On evolution of stationary processes near the origins of excitation

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ON EVOLUTION OF STATIONARY PROCESSES NEAR THE ORIGINS OF EXCITATION
'-Sergiyenko L.S., 2Nesmeyanov A.A.
1National research Irkutsk state technical university, Irkutsk-
2East-siberian institute of Ministry of Internal Affairs of Russia, Irkutsk, e-mail: lusia_ss@mail. ru
In three-dimensional Euclidean space we study elliptic system of four equations in quotient derivatives of the fourth order with four unknowns that parabolically degenerates on the coordinate axis. In certain classes of function smoothness an existence of the unique (with precision up to random constant) limited solution of the system that meet the fixed boundary terms in cylinder, symmetric to the line of degeneration, is proved. The proof is carried out through the reduction of the results of differentiation and integration of the system equations and dividing variables according to classic algorithm of Fourier. While calculating coefficients of the line that represent the solution of common linear differential equation of the second order with a special point in the centre of the research area that is a consequence of the studied system, special multinomials of so-called triangle form were built. The results can be used while modeling stationary processes in asymmetric solenoid speed field.
Keywords: stationary processes, excitation
Let the stationary process be presented as a linear operator equation
I rotT + B grads = 0, div T = 0, (1)
where components u, v, w of the vector T, and scalar function s are dependent variables of arguments x, y, z, I and B are given matrixes
1 0 0 1 0 0
0 1 0, B = 0 1 0
0 0 1 0 0 /
/= (x 2 + y2)'-
2 i /"2. •
With positive values of parameters l, k, r [1−3, 6].
As we approach the origin of the excitation, the studied fixed process starts to alter its structure — elliptic system (1) parabolically degenerates in the multiplicity x2 + y2 = 0, and for it common classic setting of problems of modern mathematical physics become incorrect [7].
The first boundary problem.
Let us study the behavior of the system (1)
u + v + w = 0, w — u — s = 0,
x y z 7 z z y 3
v — u + f (x, y) s = 0, s — v + w = 0.
z y s? s s z 7 x z y
Near the degeneration line x = y = 0 that is contained in cylinder
D = {(x, y): x2 + y2 & lt- R2, 0 & lt- z & lt- z0}
with side surface r, upper ^ and bottom r bases. Let us define D0 as a part of axis OZ, that lies in D.
Problem 1.
Find all conditions of existence and uniqueness in area D of the limited on the multiplicity of degeneration D solution of the system (1).
While l & gt- 0, k = r = 0, conditions of the correctness of the problem are defined in [4]. In the presented work under terms
vlr0=h' WL=S,
Hrour! & quot- 0'Ir = Q' (2)
where h e C0HX (ar 0) — ge C1+a (3Z& gt-) —
& lt-?eC2® — & lt-,=0-The following is proved.
Theorem 1. While l = k = 1 u r = R in classes of the function smoothness
(m, v) e C1 (p/ D0) n C (p/ D0),
wsC2(D)nC (p),
seC2(D/D0)nC (D/D0)
exists a limited near the degeneration axis solution (u, v, w, s) of the problem (1)-(2), where u and v are defined with a precision up to random constant summand, and a, w, and s — in a single way.
Through a reduction of the results of differentiation and integration of the equations (1) according to the corresponding variables we obtain a system:
+Ax y) szz = °,
Aw = 0, s + s
xx yy
z
u (x, y, z)= /[w, (x, y, %)-sy (x, y, ?)] d$ + J (y, z) —
0
z
u (x, y, z)= I[wy (x, y, 5)+ sx (x, y, ?)] + Q (y, z).
Functions J (y, z) and Q (y, z) are defined
Under boundary terms b (0) = b (z0) = 0 h
7 / / 7 / J n n 0
precisely up to the random constant summand X = nnz/z n = 1, 2, 3, …, from the first equa-
from the system
Q,+J& gt-=-Sr--
QI-J, = f (x, y) lims,
tion (6) we find
K{z)= sin
nnz
(7)
Then, from the second equation (6) in polar The main point in the proof of the theorem coordinates (9, p) we define a solution of the
is:
Lemma 1. Boundary problem
view
2, 2
x + v
s" + H-^=0- (3) We obtain the system
a = O (9)^ (p).
n n'-T/ mMZ
XX yy
s
R2+x2+y2
r0ur! «5lr «
«ec2®, ?|ir=o.
(4)
px+pf: —
Has a unique solution in the cylinder D that is limited while (x2 + y2) ^ 0.
i?2+p2+Y& quot-
*.= 0. (8)
From the first equation (8) while y = m2,
The proof is carried out according to the m, 0, 1, 2-^ single periodic solution in shape
classic algorithm of Fourier where at first vari- harmomc superp°siti°ns
ables z and (x, y) are divided. The solution is O (9) = A cos (mffl) + B sin (mffl). (9)
built as line nm nm nm
*(*. 7» z) = EMzK (*& gt- y) —
We obtain
n=0
2. 2
Aa — ««.= 0.
«R2+x2 + y
2 n
With each focused n the second equation of the system (8) always has an integral as a line
(5) (00
'-J/m"(p)=pm I + XY^P4^'- ¦ (10)
V '=0)
That is absolutely and equally met in circle |p| & lt- R under whole values of parameter m.
(6) To calculate coefficients of the degrees of the sedate line (10) let us build recurrent formulas
y4+2, =(-!)'-77:
4(i + 2)(i + 2 + m)
a,-2
i+2
j=0
ii
v2y
where
and while i & gt- 2
nn
2i+ 3+(-l)
^ _ ' a& lt- _ ^ '00 =10 = ^
1−2-2J T, 1, j
i'-» («) — 1 i in
i n-0 (2+ 2-n + T,+1_,) (2 + 2x + V-,)'-'-
The solution of the equation (3) in cylindric coordinates will look as:
s (p,& lt-p, 0=? IZmnsin~VAmncosH0+ Bmn sin (mq& gt-)]|.
mfi=0 [ Tmn R) ZQ J
Degeneration into trigonometric line of the given function on the surface T function ?(& lt-P^)=? sin-[Amn cos (& gt-mp)+ sin (/n& lt-p)].
m, n — 0
0
(11)
(12)
(13)
(14)
Let us define the values of Fourier coefficients:
A ^ f / • IffltZ 1
A^m =-- Jtf (cp, z) sm----------------------t/r, m = 0,1,2,
7E/k0 r Z0
Am =
2 r / wnz /
J #(& lt-p, z) sin----cos (/n (p)Jr, n = 1,2,3,… ,
nRz,
0 r
Bftnt n
7li? z0 r
2 sin m7tZ sin (mcp)jr, n, m = 0,1,2,3,…
(16)
The convergence of the line (15) is proved with a principle of maximum for elliptic equations.
'- (-2−2/
Multiple polynomials
To boost the process of calculating coefficients of Pj (m) line (10) let us introduce auxiliary functions [5]
Qi, i — 2Tl=o 2T^o-& quot-2x|+1=o FU (2 + 2r (+ t/+1
(17)
That represent simplified modification of mands of function Q t with similar values of
the multiple polynomials (13). / can be written as block matrixes of triangle
Then for whole nonnegative values / & gt- 1, shape. For example, while / = 1, i = 5, 6, 7 h i & gt- 2 + 2/, Ti sequence of multipliers into sum- / = 2, i = 6, 7, 8 we have summs
Q5 j = 2−4 + 2−5 + 3−5, Q61 = 2−4 + 2−5 + 3−5 + 2−6 + 3−6 + 4−6,
Q71 = 2−4 + 2−5 + 3−5 + 2−6 + 3−6 + 4−6 + 2−7 + 3−7 + 4−7 + 5−7,
Q72 = 2−4-6 + 2−4-7 + 2−5-7 + 3−5-7,
Q82 = 2−4-6 + 2−4-7 + 2−5-7 + 3−5-7 + 2−4-8 + 2−5-8 + 3−5-8 + 2−6-8 + 3−6-8 + 4−6-8,
to which will correspond the matrixes
05,1
r2-A 2−5^
0 3−5
' 06, i
07,1
V /
r2-A 2- 5 2−6 2-V
0 3- 5 3−6 3−7
0 0 4−6 4−7
0 0 0 5−7
— 07,2
•4 2−5: 2−6
0 3−5 i 3−6 5
0 0 4−6
r2- 4−6 2−4-7: 3−5 ¦7& gt-
V 0 2−5-7 i 0 & gt-
& quot-2−4-6 2−4-7 2−4-8 3−5-7 3−5-8| 4−6-8& quot-
08,2 0 2- 5−7 2−5-8 0 3−6-8'- 0 1
, o 0 2−6-8 0 o o
In connection with this characteristic functions Pn (n) are called mu/tip/e mu/tinomina/s of triang/e form.
Resume
While modeling physical processes in extreme conditions the most basic and difficult stage is the correct setting of an objective. In this work we have obtained the terms that provide for existence and uniqueness of the solution of the first boundary for degenerating on the line elliptic equations. During the problem research a special function class has been built and called multiple polynomials of triangle form.
References
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