ABOUT A METHOD OF RESEARCH OF THE NON-LOCAL PROBLEM FOR THE LOADED MIXED TYPE EQUATION IN DOUBLE-CONNECTED DOMAIN. O.Kh.

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1 Bulletin KRASEC. Phys. & Math. Sci, 014, V. 9,., pp ISSN MSC 35M10 DOI: / ABOUT A METHOD OF RESEARCH OF THE NON-LOCAL PROBLEM FOR THE LOADED MIXED TYPE EQUATION IN DOUBLE-CONNECTED DOMAIN O.Kh. Abdullayev National University of Uzbekistan by Mirzo Ulugbeka, , Uzbekistan, Tashkent c., VUZ gorodok st. obidjon.mth@gmail.com In the present paper an existence and uniueness of solution of the non-local boundary value problem for the loaded elliptic-hyperbolic type euation on the third order in double-connected domain was investigated. The uniueness of solution was proved by the extremum principle for the mixed type euations, and existence was proved by the method of integral euations. Key words: loaded euation, elliptic-hyperbolic type, double-connected domain, an extremum principle, existence of solution, uniueness of solution, method of integral euations Introduction We shall notice that with intensive research on problem of optimal control of the agroeconomical system, regulating the level of ground waters and soil moisture, it has become necessary to investigate a new class of euations called "LOADED EQUATIONS". Such euations were investigated in first in the works by N.N. Nazarov and N.Kochin, but they didn t use the term "LOADED EQUATIONS". For the first time, the most general definition of the LOADED EQUATIONS was given and various loaded euations were classified in detail by A.M. Nakhushev [1]. Let s notice that non-local problems for the loaded elliptic-hyperbolic type euations in double-connected domains have not been investigated. In the given paper, uniueness of solution of the non-local boundary value problem for the loaded elliptic-hyperbolic type euation in double-connected domain was proved, and the method of the solvability of the investigated problem was presented. Abdullayev Odidjon Khayrullaevich Ph.D. Phys. & Math. Associate Professor Dept. of Differential euations of the National University of Uzbekistan named after Ulugbek, Uzbekistan. Abdullayev O.Kh.,

2 ISSN Abdullayev O.Kh. The statement of the Problem Let s consider the euation u xx + sgnxyu yy + Rx,y = 0 1 in double-connected domain of Ω, bounded with totwo lines: and characteristics: σ 1 : x + y = 1 ; σ : x + y = ; atx > 0, y > 0; σ 1 : x + y = 1 ; σ : x + y =, atx < 0, y < 0; A j A j : x y = 1 j+1 ; B j B j : x y = 1 j 1, j = 1, at x y < 0, of the euations 1, where A 1 1;0, A 0;1, A 1 0; 1, A 1;0, B 1 ;0, B 0;, B 1 0;, B ;0, 0 < < 1, j = 1,. Rx,y = 1 sgnxy R 1 x,y = 1 + sgnxy + x Let s enter designations: [λ 1 R 1 x,y + λ R x,y], λ 1,λ = const > 0; ux,0, R x,y = 1 + sgnxy + y u0,y. 1 j E j ; + 1 j 1, E j 1 j ; 1 j, C j 1 j 1 ; 1 j, Ω 0 = Ω x > 0 y > 0,Ω 0 = Ω x < 0 y < 0, 1 = Ω x + y > y < 0; 1 = Ω x + y < y < 0, = Ω x + y > x < 0, = Ω x + y < y > 0, D 1 = Ω < x + y < y < 0, D = Ω < x + y < y > 0, D 0 = Ω 0 1, D 0 = Ω 0 1, I j = {t : 0 < t < }; I + j = { t : 0 < 1 j 1 t < 1 }, { x at j = 1, where t = y at j =. x + 1 Let s designate, through θ 1 ; x 1 y 1, θ ; y + 1 points of intersections of characteristics of the euation 1 with leaving points x,0 A 1 B 1 and 0,y A B with characteristics A 1 A 1 and A A, accordingly. 4

3 About a method of research of the non-local problem... ISSN In the domain of Ω the following problem is investigated Task. Problem A. To find function ux, y with following properties: 1 ux,y C Ω; ux, y is the regular solution of the euation 1 in the domain of Ω\xy = 0\x+y = ±, besides, u y CA 1 B 1 A B, u x CA B A 1 B 1 at that u x0,t, u y t,0 can tend infinity of an order of less unit at t ±, and finite at t ±1; 3 ux, y satisfies gluing conditions on lines of changing type: u y x, 0 = u y x,+0, in regular intervals onx,0 A 1 B 1 A B, u x 0,y = u x +0,y, in regular intervals on 0,y A 1B 1 A B, 4 ux, y satisfies to boundary conditions: ux,y σ j = φ j x,y; x,y σ j, ux, y σ j = φ j x,y; x,y σ j, 3 d dx uθ 1x = a 1 xu y x;0 + b 1 x, < x < 1, 4 d dy uθ y = a yu x 0;y + b y, < y < 1, 5 ux, y = g j t, t I j, 6 B j B j where φ j x,y, φ j x,y, g jt, a j t, b j t, j = 1, is given function, at that: g j = g j, φ,0 = g 1, φ 0, = g, φ 0, = g 1, φ,0 = g, } 7 φ j x,y = xy γ φ j x,y; φ j x,y C σ j, < γ < 3 8 φ j x,y = xy γ φ j x,y; φ j x,y C σ j, < γ < 3, 9 g j t C I j C I j, 10 a j t, b j t C [,1] C p, ; 11 Theorem. If conditions 7-11 and a 1 x > 1, a y > 1 ; 1 are satisfied, then the solution of the Problem A exists and it is uniue. 5

4 ISSN Abdullayev O.Kh. Proof. Note, that the solution the euation 1 in hyperbolic domains looks like: ux,y = f 1 x + y + f y x + λ ux,y = f 1 x + y + f x y + λ y y y tz0,tdt,aty > 0, x < 0; 13 t yz0,tdt,atx > 0, y < 0; 14 Owing to 13 we will receive, that the solution of Cauchy problem of the euation 1 satisfies conditions u0,y = τ y and u x 0,y = y in the domain of, looks like: ux,y = τ x + y + τ y x λ y x + 1 y+x y x y x tτ tdt + λ tdt λ y y+x x + y tτ tdt y tτ tdt. 15 From here, by virtue, 5 we will receive a functional relation between τ y and y from the domain of on the piece A B : i.e., 1 τ y + 1 y y λ τ tdt + λ a y 1 y = τ y + λ y+1 y+1 τ tdt = ya y + b y, τ tdt + λ y τ tdt b y. 16 Confirm, that a functional relation between τ 1 x and 1 xwe will obtain by the same method, from the solution of Cauchy problem for the euation 1 satisfying to conditions ux,0 = τ 1 x; u y x,0 = 1 x in the domain of 1, and with the account of conditions 4, which presented on the form: a 1 x 1 1 x = τ 1x + λ 1 τ 1 tdt + λ 1 y τ 1 tdt b 1 x. 17 The uniueness of solution of the Problem A It is known that, if the homogeneous problem has only trivial solution than a solution of the accordingly non-uniform problem is uniue, therefore, we need to prove that the homogeneous problem has only trivial solution. 6

5 About a method of research of the non-local problem... ISSN Let b 1 y b x 0 then, from 16 and 17, accordingly, we will receive: a y 1 y = τ y + λ y+1 τ tdt + λ y τ tdt, 18 and a 1 x 1 1 x = τ 1x + λ 1 τ 1 tdt + λ 1 x τ 1 tdt. 19 Lemma 1. If b 1 y b x 0 and satisfied conditions 1 that solution ux,y of the euation 1 reaches the positive maximum, and the negative minimum in closed domain of D 0 only on σ 1 and σ. Proof. According to the extremum principle for the hyperbolic [] and elliptic euations [3], the solution ux, y of the euation 1 can reach the positive maximum and the negative minimum in closed domain of D 0 only on A B A 1 B 1 and σ 1 σ. We need to prove, that the solution ux,y of the euation 1 can t reach the positive maximum and the negative minimum in A B A 1 B 1. Let the function ux,y u0,y = τ y reach the positive maximum the negative minimum on some point ofy 0 A B, then from 18 and based on 1 we will receive y 0 = λ a y 0 1 y 0 +1 τ tdt + y 0 τ tdt > 0 y 0 < 0. Therefore, in the view of to the gluing condition, we have + y 0 > 0 + y 0 < 0, and it contradicts the known Zareba-Zero principle [3], according to which in a point of a positive maximum a negative minimum should be y 0 < 0 y 0 > 0. Thus, ux,y does not reach the positive maximum the negative minimum on the point of y 0 A B. Owing to 19 and 1, it is similarly proved that the function ux,y does not reach the positive maximum and the negative minimum in the interval of A 1 B 1. The Lemma 1 is proved. As τ y does not reach the positive maximum and the negative minimum in the interval of A B then, τ y = const in the interval of A B. Further by virtue ux,y C Ω, we will receive τ 1 = φ 1 0,1 0, τ = φ 0, 0 at the φ j x,y 0. From here, as τ y C A B we will conclude that, τ y 0 in A B. Therefore, if b y 0 and a y > 1, then from 18 we will get y 0. Also, by the same method, we can get τ 1 x 0 in A 1 B 1 and 1 x 0. Hence, owing to uniueness of solution of the Cauchy problem for the euation 1, we will have ux,y 0 in domains 1 and. Conseuently, in the view of the Lemma 1, at φ 1 x,y φ x,y 0 we will deduce, that ux,y 0 in closed domain of Ω 0. Thus, ux,y 0 in D 0. Further, owing to condition 7 and considering the uniueness of solution of the Gaursat problem atg t 0, t I j j we will receive ux,y 0 in D j j = 1,. Takes place: 7

6 ISSN Abdullayev O.Kh. Lemma. The solution ux, y of the euation 1 reach the positive maximum and the negative minimum in closed domain D 0 only on A B A 1 B 1, σ 1 and σ. Lemma will be proved similarly as Lemma 1 On the basis of Lemma, on account of 3 at φ j x,y 0, we will receive ux,y 0 in the domain of D 0. Thus, the solution of homogeneous Problem A is identically eual to zero in the domain of Ω. The uniueness of solution of the Problem A is proved. The existence of solution of the problem I It s known, that the solution ux,y of the Nyman problem Problem N [] satisfying conditions: 1 ux,y CD 0 C D 0 \xy = 0 C 1 D 0 is the solution of the euation 1; u y CA 1 B 1, u x CA B, at that u x 0,t, u y t,0 can tend to infinity of an order of less unit at t, and finite at t 1; 3 satisfies to boundary conditions j=1, and u x 0,y = y, u y x,0 = 1 x is exists and is uniue and represented in the form [] ux,y = φ 1 ξ,η n Gξ,η;x,ydS φ ξ,η Gξ,η;x,ydS+ n σ 1 σ tgt,0;x,ydt + + tg0,t;x,ydt, 0 where Gξ,η;x,y is Green s function of the problem N for the Laplase euation in the domain of Ω 0, which looks like: Gξ,η;x,y = 1 π ln θ lnv+ln µ 1 πir θ lnv+ln µ 1 πir θ ln v+ln µ 1 πir θ ln v+ln µ 1 πir θ lnv ln µ 1 πir θ lnv ln µ 1 πir θ ln v ln µ 1 πir θ ln v ln µ 1, 1 πir where v = ξ +iη, v = ξ iη, µ = x+iy, µ = x iy, r = πi 1 ln, i = 1, θ 1 ξ = θ 1 ξ 1 r is theta function. From 0 y = 0 andx = 0, we will get functional relations between τ + y, + y and τ 1 + x, + 1 x, respectively, on the pieces A B and A 1 B 1 getting from the domain Ω 0 : τ 1 + x = 1 k 1 φ k ξ,η k=1 σ k + + τ + y = 1 k 1 φ k ξ,η k=1 σ k n Gξ,η;x,0dS tgt,0;x,0dt+ tg0,t;x,0dt. n Gξ,η;0,ydS tgt,0;0,ydt+

7 About a method of research of the non-local problem... ISSN tg0,t;0,ydt, 3 After differentiating eualities by x and 3 by y, we obtain 1 1 x = τ t Gt,0;x,0 x dt + + t G0,t;x,0 dt + F x 1x, 4 where F 1 x = 1 k 1 ϕ k ξ,η ngξ,η;x,0ds, and k=1 σ k 1 y = τ t Gt,0;0,y y dt + + t G0,t;0,y dt + F y y, 5 where F y = 1 k 1 φ k ξ,η ngξ,η;0,yds. k=1 σ k Further, the euations 16 and 17 we will rewrite in the form: τ jt + λ j t+1 Kt,zτ jzdz = F j t, 6 where 5t + 1 F j t = a j t 1 j t + b j t + λ j τ j 3, 7 Kt,z = { 5y+1 3z, z t; t+1 z, t z, t = t+1 { x at j = 1, y at j =. Note that we will search the function j t from the class of C,1, which can tend to infinity of an order of less unit at t, and finite at t 1. From here and owing to account 11, we will decide F j t C,1 and that F j t can tend to infinity of an order of less unit at t, and finite at t 1. Hence, by virtue K j t,z const, we will conclude that the euations 6 are the Volterra integral euations of the second kind which uneuivocally solved by the method of consecutive approach [] and its solution is represented in the form: τ jt = λ j t+1 8 R j t,z,λ j F j zdz + F j t, 9 where R j t,z,λ j are resolves of the kernels K j t,z, and that R j t,z,λ j const, j = 1,. 30 9

8 ISSN Abdullayev O.Kh. Further, having excluded τ j t from the relations 4 and 9, we will receive system of the integral euations a 1 x 1 1 x + λ 1 R 1 x,z,λ 1 1 zdz 1 + Gt,0;x,0 1 t x dt = Φ 1 x y+1 31 a y 1 y + λ R y,z,λ zdz 1 + G0,t;0,y t y dt = Φ y where Φ 1 x = Φ y = + t G0,t;x,0 dt λ 1 x b 1 x = b 1 x + λ 1φ,0 1 + t Gt,0;0,y dt λ y y+1 R 1 x,z,λ 1 b 1 zdz b 1 x + F 1x, 3 5x 6 + 1; b y = b y + λ φ 0, 5y Gt,0;x,0 = G0,t;0,x = K 1 x,t, K 1 x,t = 1 π [ lnt lnx ln R y,z,λ b zdz b y + F y, 33 ] + K 1x,t + K1x, t 34 Gt,0;0,x = G0,t;x,0 = K x,t = K 1x, it, 35 K1x,t = ln 1 tx t x + ln 1 n tx n=1 t n x + ln 1 n tx t n x. 36 Further, after some simplifications, from we can get the system of singularity integral euation concerning 1 x and y which represented onin the form [] 1 x = 1 ta 1 x,tdt + Φ 1 x x = ; 37 ta x,tdt + Φ x where Φ j x = Φ jx a j x 1 ; A j x,t = R λ j x,z,λ j j a j x 1 K 1 x,t a j x 1, z ; K 1 x,z a j x 1, z 1, 38 K 1 x,t = ln t xln + Kx,t + Kx, t, 39 10

9 About a method of research of the non-local problem... ISSN [ Kx,t = 1 1 π t x t 1 tx + n=1 n t n x Further, we will investigate the function Φ 1 x: 0 Φ 1 x = + t K 1 x,itdt λ 1 Gξ,η;x,0 + φ 1 ξ,η x ξ σ 1 Gξ,η;x,0 φ ξ,η x ξ σ = φ 1 ξ,η ξ where βx = λ 1 + t K 1 x,itdt Gξ,η;x,0 η φ ξ,η x ] n t 1 n tx n t 1 n tx + n t n. 40 x dη dη + ξ φ 1 ξ,η 1 ξ dξ + 0 R 1 x,z,λ 1 b 1 zdz b 1 x+ Gξ,η;x,0 φ 1 ξ,η x η σ 1 Gξ,η;x,0 φ ξ,η x η σ x Gξ,η;x,0 ξ dξ + dξ = dξ ξ Gξ,η;x,0 φ ξ,η ξ x ξ Gξ,η;x,0 η dξ + βx = = α 1 x + α x + α 3 x + α 4 x + α 5 x + βx R 1 x,z,λ 1 b 1 zdz b 1 x. Owing to account 9, 11, 1, 30, 39 and 40, we will get see []: βx const, α 1 x const, α x const, α 3 x const, dξ + α 4 x x γ 3 const, α 5 x x γ 3 const, < γ < Hence, by virtue 1 and 41, we will conclude that Φ 1 x C,1, and Φ 1 x can tend to infinity an order of less one at x, and at x 1 it is limited. It is similarly proved, that Φ y C,1, and Φ y can tend to infinity an order of less one at y, and at y 1 it is limited. Thus, the system of integral euation 37 is reduced to the Fredholm integral euations of the second kind, by the known method Karleman-Vekua [4], just as in works [], [3]. Note, that uniue solvability of the Fredholm integral euations of the second kind follows from the uniueness of solution of the Problem A and from the theory integral euations. 11

10 ISSN Abdullayev O.Kh. Solving the system of integral euations 37, we will found 1 x and y [], further owing to account 4 and τ 1 = φ,0, τ = φ 0, from 9 we will find τ j t j=1,. Hence, after having found τ 1 x and 1 x, τ y and y, the solution of the Problem A can be restored in the domain of Ω 0 as the solution of the Problem N 0, and in domains j j=1, as the solution of the Cauchy problem. The solution of the Problem A in domains of D j j=1, can be restored as a solution of the Gaursat problem with conditions 6 and ux,y B j E j = h j t, where h j t j=1, are traces of solution of the Cauchy problems in domains j j=1,, on the line x + y =, and reciprocally in domains j j=1, as the solution of the Cauchy-Gaursat problem with conditions u y x,0 = 1 x; u x0,y = y, 1 < x,y < and ux,y B j E = h j t, where h j t j j=1, are traces of solution of the Gaursat problems in domains D j j=1,. And finally the solution of the Problem A can be restored in the domain of Ω 0 as the solution of the problem N, similarly as 0. The Theorem is proved. References 1. Nakhushev A.M. Nagruzhennye uravneniya i ih prilozheniya [The loaded euations and their applications]. Moscow, Nauka Publ., p.. Abdullaev O.Kh. Kraevaya zadacha dlya nagruzhennogo uravneniya elliptiko-giperbolicheskogo tipa v dvusvyaznoj oblasti [Boundary value problem for a loaded euation elliptic-hyperbolic type in double connected domain]. Vestnik KRAUNC. Fiziko-matematicheskie nauki Bulletin of the Kamchatka Regional Association «Education-Scientific Center». Physical & Mathematical Sciences, 014, vol. 8, no., pp Bitsadze A.V. Differential euations of Mixed type. New York, Pergamon Press, p. 4. Ìuskheleshvili N.I. Singulyarnye integral nye uravneniya [Singularity integral euations]. Moscow, Nauka Publ., p. Original article submitted: