ON A BOUNDARY VALUE PROBLEM FOR AN ODD-ORDER EQUATION WITH MULTIPLE CHARACTERISTICS Текст научной статьи по специальности «Математика»

Vestnik KRAUNC. Fiz.-Mat. nauki. 2022. vol. 38. no. 1. P. 28-39. ISSN 2079-6641

MSC 35G30, 35Q53

Research Article

On a boundary value problem for an odd-order equation with

multiple characteristics

O. T. Kurbanov

Tashkent State Economic University, Uzbekistan, 100003, Tashkent, Chilanzar district, Ave. Islam Karimov, 49 E-mail: odil69@inbox.ru

A nonlinear boundary value problem for a third-order nonlinear equation with multiple characteristics is studied in the article in a curvilinear domain. The unique solvability of the problem is proved. The uniqueness of the solution of the boundary value problem is proved by the energy integral method using some elementary inequalities. An auxiliary problem is considered for the existence of a solution, for which the Green function is constructed. By solving an auxiliary problem, the original problem is reduced to a system of Hammerstein integral equations. The solvability of a nonlinear system is proved by the contracting mapping method.

Key words: nonlinearity, uniqueness, existence, system of Hammerstein equations.

For citation. Kyp6aHOB O. T. On a boundary value problem for an odd order equation with multiplec haracteristics. Vestnik KRAUNC. Fiz.-mat. nauki. 2022,38: 1,28-39. d DOI: 10.26117/2079-6641-2022-38-1-28-39

The content is published under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/deed.ru)

© Kurbanov O.T., 2022

Introduction

The following equation refers to poorly studied third-order equations:

called an equation with multiple characteristics [4].

Equation (M) is often used in various problems of physics and mechanics, which are of great theoretical and applied importance. This equation contains the well-known Korteweg de Vries equation (KdV)

Funding. The study was carried out without financial support from foundations.

d DOI: 10.26117/2079-6641-2022-38-1-28-39 Original article submitted: 09.02.2022

Revision submitted: 16.03.2022

Uxxx - Uy = f (x, y, u(x,y ) ,Ux (x,y ), Uxx (x,y )),

(M)

uy + uux + |3uxxx = 0 (KdV)

which is the object of research by many authors and occupies an important place in the nonlinear wave propagation in weakly dispersive media [1, 2, 3].

The Korteweg de Vries equation (KdV) describes the evolution of weakly nonlinear long-wave excitations in a medium with dispersion in the high-frequency region. The KdV equation arises in the study of many physical systems, such as gravitational waves in shallow water, ion-acoustic waves in plasma, Rossby waves in a rotating fluid, waves in electrical circuits containing nonlinear elements, etc. [1].

Some characteristic features of wave propagation in dispersive media can be traced already in the linear approximation [2]

uy + |3uxxx = 0. (LKdV)

The (LKdV ) equation describes sufficiently long waves in media where limit Y (phase velocity) as k —> 0 has a finite value (weakly dispersive waves). The (LKdV ) equation is called the linearized Korteweg de Vries equation [2,3].

The study of boundary value problems is also relevant for an odd-order equation with multiple characteristics. Some linear boundary value problems for a linear equation with multiple characteristics of the third order were studied in [4, 5]. In [5], a linear boundary value problem for a nonlinear equation with multiple third-order characteristics was studied by the method of successive approximations. A nonlinear boundary value problem for a linear equation with multiple third-order characteristics was studied in [6]. In [7], a problem was studied for a nonlinear equation with multiple third-order characteristics with nonlinear boundary conditions. In this article, we study a nonlinear boundary value problem for a nonlinear equation with multiple characteristics in a curvilinear domain.

Formulation of the problem

Problem A. It is required to determine function u(x,y) in domain D = {(x,y);h/| (y) < x < h.2(y),0 < y < 1} the function has the following properties:

1)u(x,y) g C3'1 (D) n C2'°(x = h2(y),0<y < 1) n C1'0(D)

2)which is a regular solution to the following equation

L(u) = uxxx - uy = f (x,y,u(x,y )), (1)

in domain D;

3)satisfying the following conditions

u(x,0) = u0(x),h1 (0) < x < h2(0), (2)

u(hi (y),y) = cp(y),0 < y < 1, 29

ux(hi(y),y)= ^(y),0 < y < I, (4)

Uxx(h2(y),y) + aux(h2(y),y) = g(u(h2(y),y),y),0 <y < I, (5)

as well as natural matching conditions at angular points: u0(h1(0)) = ^(0),u/0(h1 (0))= 4(0), u//0(h2(0)) + au/0(h2(0)) = g(u(h (0),0),0).

Uniqueness of the solution

The following theorem holds

Theorem. Let hr(y ) G C1 [0,1],r = 1,2 and g(u(x,y ),y)'f(x,y'u(x,y)) be continuous functions of their arguments 0 < y < 1 for any |u| < oo, that satisfies the following conditions

Ig(u1,y)- g(u2,y)| < l(y)|u1 -u21, (6)

|f(x,y,ui)-f(x,y,U2)| < L(x,y)|ui -U2l, (7)

l(y) + h/2(y)+ a2 < 0, (8)

a3 - ß + L(x,y ) < 0. (9)

Then the solution of problem (1) - (5) is unique.

Proof. The uniqueness of the solution to the problem is proved by the energy integrals method, using some elementary inequalities. Let there be two solutions of the considered problem ui and u. Consider their difference w = ui — u2. Forw, we get the following problem:

L(w) = wxxx — wy = f(x,y,ui(x,y)) — f(x,y,u2(x,y)), (1o)

w(x,0)= 0,h1 (0) < x < h2(0), (20)

w(hi(y),y)= 0,0 < y < 1, (30)

w

:(hi(y),y) = 0,0 <y < I,

(40)

uxx(h2(y),y) + aux(h.2(y),y) = g(ui (h2(y),y),y) - g(u2(h2(y),y),y),0 < y < 1. (5o)

Integrating the identity

v^L(^) = - ^y) = v^(f(x,y,ui(x,y))- f(x,y,U2(x,y))) (10)

over domain D, where v = exp(-ax — (3y),a > 0, (3 > 0, with boundary conditions (2o) - (50), we have

1

[g(ui (x,y),y) -g(u2(x,y),y)]v^|x=h2(y)dy -2 V^Jx=h2(y)dy+

0

1

+2

1

3a

[a + h/2(y)]vw lx=h2(y) dy - -y

2

v^xdxdy +

0

1

+2

D

21

[Vy - Vxxx]^ dxdy --

h2(y)

v^ |y=i dx =

D

hi (y)

[f(x,y,ui (x,y))- f(x,y,U2(x,y))]dxdy.

(11)

D

We introduce the following notation:

h2(y)

1 = 2

1

V^x|x=h2(y)dy

2

2 3a

vw |y=i dx + —

2

v^xdxdy > 0.

(12)

hi (y)

D

With conditions (6) - (7) from (13), we have

1 s 2

[l(y) + a + h/2(y)]vw lx=h2 (y)dy + -

[a3 - ß - L(x,y)]Wdxdy. (13)

D

Taking into account the condition of the theorem, the expressions in brackets on the right side of inequality (13) are nonpositive: I < 0. With (12), 0 < I < 0, whence it follows that I = 0.

Then from (12) we obtain the following conditions:

^(h.2(y),y) = 0, w(x,1)= 0,

^x(x,y) = 0.

Hence we have ^(x,y) = p(y), (x,y) e D.

Since ^(h2(y),y) = 0,0 < y < 1, then p(y) = 0.

Due to the continuity of ^(x,y) in D, we have ^(x,y) = 0.

Existence of solution

Before proceeding to the proof of the problem of solution existence, it is necessary to study the following auxiliary problem.

Problem B. It is required to determine in domain D a regular solution

u(x,y ) G C3,i(D) n C2,0(x = h2(y),0<y < i) n C',0(D)

-2,0

i,0/

of equation

L(u) = uxxx - uy = f (x,y ),

satisfying the following conditions

(ii)

u(x,0) = u0(x),hi(0) < x < h2(0),

(2)

u(hi (y),y) = p(y),0 < y < i,

ux(hi(y),y)= ^(y),0 < y < i,

uxx(h2(y),y)= 4i(y),0 < y < 1 The solution to this problem has the form:

(3)

(4) (5i)

u(x,y) = -n

ii

n

n

i

+-n

h2(0)

Gu|n=0d^ - -

n

G(x,y; ^,-n)f(^,n)d^dn.

(i4)

hi(0)

D

here G(x,y;^,n) = U(x,y;^,n)- W(x,y;^,n), where W(x,y;^,n) - be any regular solution of equation

+ ©n = 0,

U(x,y; is the fundamental solution of equation [4]

— CD-p = 0

i x - ^ U(x,y;= 7-ttttfh-tttt),y >n,x =

(y -n)i/3 (y -n)i/3

V (x,y; =

—!t/t),y >n,x = ^

(y - n)i/ (y - n)i/3

i

where

f(t) =

cos(A3 - At)dA,

^(t) =

(exp (-A3 - At) + sin (A3 - At)) dA, t = x-E

(y - n),/3'

Functions f(t) and cp(t), called the Airy functions, satisfy the following equation

z//(t) + 3 z(t) = 0.

<x> 0

, ^ 2n

f(t) = T'

0

r ^ n

f(t) = 3'

f(t)= n,

^(t) = 0.

The following estimates hold for the functions U(x,y; V(x,y; £,,n)

lU(x,y;E,n)l <

K

(y - n)i/3'

,9i+jU(x,y;E,n), ^ |x- El

I--A i-A -1 < Ci

9xi9yj

2i±6j-1 Iy -nl 4

,9i+jV(x,y;E,n), ^ lx-El2^

I--A i-A -1 < C2

for x-E

(y-n)1/3

9xi9yj

-» +oo,i + j > i'Ci > 0,C2 > 0,

|9i+jU(x,y; E,n) |<

2i±6j-1 ly - nl 4

C3

9xi9yj

i±3j±1

ly - nl 4

C |x-£|2 C4-T

e y-ni2

for

x-E

(y-n)1/3

—oo,i + j > i,C3 > 0,C4 > 0.

Note that the same estimates are valid for function G(x,y;E,n) as for function U(x,y; E,n).

Function G(x,y;E,n) we call the Green function of problem B. We now turn to problem A.

Theorem. Let, along with the conditions of the uniqueness, the following conditions be satisfied

uo(x) e [Hi(0),H2(0)],^(y) e C2[0,1],p(0) = g C1 [0,1],

9f(x,y,u) e C(D),f(x,0,u(x,0)) = 0 3y

and y G [0, i ] for any |u| < oo, the inequalities hold

|g(u,y)| < Mi,|gy(u,y)| < M2,|gu(u,y)| < M3,

And for (x,y) e D and |u| < 00

|f(x,y,u)| < M4,|fx(x,y,u)| < M5,|fy(x,y,u)| < M5,|fu(x,y,u)| < M7.

Then the solution to problem (1) - (5) exists.

Proof. According to (14), we find the solution to problem A in the following form

u(x,y) = -n

G(x,y; h2(n),n)g(u(h2(n),n))dn -n

G(x,y; H2(n),n)aux(H2 (n),n)dn-

[G^ + h/i (y) G]u|^=hl (n) dn + n (n) dn+

1

+-n

0

h2(0)

Gu|n=0d^--

1 n

G(x,y; ^,n)f(^,n,u(^,n))d^dn.

(15)

h1 (0)

D

Let

u(h2(n),n) = T(n),ux(h2(n),n) = v(n).

Then, from (16) we have

u(x,y) = -n

G(x,y;h2(n),n)g(T(n),n)dn - -

n

G(x,y; h2(n),n)v(n)dn-

[G^ + H/,(y)G]|e=hl(n)P(n)dn + n Gtk=h1(n)^(n)dn+

1

+-n

0

h2(0)

G|n=oUo (E)dE - -1 n

G(x,y; E,-n)f(^,n,u(E,n))dEdn,

hi(0)

D

or

u(x,y) = -n

a

G(x,y; h2(n),n)g(T(n),n)dn -n

D

where

H(x,y) = -n

1 f 1

[G^+H/i(y)G]|e=hl(n)P(n)dn + n Gk=h (n)^(n)dn + -

h2(0)

n

(17)

G(x,y; h2(n),n)v(n)dn-G(x,y; E,n)f(^,n,u(E,n))d^dn + H(x,y), (18)

GL=0U0(E)dE.

0 0 h,(0) Now passing to limit x —> h.2(y), according to notation in (16), from (18) we have

T(y) = 1

G(h2(y),y; h2(n),n)g(T(n),n)dn - a

n

G(h2(y),y; h2(n),n)v(n)dn-

G(h2(y),y; E,n)f(^,n,u(E,n))d^dn + H(h2(y),y).

(19)

D

Further, differentiating with respect to x (18), passing to limit x —> h,2(y), according to notation in (16), we have

v(y) = n

a

Gx(h2 (y),y; h2(n),n)g(T(n),n)dn -n

Gx(h2(y),y; h2(n),n)v(n)dn-

Gx(h2(y),y; E,n)f(^,n,u(E,n))d^dn + Hx(My),y).

(20)

D

System (18) - (20) is a system of Hammerstein's nonlinear integral equations with respect to u(x,y),T(y) and v(y). We will prove the unique solvability of this system using the contracting mapping principle.

Let Gg— be a set of functions F = {u(x,y ),T(y), v(y)}, that are continuous in domain D0 = {(x,y);hi (y) < x < h2(y),0 < y < 0} and have on the interval 0 < y < 0 , bounded norm ||F|| = ||u|| + 11tM + ||v||, where lluN = max |u|, IItM = max |t|, IIvM = max |v|.

(x,y)GD 0<y<0 0<y<0

Let G0,n denote subset {F : F G Ge, |F| < N} of set Ge.

Denoting the right-hand sides of (19), (20), (21) by Ai(u,T, v),i = i,3, , respectively, we define the mapping A = {Ai (.), A2(.), A3(.)}.

The mapping A is well defined (since all integrals on the right-hand sides of (19)-(21) exist).

We show that for some 0 and N > 0, , for 0 < y < 0, , operator A transforms G0,n into itself. That is, the inequalities |Ai < y,i = i,3, are valid when (u,T,v) G G0,n . . To do this, assume that Ai(u,T,v),i = i,3 are defined in G0,n, respectively.

From relation (19) we obtain

|Ai(u,T,v)| < {[^KM + ^MVM + !i2M1 (h2(y) — Hi(«)) + 2Kll^MMh/i(y)|]ei/i2+

, rM^MMh/i(y)^Q1/3 , H^L w un iQi/i2 , 4Ci II I inni/4..........

+[---K0/ + (x — hi (y))C5]0/ +--M4M}0/ + ||u0|| + |M|.

3n 1 3n 1 n 1

ForNi we take Ni = 3(||uo|| + ||p|| +1) , a 0i , and 0i is chosen so that the following inequality holds

{[ EM + ||v|| + EM (h2(y)— hi(y)) + g B^lBh/i(y)«]e5/12+ +[ MMM Kei/3+M (x—hi(y))C5]e1/12+^ B^«}e1/4 < 1.

3n 1 3n 1 n 1

N1 3

Likewise, from (20) and (21) we have

Then the relation |A11 < N1 holds

N2 N3

|A2|< "3p|A3|< -33.

| A2 |< A3|< ^

Setting N = maxNt and 0 = min0i,i = 1,3, for 0 < y < 0 we prove that the operator A maps the set G0,n into itself. Let us show that with an appropriate choice of 0, operator A is contractive. We have

|A1 (u,T,v)- A2(u*,t*,V)| <n

y

1

|g(T(y),y)- g(T*(y),y)|G(x,y; h2(n),n)dn+

y

-n

|v(y)- V(y)|G(x,y; h2(n),n)dn +1

n

0D

|f(^,n,u(^,n))-f(£„n,u*(£,n))|G(x,y;^,n)d£,dn <

< {||t — t*|| + ||v — v*|| + ||u—u* }i,e2/3, l, = max{ 3Ki,3oK,3K(H2(v) — H,(y)) L

We choose e, so that the inequalities I, e2/3 < 1 hold. Likewise, for A2(u,T,v) and A3(u,T,v) we have

|A2(u,t, v) — a2(u*,t*,v*)| < {||t-t*|| + ||v-v*|| + ||u-u*jl292'

2/3

|A3(u,T,v) — A3(u*,T*,v*)| < {||t — t*|| + ||v — v*|| + ||u—u*}l3e,/4.

For e = mine^i = ,,3, operator A(u,T,v) is a contraction mapping. Then, by virtue of the contraction mapping principle, it has a single fixed point (u,T,v) e Ge,N. We assume that e is chosen so as to ensure the compressibility of operator A(u,T,v) and that operator A(u,T,v) maps Ge,N to itself.

Therefore, (u,T,v) e Ge,N is a solution of system (19) - (21) for 0 < y < e.

Let us establish estimate ux(x,y) b D in , which ensures the global solvability of the problem posed [9].

We have

Ux(x,y) = -n

Gx(x,y; h2(n),n)g(u(h2(n),n)dn -n

Gx(x,y; H2(n),-n)aUx(H2(n),-n)dn-

[G^x + h/1 (y )Gx]u|e=h1 (n) dn + n GExUE|E=h1 (n) dn +

1

+ -n

h2(0)

Gx(x,y; E,0)u0^dE--n

hi(0)

We have

Gx(x,y; E,n)f(^,n,u(E,n))dEdn = I1 +12 +13 +I4+15 + Ig.

D

|I1| <

CM1

n

dn 4CM1 ^ „

1 <-1 № < K1,

(y -n)3/4" n

|I2I<

a|v|C1

n

dn all VI C1 w

1 < 11 11 1 ^ < K2.

(y -n)3/4" n

The remaining integrals are estimated in a similar way. Let maxKi = K,i = ,,6, then we have

lUxll <K.

Thus, we conclude that the solution of problem (1)-(5) can be extended to [0, 1] in y.

Competing interests. The author declares that there are no conflicts of interest with respect to authorship and publication.

Contribution and responsibility. The author contributed to the writing of the article and is solely responsible for submitting the final version of the article to the press. The final version of the manuscript was approved by the author.

References

1. Korteweg D. J., De Vries G. On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves, Phil. Mag., 1985. vol. 39, pp. 422-443.

2. Baranov V. B., Krasnobaev K.V. Gidrodinamicheskaya teoriya kosmicheskoy plazmy [Hydrody-namic theory of space plasma]. Moscow: Nauka, 1977 (In Russian).

3. Karpman V. I. Nelineynyye volny v dispergiruyushchikh sredakh [Nonlinear waves in dispersive media]. Moscow: Nauka, 1973 (In Russian).

4. Cattabriga LUn problem al contorno per una equazione parabolica di ordin dispari., Amali della Souola Normale Superiore di Pisa a Matematica, 1959. vol. XIII. Fasc, no. II. Series III., pp. 163203.

5. Juraev T. J. Krayevyye zadachi dlya uravneniy smeshannogo i smeshanno-sostavnogo tipov [Boundary value problems for mixed and mixed-composite types equations]. Tashkent: Fan, 1979 (In Russian).

6. Abdinazarov S., Khashimov A. R. On boundary value problems for equations of the third order with multiple characteristics and discontinuous coefficients, Uzb. Mat. Jour, 1993. vol. 1, pp. 3-12 (In Russian).

7. Khashimov A. R. Nonlinear boundary value problems for the equation of the third order with multiple characteristics, Uzb. Mat. Jour, 1993. vol.2, pp. 97-102 (In Russian).

8. Kurbanov O. T., Kholboev B. M. On a non-linear boundary value problem for an odd-order equation with multiple characteristics, Uzb. Mat. Jour, 2003. vol.3, pp. 35-40 (In Russian).

9. Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N. Lineynyye i kvazilineynyye uravneniya parabolicheskogo tipa [Linear and quasilinear equations of parabolic type]. Moscow: Nauka, 1967 (In Russian).

Kurbanov Odilzhan Tukhtamuradovichfa - PhD (Phys. & Math.), Associate Professor, Associate Professor, Dept. Apl. Math., Tashkent State University of Economics, Tashkent, Uzbekistan, ORCID 0000-0003-1360-325x.

o ÉÉ

Вестник КРАУНЦ. Физ.-мат. науки. 2022. Т. 38. №. 1. С. 28-39. ISSN 2079-6641

УДК 517.956 Научная статья

Об одной краевой задаче для уравнения нечетного порядка с

кратными характеристиками О. Т. Курбанов

Ташкентский государственный экономический университет, Узбекистан, 100003, Ташкент, ул. Ислама Каримова, 49 E-mail: odil69@inbox.ru

Для нелинейного уравнения с кратными характеристиками в криволинейной области исследована однозначная разрешимость одной краевой задачи.

Ключевые слова: нелинейность, единственность, существование, система уравнений Гаммерштейна.

d DOI: 10.26117/2079-6641-2022-38-1-28-39

Поступила в редакцию: 09.02.2022 В окончательном варианте: 16.03.2022

Для цитирования. Kurbanov O. T. On a boundary value problem for an odd-order equation with multiple characteristics // Вестник КРАУНЦ. Физ.-мат. науки. 2022. Т. 38. № 1. C. 28-39. DOI: 10.26117/2079-6641-2022-38-1-28-39

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© Курбанов О. Т., 2022

Курбанов Одилжан Тухтамурадович& - кандидат физико-математических наук, доцент, доцент кафедры прикладной математики, Ташкенский государственный экономический университет, Ташкент, Узбекистан, СЖСГО 0000-0003-1360-325х.

Oi ii

Финансирование. Исследование выполнялось без финансовой поддержки фондов.