The loaded parabolic-hyperbolic equation and its relation to non-local problems Текст научной статьи по специальности «Математика»

The loaded parabolic-hyperbolic equation and its relation to non-local problems

U. I. Baltaeva

National University of Uzbekistan named after Mirzo Ulugbek Differential equations and Mathematical physics, 100174, Tashkent, Uzbekistan

[email protected]

DOI 10.17586/2220-8054-2017-8-4-413-419

In this present paper, unique solvability is proved for the boundary value problems for the loaded differential equations associated with non-local boundary value problems, for the classical partial differential equations.

Keywords: parabolic-hyperbolic type equations, non-local problems, loaded equation, boundary value problems, gluing condition, integral equation.

Received: 25 July 2017 Revised: 3 August 2017

1. Introduction

Research on equations of parabolic-hyperbolic and hyperbolic-elliptic types has been conducted heavily. Since, there are both theoretical and practical uses for their applications, especially in the development of principal parts of partial differential equations. Moreover, in recent years it has become increasingly important to investigate a new class of equations, known as loaded equations, as a direct result of issues with the optimal control of the agro-economical system, long-term forecasting and regulation of groundwater levels and soil moisture [1-4].

Basic questions on the theory of boundary value problems for partial differential equations still exist, and are equivalent to that of the boundary value problems for loaded differential equations. However, the existence of the loaded part operator A does not often make it possible to directly apply the known theory of boundary value problems, for the classical partial differential equations. In view of this, searching for the solutions of loaded differential equation using preassigned classes, it may be possible to reduce to them to new problems, for non-loaded equations(see [4-6]).

At the same time, mixed type equations appear in some problems of nanophysics and are present in some micro- and nanoflow models. Particularly, parabolic-hyperbolic equations are considered in some models of spinodal decomposition (so-called hyperbolic diffusion, see, e.g. [7,8]), and also described flow in thin viscous layers subjected to ultrasonic acoustic field [9].

The present work is devoted to the formulation and investigation of the boundary value problems for a loaded equation of the mixed parabolic hyperbolic type:

d

— (Lu) = ^2ai(x)D oX [uy (x, 0) - u(x, 0)], (1)

=1

where

1 - sgny 1 + sgny

Lu = uxx----Uyy----Uy - Au ,

1 f f (t)dt

D0X f (x) =\ r(-ai) 0 (x - t)1+ai d

ai < 0,

¿DOX-1 f (x), 0 <ai < 1,

integro-differential operator [4]. D-XaiD^Xf = D00xf = f(x). We assume, that an < an-1 < ... < a1 = a < 1 and coefficients ai = ai(x) e C1[0,1] n C3 (0,1), A, ^ are given real parameters, and A > 0.

Furthermore, we will discuss how the boundary value problem for the loaded differential equation is associated with non-local boundary value problems, for classical partial differential equations. In our investigations, we have formulated first-kind boundary value problems (such as the Tricomi problems) for the linearly loaded, integro-differential equation of the third order, with mixed operators.

2. First kind of boundary value problems for a loaded equation of the mixed type

Let c R2 be a domain bounded by the segments AB, BB0, AA0, A0B0 of straight lines y = 0, x = 1, x = 0, y = h, respectively when y > 0. is a characteristic triangle bounded by the segment AB the axis OX and two characteristics:

AC : x + y = 0, BC : x - y = 1,

of equation (1) for y < 0.

Let us introduce the following notation:

I = {(x, y) : 0 < x < 1, y = 0}, Q = U Q2 U I. Problem 2.1. Find a regular solution u(x, y) equation (1) in Q, which has continuous derivatives wx(wy), up to AAo U AB U AC (AB U AC), and satisfies the boundary value conditions:

u(0,y)= ^i(y), u(1,y) = ^2(y), Ux(0,y)= ^3(y), 0 < y < h, (2)

= ^2(x), 0 < x < 1, (3)

u(x, —x) = ^l(x),

dn

and the gluing conditions:

uy (x, +0) = uy (x, —0), (x, 0) G I, where ^1(y), y>2(y), ¥>3(y), ^1(x) and ^2(x) are the given functions.

Problem 2.2. Find a regular solution u(x, y) equation (1) in Q, which has continuous derivatives wx(wy), up to AA0 U AB U BC (AB U BC), and satisfies the boundary value conditions (2),

= ^2(x), 1 < x < 1, (4)

y=x-1

, "its du(x,y)

u(x, x — 1) = -¡ii(x),

dn

and the gluing condition:

(x, +0) = (x, -0), (x, 0) € I,

where ^(x) and ^2(x) are the given functions.

We note that physical characteristic problems may reduce to equations which include, some composed operator in the loaded part. The unique solvability of the analogs of the Tricomi for the loaded differential equations was investigated in [5,10], for which the loaded part contains the imputation or imputation operators of the unknown function.

An important aspect of the investigation of the boundary value problems, is shown by the following theorem. The following theorem holds.

Theorem 2.1. Any regular solution of equation (1) (when y = 0) is represented in the form:

u(x,y) = z(x,y) + w(x), (5)

where z(x, y) is a solution to the equation:

d ( _ 1 - sgny _ 1 + sgny

dx 2 Zyy 2 '

w(x) is a solution of the following ordinary differential equation:

zxx--^-zyy--7.-zy — Az ) = 0, (6)

(x) — Aw (x) — = ^^^¿(x^^X [zy (x, 0) — z(x, 0)]. (7)

^ ( X / ixi - ( /. r il.: I x I I / o ^

= 1 ¿=1

Proof of Theorem 2.1.

Let u(x, y), represented by formula (5), be the solution of equation (1) for y < 0. Then, substituting:

d n d dx (uxx - Uyy - Am) - y^a^x)^ K(x, 0) - u(x, 0)] = — (zKK - - Az) + i=1

n n

+w'''(x) - Aw'(x) - y aj(x)D0Xw(x) - y ^^ aj(x)D0X [zy (x, 0) - z(x, 0)] = 0, i=i i=i

satisfies equation (1) for y < 0.

Then, vice versa, let u(x,y) be a regular solution to equation (1), and w(x) be a certain solution:

n n

w'''(x) — Aw'(x) — j^^ aj(x)D0Xw(x) = jJ^aj(x)DaX[uy(x, 0) — u(x, 0)]. (8)

i=i i=i Let us prove the validity of the relation (5). Manifestly, the function:

x n

u(x, y) = z(x, y) + J (cosh VA(x — t) — 1)^""^ a,(x)Da [uy(t, 0) — u(t, 0)]dt, o i=1

is a solution to equation (1), where z(x, y) is a solution to equation (6), and the function:

X

u(x,y) = — J J (1 — cosh VA(x — t))^ ai(t)Da? [uy (t, 0) — u(t, 0)]dt,

o i=1

is a partial solution to equation (1). Hence, (1) highlights the validity of the representation (5), i.e. u(x, y) = z(x, y) + w(x).

It follows from the latter representation that u(x, 0) = z(x, 0) + w(x). Then, (8) provides:

nn

w'''(x) — Aw'(x) — j aj(x)DaXw(x) — J aj(x)D0X [zy(x, 0) — z(x, 0)] = 0, i=1 i=1 and the function z(x, y) = u(x, y) — w(x), satisfies equation (6) for y < 0. Analogously proved in the case for y < 0. Theorem 2.1 is proved.

Now, invoking that the function ae^x + 6e-^x + c satisfies equation (6), we can assume without loss of generality that:

w(0) = w'(0) = w''(0) = 0, (9)

when studying Problem 2.1 and Problem 2.2.

Let us solve the Cauchy problem for equation (7) with the conditions (9), with respect to w(x). Assuming

that:

nn

/(x) = J^aj(x)DXw(x) + j^aj(x)DaX [zy (x, 0) — z(x, 0)], i=1 i=1 we write equation (7) in the form:

X

1

w(x) = -A

o

(x) = 1J (cosh VA(x -1) - /(t)dt.

o

The last equality with respect to designation and after some transformation becomes:

X

w(x) - vj K(x,t)w(t)dt = F(x), 0 < x < 1,

(10)

where

X

K(x, t) = ————t [ Aj(t)(t — s)-1-a fcosh VA(x — t) — l) dt, a, < 0, (11)

r(—aj)J V y

o

x

K(x,t) = Ar(11— ^ J d [Ai(t)(t — (cosh VA(x — t) — l)

dt, 0 < a, < 1, (12)

^ n

F (x) = JJ (cosh VA(x — t) — 1) ^ ai(t)Dgx [zy (t, 0) — z(t, 0)]dt. (13)

o i=1

The recurring index i = 1,2, ...n, implies summation. Therefore, solving the next equation with respect to [11], we have a solution to the Cauchy problem for equation (7) with the conditions (9):

x n

w(x) = I P(x,t)£ aj(t)DSX[zy (t, 0) — z(t, 0)]dt, (14)

n j=1

where

P(x,t) = j jcosh VA(x — t) — 1 + y J ^cosh VA(s — t) — lj R(x, s; y)ds | . (15)

R(x, s; y) is the resolvent of the kernel K(x,t), when studying Problem 2.1 and Problem 2.2. Thus, by virtue of the representation (5), equation (1) Problem 2.1 and Problem 2.2, in view of (9), are all reduced to the following Problem NP1 and Problem NP2 for equation (6).

We investigate the following problems: Problem NPi. To find a regular solution to equation (6) in the domains and Q2, which has continuous derivatives wx(wy), up to AA0 U AC (AC), and satisfies the boundary conditions:

z(0,y) = ^(y), z(0,y) = ^(y) — w(1), 0 < y < h, (16)

zx(0,y) = (y), 0 < y < h, (17)

«¿(tJDX [zy (t, 0) — z(t, 0)]dt, 0 < x < 1

¿=1

? " 1 z (x, —x) = ^1(x) — P (x,t)J^ ai(t)D0X [zy (t, 0) — z(t, 0)]dt, 0 < x < ^, (18)

x

I z. It 0 1 — ZI t Ulld.t \ f V. TV.

0 i=1 x n

dZ(dn-T) = ^(x) - J PX (x,t)£ a^)^ [Zy (t, 0) - z(t, 0)]dt, 0 < t < 1, (19)

0 i=1

and the gluing condition:

zy (x, +0) = zy (t, -0), (x, 0) G I, where ^1(y), y>2(y), ^(y), -^(x) and ^2(x) are given functions, such that ^1(0) = ^1(0).

Problem NP2. To find a regular solution of equation (6) in the domains and n2, has continuous derivatives wx(uy), up to AA0 U BC (BC), and satisfies the boundary conditions (16), (17),

X n :

z(x,x - 1) = ^(x) - P(x,t)^ ai(i)Dgx [zy(t, 0) - z(t, 0)]dt, ^ < x < 1, (20)

0 i=1 x n

dz(XdXn- 1) = ^2(x) - |Px(x,t)£ ai(t)D0ax[zy(t, 0) - z(t, 0)]dt, 1 < x < 1, (21)

and the gluing condition:

0 ¿=1

zy (x, +0) = zy (x, -0), (x, 0) € /,

where ^(x) and V>2(x) are given functions, such that ^2(0) = ^(1).

Since Problem 2.1 and Problem 2.2 were reduced to the equivalent non-local Problems NP1 and NP2 for the third order mixed type equation (6), we may conclude that Problems 2.1 and 2.2 have a unique solutions, as a direct result of the unique solvability of the non-local problem. The early 1970's marked the beginning of the systematic study of the third and higher order, mixed and mixed composite type PDEs; which contain parabolic-hyperbolic, hyperbolic-elliptic and elliptic-parabolic operators in the main part. This subject has been studied extensively and has been developed by numerous mathematicians [12-14], due to, the increasing popularity of its connection with non-local problems, and the appearance of these PDEs in mathematical models for various problems which possess physical traits. For instance, while studying problems of moisture transfer in soils, heat transfer in heterogeneous media, diffusion of thermal neutrons in inhibitors, simulation of various biological processes, phenomena etc [6,15-18]. The following theorems hold.

Theorem 2.2. If A > 0 and ^1(0) = ^1(0),

(y) € C 1[0, h], (j = 1, 2), ^s(y) € C[0, h] n C 1(0, h),

^1(x) € C1 [0, 1/ 2] n C3 (0,1/2), (x) € C [0, 1/2 ] n C2 (0,1/2), then there exists a unique solution to the Problem 2.1 in the domain Q.

Theorem 2.3. If A > 0 and ^2(0) = ^1(1),

^ (y) e C 1[0, h], (j = 1,2), ^s(y) e C[0, h] n c 1(0, h), ^(x) e C 1[1/2,1] n C3 (1/2,1), ^2(x) e C[1/2,1] n C2 (1/2,1), then there exists a unique solution to the Problem 2.2 in the domain Q.

3. Existence of solution to Problem 2.1

Let us introduce the notation:

u(x, ±0) = t(x), uy(x, ±0) = v(x),

now with regards (5) and gluing conditions, we get:

(x, ±0) = t(x) — w(x), 0 < x < 1, dz(x,y) = v(x), 0 <x< 1.

dy

(22) (23)

y=±o

A solution of the Darboux problem for the equation (6) with the boundary value conditions (18), (19) and (22) in the domain can be represented as:

z(x y) = t2(x + y) + ( y ) — ^ x + y

œ+y

—Ay J t *(t)-T1 y/ A(x — y — t)(x + y — t) dt sinVA (y — t) t)dt+

oo

x — y

y

+y^|cos VA(y — t)^1(—t)dt + A y g1 (x — y, x + y,t) ^1(t)dt—

oo

x + y x — y

2 2

—A J g 1 (x + y, x — y, t) ^ (t)dt + V2 J g2 (x — y, x + y, t) ^ (t)dt—

oo

x —y

— V2 J g2 (x + y,x — y,t) ^(^—^2(0) {sin VA ^

x—y

— sin\/A(—] + sin\/Ay + A(x + y) I T \JA(x + y)(x — y — 2t) sinVAtdt—

x+y 2

where

g 1 (x,y,t) = V^sinVA(t — x j + y/1 v/Ay(x — 2t)

—A(x — y) J ^/A(x — y)(x + y — 2t)

o

x/2

sin

x/2

+ VAy / sin VA(t — s)iT1 \/Ay(x — 2s)

g2(x, y, t) = cosv^t — x j + AW cosv^t — s)i"1 -\/Ay(x — 2s)

^1(x) = ^1(x) — W(x), ^2 (x) = ^2 (x)--pw'(x),

ds,

V2

t 2(x) = t (x) — w(x),

x n

5(x) = J / (1 — cosh VA(x — t)) ^aj(t)Dati[v(t) — t(t)]dt,

(24)

ds, (25)

(26)

y

y

-T1(x) = I1(x)/x, I1(x) is the modified Bessel function (Bessel function of the first kind with imaginary argument [19]).

Differentiating (24) with respect to y and passing through the limit y ^ +0 with the preceding notations and (23), we obtain first functional relation between the function t(x) and v(x), transferred from the to AB:

x n

v(x) + jJ L1(x,i)^ aiD^ [v(t) - t(t)]dt- (27)

x/2

M

L2(x,t)^ aiDati [v(t) - t (t)]dt =t'(x) - A/ t(î)_T1 —A(x - t) dt + f(x)

where

L1(x,t) = —^ sinh—A(x - t) + i - cosh —A(s - t)j jT:l —A(x - s)

ds,

Va

t

L2(x, t),f (x) are continuously differentiable functions.

Further, from (6) considering the property of Problem 2.1, (22) and (23), passing through the limit we obtain:

t''(x) - At(x) = k + v(x) + w''(x) - Aw(x), (28)

where k is an unknown constant to be defined.

The equality (28) is a second functional relation between t(x) and v(x), transferred from the domain to AB.

Solving equation (28) with respect to t(x) with the conditions:

t (0) = ^(0), t ' (0) = ^(0), (29)

we have

x

t(x) = J SlnW^(x t) v(t) dt - UJ (1 - cosh VA(x - t)) ^ «¿D^t [v(t) - t(t)] dt-

0 0 i=1

— (1 - cosh \/Ax^ + ^1(0) cosh VAx +—=^3(0) sinh VAx. (30)

A V /

Thus, we get (27) and (30) system of the equations for the functions t(x) and v(x). If we write relation (30), in the form

x „ x

and assuming that:

T (x) - (1 - cosh —A(x - t)) aiD^t t (t) dt = j sinh t) v(t) dt-

0 i=1 0

x n

j (1 - cosh —A(x - t)) ^ aiDaiiv(t) dt-0 i=1

— - cosh — Ax^ + ^1(0) cosh —Ax +—=(0) sinh — Ax, A V /

x . /— x n

$(x) = I sinh VA(X - t) v(t) dt - jf - cosh —A(x - t)) ^ aiDaiv(t) dt-

J y A J . _ 1

(31)

x) = J -\-^v(t) dt -^y (^1 - coshVA(x - t)j2^ «¿D^v(t) dt- (32)

0 0 i=1

— (1 - cosh \/Ax^ + ^1(0) cosh \/Ax +—=^3(0) sinh \/Ax, A V / a/^

for ai < 0 and 0 < ai < 1 applying the Dirichlet permutation integration formula and corresponding to change of

variables s = t + (x - t)v, we obtain the second kind Volterra type integral equation:

x

t(x) - uj K(x, t)T(t) dt = $(x), (33)

0

x

x

where

1

~ (x - t)-a f Ai (1 - cosh^ (x - t)(1 + v)]) j t

K (x't) =- -Âfi-oyi-vi+o-dv,at a < °,

0

and at 0 < ai < 1,

i

K (x, t) = ^TTay / v-ai dvlAi (l - cosh[^Â2 (x - t)(1 + v)j)

dv.

From the representations of the functions K(x. t) and $(x), based on the general theory of integral equations [11], one can be certain that (33) has a unique solution, which is represented as:

x

t<x>=*<x>+/^.Wl-".0sx s 1. (34)

0

where R(x.t) is the resolvent of the kernel K(x.t). Substituting (34) in (27), consider (32) and the sewing condition, we may obtain the integral equation with a a shift, with respect to v(x), by which a kernel exists with weak singularity. Therefore, by using successive approximations [11], with regards to the conditions of the theorem and (32), we can uniquely define the function v(x) at A = 0, a = a,j(x) G C[0.1] n C3(0.1). Upon determining t(x) and v(x), we find the functions w(x) from (14). Hence, by virtue of the condition t(1) = y>2(0) — w(1), k are determined uniquely.

After finding t(x) and v(x), the solution of the Problem NP1 defined in Q2 by formula Cauchy-Goursat (24) or Cauchy in [19] and in the domain Q1 we arrive to the problem for a non loaded equation of the third order [12].

Thus, we now know all particulars, such that the solution of Problem 2.1, in the domains and Q2 can be constructed from (5).

Problem 2.1 is uniquely solvable. Thus, Theorem 2.2 is proved. Analogously, one can also prove the uniqueness of the solution to Problem 2.2.

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