unique solution to the problem are solvability conditions for the equivalent nonlocal boundary value problem with the integral condition. Considering an example of the unique solvability of a nonlocal boundary value problem with an integral condition and verify the conditions of the theorem using the MatLAB software environment. Keywords: solvability, nonlocal condition, integral condition, hyperbolic systems of equation, Friedrichs, only decision, MatLAB, parameterization method.
О РАЗРЕШИМОСТИ КРАЕВОЙ ЗАДАЧИ С ИНТЕГРАЛЬНЫМ
УСЛОВИЕМ Танкеева А.К.
Танкеева Айгерим Киевна - магистр естественных наук, кафедра естественнонаучных дисциплин, Военный институт Сил воздушной обороны им. дважды Героя Советского Союза Т.Я. Бегельдинова, г. Актобе, Республика Казахстан
Аннотация: исследуется краевая задача с интегральным условием для системы дифференциальных уравнений в частных производных. Для решения рассматриваемой задачи используется метод параметризации. Установлены признаки однозначной и корректной разрешимости нелокальной краевой задачи для системы гиперболических уравнений со смешанной производной в терминах
специальной матрицы, составленной по исходным матрицам уравнения и матрицам граничного условия. Гиперболическая система уравнений сводится к линейной задаче семейства обыкновенных дифференциальных уравнений. Коэффициентные достаточные условия существования единственного решения задачи являются условиями разрешимости эквивалентной нелокальной краевой задачи с интегральным условием. Рассматривается пример на однозначную разрешимость нелокальной краевой задачи с интегральным условием и проверяется условие теоремы с использованием программной среды MatLAB.
Ключевые слова: разрешимость, нелокальное условие, интегральное условие, гиперболические системы уравнения, Фридрихс, единственное решени, MatLAB, метод параметризации.
УДК 3054
Nonlocal boundary value problems with integral conditions for some classes of partial differential equations have been studied by many authors, note [1,2], where you can find a review on the theory of boundary value problems with nonlocal constraints for partial differential equations and a bibliography on these problems.
On the strip П = {(x, t): t < x < t + c,0 < t < T}, T > 0, с > 0 we consider a boundary value problem with an integral condition for a system of partial differential equations
Du = A(x, t)u + f (x, t), u e Rn, (1)
J at(.
, s)u(x, s )ds = d (x), (2)
j x, s u(x, s as = d (x
0
a a
where D =--1--; (n x n ) -matrices A(x, t), K(x, t ) and n -vector-function
at a* ' v ' v '
f (x, t) are continuous in x and t on Q ; n - vector-function d(x) is continuous on [0,©].
We denote by C(q, Ä") the space of function u is continuous in X and t :
( \ n
x, t)I; ||A|| = max IIA(x, t)I = max max V la,.. (x, t)
■■ (x,t )=Q" ...... (!,t>Q V 711 MeQ i=\Tn j^l
The purpose of this paper is to establish sufficient coefficient solvability conditions in the broad sense of a nonlocal boundary value problem with integral condition (1) - (2) for a system of partial differential equations.
The function u(x, t) continuous on Q is called the solution of the boundary value
problem with the integral condition for the system of partial differential equations (1) under
condition (2) in the broad sense according to Friedrichs [3], if the function u(x, t) is
continuously differentiable with respect to the variable ^ along the characteristic, satisfies the family of equations and the integral condition (2 ).
A nonlocal boundary value problem with the integral condition (1) - (2) is called
uniquely solvable in the broad sense, if for any f (x, t) e c(q, Rn),
d(x) e C([0,©] Rn) it has a unique solution u(x, t) e C(Q, Rn) that is continuously differentiable with respect to the variable t along the characteristic.
By
c(h , Rn ) we denote the space of functions u : H ^ Rn with norm
||u|| = max max||u(E,t)| continuous in E and т .
11 110 ie[0,®]T6[0,T f ™ ^
We reduce problem (1) - (2) for a hyperbolic system of equations in the region H = {(E, t) : 0 < E < o>,0 < т < T}, T > 0, c> 0 to the linear problem of a family of ordinary differential equations:
= A(E,T)~(E,T)+/(E,T), te[0,T], (3)
от
T
J K (e,t)~(e,t)t = d(E), E^M, (4)
0
where ~(E,t) = u(E + T,t), A(E,t) = A(E + t,t) , ~(E,t) = K (E + t,t) ,
f (E, t) = f (E + T, t) (n x n)- matrices a(e, t) , k(e, t) and n -vector-function
f (x,t) continuous in E and т onH ; n -vector-function d(E~)- continuous on [0,c].
To find a solution to a family of boundary value problems with the integral condition (3) - (4), the parameterization method [5] is used: take a step h > 0: Nh = T and split
__N
H = [0,c] U U [(r - l)h, rh).
r=1
Denoting by ur (E, t) , r = 1, N the restriction of the function u(E,t) on [0, c]x[(r -l)h,rh), taking Ar(E) as the value of the function ur(E,t) for т = (r — l)h, r = 1, N, producing on every area of breaking up замену ur (E,t) = ur (E,z) — Xr (E) we obtain a boundary value problem with functional parameters \(E\ In the problem with functional parameters, the initial conditions appeared. Solving a special Koshi problem of a family of boundary value problems, we determine ur (E, t) from the integral equations:
т t T
ûr (E,t) = jA(E,r)ûr (E,r)dT+ jA(E,r)Ar {E)dT+ j f (E,r)dT . (5)
(r-l)h (r-l)h (r-l)h
Substituting ur (e, t) into the right-hand side of (5) repeating this process V times, passing to the limit as t ^ rh - 0, r = 1, N, E G [0,T], we find lim ur (E,t) .
T^rh-0
Using them in the boundary conditions and in the conditions for gluing the solutions in the internal lines of the partition, multiplying both sides by h > 0, we have a system of equations with respect to the parameters Âri (e) ( r = 1, N, i = 1, n) :
a (E, h)A{E) = -Fv (E, h) - Gv (E, h, u ).
"0 0 0 ... 0 kVn(e,T# + L'N(E,h)y
I + Lv1 (E,h) -I 0 ... 0 0
Qv(E,h)= 0 I + Lv2 (E,h) -1 ... 0 0
0 0 ... I + L,n-i (E, h) ......... -1
Theorem 1. Suppose that for some h > 0 : Nh = T and v, V = 1,2,..., (nN X nN) -the matrix Qv (£, h) is invertible for all ^ £ [0, c\ and the following inequalities hold:
llfevfe,h)\-1|| <7v(h),
^ -1 -a&h -...-M^
h)=/v(h)max
v!
rh
x |maxK(^,r)dT<a< 1,
(r-l)h
^ ' o = const..
«Ml
where a(f)= maxll
v 7 tg[0,T ]ll
Theorem 2. Let the conditions of Theorem 1 be satisfied. Then the nonlocal boundary value problem with the integral condition (1)-(2) has a unique solution
u*(*, t)g C(Q, Rn ).
Example. On the strip H = {(£, t) :0 <%<a>,0 <T< T}, T > 0, consider a family of ordinary differential equations
^ = Â(Ç,T)U(Ç,T)+J(Ç,T), T G [o,T], (1)
с > 0 we
дт
1
J = d(g), £е[0,с],
(2)
Г 2 0 0 > Г 3 0 01
where и (£, т) = и(£ + т, т), A = 0 2 0 , к = 0 2 0
v 0 0 2 у v 0 0 1,
-(n x n) -
matrices and n -vector-function f (£,r) = f + t,t) = 0; n - vector-function d(£) = 0 is continuous on [0, co].
Task. Find Q3 (£, h) , Q— (£, h) and evaluate this matrix.
Decision. Using the initial data, we construct the matrix Qv (£, h) for v = 3, h = 0.3:
h) =
0 0 0 0 0 0 0.27 0 0
0 0 0 0 0 0 0 0.18 0
0 0 0 0 0 0 0 0 0.37
1.6 -1 0 0 0 0 0 0 0
0 1 -1 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0
0 0 0 1.96 -1 0 0 0 0
0 0 0 0 1 -1 0 0 0
0 0 0 0 0 1 -1 0 0
Let us show the feasibility of the conditions of the theorem using the MatLAB 7.10.0 software environment. We introduce the input data of the problem under consideration, as
well as the necessary formulas. We calculate Qv (£, h) and qv (£, h) for v = 3.
£". it- So s<ll T«W I.1", ."I PJ.IIIH hilltop H*lp
: Q d * 4 * *ï <- » rib « C^F^te CAlh^i^r^^n^NWTLM - 0 ¿Г
: D t3 У Л Ч Ш t Г" ' fcl-lil«Vat|!wt -м /*
"8 CE - 10 ♦ I - 1.1 «Sйй О.
1 Q=[0000000.27 00; 00000000.18 0; 00000000 0.3726; 1.6 -1000000
2 01-1000000; 001-100000; 0 0 0 1.Э6 -10 0 0 0; 00 0 0 1-10 0 0;
3 000001 -1 0 0]
4 Z=inv (Q) -
5 y=norm(Z, 1) -
€ A=[2 0 0; 0 2 0; 0 0 2] -
7 8 a—norm(A,1) h=0. 3 L
9 v=3 1-
10 q=y*l*(exp(a*h)-l-a*h-(a*h)A(v-1)/factorial(v-1)-(a*h)Av/factorial{v> > -
11
Fig. 1. Task input data
E't Edrt CXbug El«W Btl40p a "J cî à 4 » -9 С Arf Й Te", Jn : ''.L .jr- tr ; IATUXB -la as
z -
1 1010 0 0 0.6250 0.6250 0.6250 0 3189 0 3189 0 3109
1 8896 Q 0 0 1.0000 1.0000 0 5102 0 5102 0 5102
1 8896 0 0 0 0 1.0000 0 5105 0 5102 0 5105
1 8896 0 0 0 0 0 0 5102 0 5102 0 5102
3 7037 0 0 0 0 0 0 1 0000 1 0000
3 70Э7 a 0 0 0 0 0 0 1 0000
3 7037 0 0 0 0 0 0 0 0
0 5 5556 0 0 0 0 0 0 0
0 0 5.6838 0 0 0 0 0 0
У »
/* 17 9611 lm -— «II.
Fig. 2. Finding the inverse matrix
Using the software environment, an assessment is established 03х(£, h)|| </3(h),
where y3 (h) = 17.9611. Next, we verify the fulfillment of condition of theorem 1.
Based on the calculation results in the MatLAB software environment qv (g, h) < 0.1099. Thus qv (g, h) < 0.1099 < S < 1.
References / Список литературы
1. Naxushev A.M. The bias problem for partial differential equations. M.:Nauka, 2006. 287 p.
2. Rozhdestvenskii B.L., Yanenko N.N. Systems of quasilinear equations and their applications to gas dynamics. M.: Nauka, 1968. 592 p.
3. Abdikalikova G.A. Correct solvability of a nonlocal boundary value problem. // Bulletin of the Orenburg State University, 2007. № 10 (74). P. 162-165.
4. Dzhumabaev D.S. Signs of unique solvability of a linear boundary problems for an ordinary differential equation. // Computing Journal mathematics and mathematical physics, 1989. T. 29. № 1. P. 50-66.
