Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 221, pp. 1–10.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
BOUNDARY-VALUE PROBLEMS FOR A THIRD-ORDER
LOADED PARABOLIC-HYPERBOLIC EQUATION WITH
VARIABLE COEFFICIENTS
BOZOR ISLOMOV, UMIDA I. BALTAEVA
Abstract. We prove the unique solvability of a boundary-value problems for
a third-order loaded integro-differential equation with variable coefficients, by
reducing the equation to a Volterra integral equation.
1. Introduction
The theory of mixed type equations is one of the principal parts of the general
theory of partial differential equations. The interest for these kinds of equations
arises intensively due to both theoretical and practical uses of their applications.
Many mathematical models of applied problems require investigations of this type
of equations. The first fundamental results in this direction were obtained in 1923
by Tricomi. The works of Gellerstedt, Lavrent’ev, Bitsadze, Frankl, Protter and
Morawetz, Salakhitdinov, Djuraev, Rassias have had a great impact in this theory,
where outstanding theoretical results were obtained and pointed out important
practical values.
Currently, the concept of mixed-type equations has expanded to include all possible combinations of two or three classic types of equations.
The necessity of the consideration of the parabolic-hyperbolic type equation was
specified for the first time in 1956 by Gel’fand [8]. He gave an example connected
to the movement of the gas in a channel surrounded by a porous environment. The
movement of the gas inside the channel was described by the equation, outside by
the diffusion equation [2, 3, 15, 18].
A systematic study of the third and higher order mixed and mixed-composite
type PDEs, containing in the main part parabolic-hyperbolic, hyperbolic-elliptic
and elliptic-parabolic operators began in the early seventies and intensively developed by many mathematicians [4, 5, 16, 17].
In the recent years, in connection with intensive research on problems of optimal
control of the agro economical system, long-term forecasting and regulating the level
of ground waters and soil moisture, it has become necessary to investigate a new
class of equations called as “loaded equations”. Such equations were investigated
2010 Mathematics Subject Classification. 35M10.
Key words and phrases. Equations of mixed type; loaded equation; gluing condition;
boundary-value problem; integral equation.
c
2015
Texas State University - San Marcos.
Submitted February 25, 2015. Published August 25, 2015.
1
2
B. ISLOMOV, U. I. BALTAEVA
EJDE-2015/221
for the first time by Knezer [10], Lichtenstein [11]. However, they did not use
the term “loaded equation”. This terminology has been introduced by Nakhushev
[12], where the most general definition of a loaded equation is given and various
loaded equations are classified in detail, e.g., loaded differential, integral, integrodifferential, functional equations etc., and numerous applications are described [20,
13].
Definition 1.1. An equation
Au(x) = f (x)
(1.1)
is called loaded in an n-dimensional Euclidean domain Ω if (part of) the operator A
depends on the restriction of the unknown function u(x) to a closed subset of Ω, of
measure strictly less than n.
Definition 1.2. A loaded equation is called a loaded differential equation in the
domain Ω ⊆ Rn if it contains at least one derivative of the unknown solution in a
subset of Ω of nonzero measure.
Basic questions of the theory of boundary value problems for PDEs are the same
for the boundary value problems for the loaded differential equations. However, the
existence of the loaded part operator A does not always make it possible to apply
directly the known theory of boundary value problems for equations
L(x) = f (x).
On the other hand, searching for solutions of loaded differential equation preassigned classes it might reduce to new problems for non-loaded equations.
Works of Nakhushev, Shkhankov, Borodin, Borok, Kaziev, Pomraning, Larsen,
Pul‘kina, Eleev, Dzhenaliev, Attaev, Wiener, Islomov, Khubiev et al. are devoted to
loaded second-order partial differential equations. However, we would like to note
that boundary-value problems for third-order loaded equations of a hyperbolic,
parabolic-hyperbolic, elliptic-hyperbolic types are not well studied. We indicate
only the works [6, 7, 19] in which study-case, when loaded part contain only track
or derivative track from unknown solutions. It can be explained with the absence
of the representation of the general solution for such equations; on the other hand,
these problems will be reduced to integral equations with stir [1], which are not
investigated in detail.
2. Formulating of the problem
Let Ω be a simple connected domain located in the plane of independent variables x and y, in the case y > 0, is bounded by the segments AA0 , BB0 , and
A0 B0 (A(0, 0), B(1, 0), A0 (0, h), B0 (1, h)), of the straight lines x = 0, x = 1, and
y = h, respectively, and in the case y < 0, with segments AC : x + y = 0,
BC : η = x − y = 1 originating at the point C(1/2, −1/2).
We use the following designation:
I = (x, y) : 0 < x < 1, y = 0 , Ω1 = Ω ∩ {y > 0}, Ω2 = Ω ∪ {y < 0}.
We consider a linear loaded integro-differential equation
∂
a
+ c Lu = 0,
∂x
(2.1)
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BOUNDARY-VALUE PROBLEMS
3
where
Lu ≡


L u ≡ u + a (x, y)ux + b1 (x, y)uy + c1 (x, y)u

 1Pn xx α 1

i
−
i=1 di D0x u(x, 0),
if y > 0,


L2 u ≡ uxx − uyy + a2 (x, y)ux + b2 (x, y)uy + c2 (x, y)u


 Pn
βi
− i=1 ei D0ξ
u(ξ, 0),
if y 6 0,
where a, c are given real parameters, ai , bi , ci , di , ei are given functions on Ωi
(i = 1, 2), and b1 (x, y) < 0, c1 (x, y) 6 0 on Ω̄1 ; moreover the functions a1 , b1 , c1 ,
di , a1x , a1y , b1x , b1y , dix , diy on Ω1 satisfy a Hölder condition, and a2 , b2 ∈ C 2 (Ω̄2 ),
αi
c2 ∈ C 1 (Ω2 ), ei ∈ C 1 (Ω̄i ). D0x
is integro-differential operator (in the sense of
Riemann-Liouville), αi , βi < 1, i = 1, . . . , n.
For equation (2.1) we investigate the following problems (a 6= 0).
Problem 2.1. Find a function u(x, y) possessing the following properties:
(1) u(x, y) ∈ C(Ω̄) ∩ C 1 (Ω);
(2) ux (uy ) is continuous up to AA0 ∪ AC, (AC);
(3) u(x, y) is a regular solution of equation (2.1) in the domains Ω1 and Ω2 ;
(4) u(x, y) satisfies the boundary value conditions
u(x, y)AA0 = ϕ1 (y), u(x, y)BB0 = ϕ2 (y),
(2.2)
ux (x, y)AA0 = ϕ3 (y), 0 6 y 6 h,
1
(2.3)
u(x, y)AC = ψ1 (x), 0 6 x 6 ,
2
1
∂u(x, y) = ψ2 (x), 0 6 x 6 ,
(2.4)
AC
∂n
2
where n is an inner normal, ϕ1 (y), ϕ2 (y), ϕ3 (y), ψ1 (x) and
√ ψ2 (x) are given
real-valued functions, moreover ϕ1 (0) = ψ1 (0), ψ10 (0) = 2ψ2 (0) − 2ϕ01 (0).
Problem 2.2. Find a function u(x, y), satisfying the following conditions:
(1) u(x, y) ∈ C(Ω̄) ∩ C 1 (Ω);
(2) ux (uy ) is continuous up to AA0 ∪ BC, (BC);
(3) u(x, y) is a regular solution of equation (2.1) in the domains Ω1 and Ω2 ;
(4) u(x, y) satisfies the boundary value conditions (2.2) and
1
u(x, y)BC = ψ3 (x),
6 x 6 1,
(2.5)
2
∂u(x, y) 1
= ψ4 (x),
6 x 6 1,
(2.6)
BC
∂n
2
where n is an inner normal, ϕ1 (y), ϕ2 (y), ϕ3 (y), ψ3 (x) and ψ4 (x) are given
real-valued functions, moreover ϕ2 (0) = ψ3 (0).
3. Main results
From condition (1) problems 2.1 and 2.2 it follows that
u(x, +0) = u(x, −0) = τ (x),
(3.1)
uy (x, +0) = uy (x, −0) = ν(x),
(3.2)
0
ux (x, +0) = ux (x, −0) = τ (x),
(3.3)
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B. ISLOMOV, U. I. BALTAEVA
EJDE-2015/221
where τ (x) and ν(x), are still unknown functions. Assuming that
(
u1 (x, y), (x, y) ∈ Ω̄1 ,
u(x, y) =
u2 (x, y), (x, y) ∈ Ω̄2 ,
equation (2.1) can be represented by two systems:
L1 u1 +
n
X
αi
di D0x
u1 (x, 0) = υ1 (x, y),
(x, y) ∈ Ω̄1 ,
i=1
(3.4)
aυ1x + cυ1 = 0,
L2 u2 +
n
X
βi
ei D0ξ
u2 (ξ, 0) = υ2 (x, y),
(x, y) ∈ Ω̄2 ,
i=1
(3.5)
aυ2x + cυ2 = 0,
where υ1 (x, y), υ2 (x, y) are continuous differentiable functions.
Theorem 3.1. If b1 (x, y) < 0, c1 (x, y) ≤ 0 and ai (x, y) ≥ 0 for all (x, y) ∈ Ωi ,
ϕi (y) ∈ C 1 [0, h], (i = 1, 2),
1
3
ψ1 (x) ∈ C [0, 1/2] ∩ C (0, 1/2),
ϕ3 (y) ∈ C[0, h] ∩ C 1 (0, h),
2
ψ2 (x) ∈ C[0, 1/2] ∩ C (0, 1/2),
(3.6)
(3.7)
then there exists a unique solution to the problem 2.1 in the domain Ω.
Theorem 3.2. If b1 (x, y) < 0, c1 (x, y) ≤ 0 and ai (x, y) ≥ 0 for all (x, y) ∈ Ωi ,
condition (3.6) is satisfied and
ψ3 (x) ∈ C 1 [1/2, 1] ∩ C 3 (1/2, 1),
ψ4 (x) ∈ C[1/2, 1] ∩ C 2 (1/2, 1),
(3.8)
then there exists a unique solution to the problem 2.2 in the domain Ω.
Proof of Theorem 3.1. Bearing in mind [5] that system (3.5) is reduced to the form
L2 u2 +
n
X
c βi
ei D0ξ
u2 (ξ, 0) = w2 (y) exp − x .
a
i=1
(3.9)
Hence going over to the characteristic coordinates ξ = x + y, η = x − y, we obtain
u2ξη + a3 (ξ, η)u2ξ + b3 (ξ, η)u2η + c3 (ξ, η)u2
(3.10)
1
ξ − η
c
βi
= Ei (ξ, η)D0ξ
τ + ω2
exp − (ξ + η) ,
4
2
2a
where a3 (ξ, η), b3 (ξ, η), c3 (ξ, η) depend on the coefficients of equation (3.9),
1 ξ + η η − ξ
Ei (ξ, η) = ei
,
,
4
2
2
with recurring index i = 1, 2, . . . , n implied summation. The boundary value conditions (2.3) and (2.4) is reduced to the form
η
u2 (ξ, η)ξ=0 = ψ1
, 0 6 η 6 1,
(3.11)
2
and
∂u2 (ξ, η) 1
η
= √ ψ2
, 0 < η < 1.
(3.12)
ξ=0
∂ξ
2
2
The solution of the equation (3.10), with boundary conditions (3.11) and
(u2ξ − u2η )
= ν(ξ), 0 < ξ < 1,
(3.13)
η=ξ
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BOUNDARY-VALUE PROBLEMS
5
(problem Cauchy-Goursat), is represented analogously as [14]
Z
Z η
1 ξ
c
t−τ
u2 (ξ, η) = F (ξ, η) +
dt
T (t, τ ; ξ, η) exp − (t + τ ) ω2
dτ
4 0
2a
2
t
Z ξ
+
T0 (ξ, η; t)ν(t)dt
0
Z ξ
Z
c
1 ξ
t−τ
S(t, τ ; ξ, η) exp − (t + τ ) ω2
dt
+
dτ
4 0
2a
2
t
Z ξ Z ξ
βi
τ (t)S(t, τ ; ξ, η)dτ
+
dt
Ei (t, τ )D0t
0
Z
(3.14)
t
ξ
η
Z
βi
Ei (t, τ )D0t
τ (t)T (t, τ ; ξ, η)dτ,
dt
+
t
0
where
η
ξ
F (ξ, η) = ψ1 ( ) + ψ1 ( ) − ψ1 (0)
2
2
Z ξ Z ξ
Z
+
dt
K(t, τ )S(t, τ ; ξ, η)dτ +
0
t
ξ
Z
dt
0
η
K(t, τ )T (t, τ ; ξ, η)dτ,
t
1
ξ
1
η
ξ
η
K(ξ, η) = − a3 (ξ, η)ψ10 ( ) − b3 (ξ, η)ψ10 ( ) − c3 (ξ, η) ψ1 ( ) + ψ1 ( ) − ψ1 (0) ,
2
2
2
2
2
2
ξ
Z
T0 (ξ, η; t) = 1 −
Z
a3 (t, τ ) S(t, τ ; ξ, η)dτ −
t
Z
Z
ξ
ds
t
Z
−
ξ
c3 (s, τ )S(s, τ ; ξ, η)dτ
s
Z η
ds
0
a3 (t, τ ) T (t, τ ; ξ, η)dτ
t
ξ
−
ξ
c3 (s, τ )T (s, τ ; ξ, η)dτ,
t
where S(t, τ ; ξ, η) and T (t, τ ; ξ, η) are expressed via coefficients a3 , b3 , c3 and continuous in Ω̄2 × Ω̄2 functions Sξ , Sη , Tη are continuous in Ω̄2 × Ω̄2 , and function
Tξ it can have discontinuities of the first kind on compact subsets of this domains.
More properties of these functions are established in [14].
Substituting (3.14) √
in (3.12), taking into
account that ν(0) = u2η (0, 0) = u1η (0, 0) =
1
0
0
0
ϕ1 (0) and ϕ1 (0) = 2 2ψ2 (0) − ψ1 (0) , we obtain
Z η
c τ
T (0, τ ; 0, η) exp − τ ω2 −
dτ
2a
2
0
Z
η
√
η
(3.15)
= 2 2ψ2 ( ) − 2ψ10 (0) − 4
K(0, τ )T (0, τ ; 0, η)dτ
2
0
Z
η
− 4ϕ01 (0) 1 −
a3 (0, τ )T (0, τ ; 0, η)dτ .
0
From here with regard (3.7), differentiating (3.15) with respect to η, reduction in
this integral equation of the second kind
Z η
c
τ
η
(η − τ ) ω2 −
dτ = g (η),
(3.16)
ω2 −
−
Tη (0, τ ; 0, η) exp
2
2a
2
0
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B. ISLOMOV, U. I. BALTAEVA
g (η) =
EJDE-2015/221
√
Z η
η
K(0, τ )Tη (0, τ ; 0, η)dτ
2 ψ20 ( ) − 4K (0, η) − 4
2
0
Z
η
c + 4 ϕ01 (0) a3 (0, η) +
a3 (0, τ )Tη (0, τ ; 0, η) dτ exp
η .
2a
0
From (3.7), we conclude that the kernel and g(η) are continuous. Then it leads
to a unique solutions in the class of continuous functions. Solving this, weobtain
.
ω2 − η2 in − 21 6 − η2 6 0. Therefore in instead of ω2 − η2 we can take ω2 ξ−η
2
Substituting in (3.14) the expression ω2 ξ−η
we find the solution u2 (ξ, η) in the
2
form
Z
ξ
u2 (ξ, η) = M (ξ, η) +
T0 (ξ, η; t)ν(t)dt
0
Z
ξ
+
0
Z
+
0
Z
dt
ξ
βi
τ (t)S(t, τ ; ξ, η)dτ
Ei (t, τ )D0t
(3.17)
t
ξ
Z
dt
η
βi
Ei (t, τ )D0t
τ (t)T (t, τ ; ξ, η)dτ,
t
where
Z
Z ξ
c
1 ξ
t−τ
)dτ
dt
S(t, τ ; ξ, η) exp − (t + τ ) ω2
4 0
2a
2
t
Z
Z η
c
1 ξ
t−τ
+
dt
T (t, τ ; ξ, η) exp − (t + τ ) ω2
)dτ ,
4 0
2a
2
t
M (ξ, η) = F (ξ, η) +
depend on a given function.
In η = ξ = x, setting M (x) = M (x, x), T0 (x, t) = T0 (x, x; t), τ (x) = u2 (x, x),
from (3.14) we obtain
Z x
Z x
Z x
βi
τ (x) = M (x) +
T0 (x, t)ν(t)dt +
D0t
τ (t)dt
Ei (t, τ )S(t, τ ; x, x)dτ
0
0
t
Z x
Z x
βi
+
τ (t)dt
D0t
Ei (t, τ )T (t, τ ; x, x)dτ.
0
t
Differentiating the last relation, obtain integral equation second kind relative to
ν(x):
Z x
Z x
0
βi
ν(x) +
Tox
(x, t)ν(t)ds = τ 0 (x)−
L(x, t)D0t
τ (t)dt − M 0 (x),
(3.18)
0
0
where
L(x, t) = Ei (t, x) (S(t, x; x, x) − T (t, x; x, x))
Z x
+
Ei (t, τ ) (S 0 (t, τ ; x, x) + T 0 (t, τ ; x, x)) dτ.
t
The right-hand side equation (3.18) is continuous and kernel can be discontinuous
of the first kind. Therefore ν(x):
Z x
0
βi
ν(x) = τ (x) −
L(x, t)D0t
τ (t)dt − M 0 (x)
0
(3.19)
Z x
Z t
βi
0
0
+
Γ0 (x, t) M (t) − τ (t) +
L(t, s)D0s τ (s)ds dt,
0
0
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BOUNDARY-VALUE PROBLEMS
7
0
where Γ0 (x, t) is the resolvent of the kernel T0x
(x, t). This is the first functional
relation between the function τ (x) and ν(x) transferred from the Ω2 . Present we
need obtain second functional relation between this functions. To this end equation
(2.1) at y > 0 rewrite in the form
L1 u1 ≡ u1xx + a1 u1x + b1 u1y + c1 u1 +
n
X
c αi
u1 (x, 0) = w1 (y) exp − x ,
di D0x
a
i=1
where w1 (y) is arbitrary continuous functions. Hence, considering property of the
problem 2.1, in b1 = −1, passage to the limit, we obtain second functional relation
between the function τ (x) and ν(x) transferred from the Ω1 :
τ 00 (x) + a1 (x, 0)τ 0 (x) + c1 (x, 0)τ (x) −
n
X
c α
dj D0xj τ (x) − ν(x) = ω1 (0) exp − x .
a
j=1
(3.20)
Substituting (3.19) in (3.20), results
00
0
τ (x) + p(x)τ (x) + q(x)τ (x) −
n
X
α
dj D0xj τ (x)
j=1
Z
x
Z
x
(3.21)
βi
Γ0 (x, t)τ 0 (t)dt +
Γ1 (x, t)D0t
τ (t)dt
0
0
c = ω1 (0) exp − x + m(x),
a
+
where
p(x) = a1 (x, 0) − 1, q(x) = c1 (x, 0),
Z x
Γ1 (x, t) = L(x, t) −
Γ0 (x, s) L (s, t) ds,
t
Z x
m(x) =
Γ0 (x, t)M 0 (t)dt − M 0 (x).
0
Solve (3.21) under the initial condition
τ (0) = ϕ1 (0) = ψ1 (0),
τ 0 (0) =
√
2ψ2 (0) − ϕ01 (0).
Introduce new unknown function τ 00 (x) = z(x). Then with regards the next conditions we have
Z x
√
τ 0 (x) =
z(t)dt + 2ψ2 (0) − ϕ01 (0),
0
Z x
√
τ (x) =
(x − t) z(t)dt +
2ψ2 (0) − ϕ01 (0) x + ψ1 (0).
0
Bearing mind this, we rewrite equation (3.21) in form
Z x
c z(x) +
Q(x, t; αj , βi )z(t)dt = ω1 (0) exp − x + M (x),
a
0
(3.22)
where
Z
Q(x, t; αj , βi ) = p(x) + q(x) (x − t) − Q1 (x, t; αj ) + Q2 (x, t; βi ) +
x
Γ0 (x, s)ds,
t
8
B. ISLOMOV, U. I. BALTAEVA
EJDE-2015/221
P
 n dj B(2;−αj ) (x − t)1−αj ,
αj < 0,
j=1 Γ(−α )
Q1 (x, t; αj ) = Pn d (2−α j)B(2;1−α )
1−αj
j
j
j

(x − t)
, 0 < αj < 1,
j=1
Γ(1−αj )

R
 B(2;−βi ) x Γ1 (x, s) (s − t)1−βi ds, βi < 0,
Γ(−βi ) t
Q2 (x, t; βi ) = B(2;1−β
Rx
2−βi
i)

ds, 0 < βi < 1.
Γ(1−βi ) t Γ1 (x, s) (s − t)
where B is the Beta function and Γ(z) is the Gamma function.
The Kernel and the right-hand side of (3.22) are continuous. Therefore, z(x) ∈
C[0, 1]. Solving we find z(x):
Z x
z(x) = M (x) +
R(x, t; αj , βi )M (t)dt
0
c Z x
c + ω1 (0) exp − x +
R(x, t; αj , βi ) exp − t dt ,
a
a
0
where R(x, t; αj , βi ) is a resolvent of the kernel Q(x, t; αj , βi ). Taking into account
the last equality, we obtain
Z x
c τ (x) = ω1 (0)
(x − t) exp − t
a
0
(3.23)
Z t
c +
R (t, s; αj , βi ) exp − s ds dt + M1 (x),
a
0
where
x
Z
Z
(x − t) M (t) +
M1 (x) =
0
+
√
2ψ2 (0) −
t
R (t, s; αj , βi ) M (s) ds dt
0
x + ψ1 (x).
ϕ01 (0)
Hence, by the condition τ (1) = ϕ2 (0), w1 (0) are determined uniquely. Thus, from
function τ (x) using relation (3.20) we uniquely define function ν(x). Set value
function τ (x) and ν(x) in (3.14), we obtain function u2 (ξ, η) in domain Ω2 . For
determination function u1 (x, y) in domain Ω1 reduce to problem 2.2 and
u1 (x, 0) = τ (x),
for the equation
∂
+ c (u1xx + a1 (x, y)u1x + b1 (x, y)u1y + c1 (x, y)u1 ) = F (x, y),
(3.24)
∂x
Pn
αi
∂
where F (x, y) = a ∂x
+c
i=1 di D0x τ (x) is a well-known function. Unique solvability this problem was proved in [5, §2, chapter 4]. We can conclude from these
that, there exists a regular solution of problem in Ω1 . Therefore, we can conclude
from these that, there exists a regular solution of problem 2.1.
a
Proof of Theorem 3.2. The proof for Problem 2.2 is analogous to the proof for
Problem 2.1. We omit it.
Remark 3.3. For problems 2.1 and 2.2 it is possible examine with general discontinuous gluing conditions. In this case 1, problems 2.1 and 2.2, change in the
following way: function u(x, y) is continuous in each closed domains Ω̄1 and Ω̄2 ,
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BOUNDARY-VALUE PROBLEMS
9
conditions (2), (3) and (4) it remains invariant. Indeed, the following conditions
are fulfilled:
u(x, +0) = α1 (x) u(x, −0) + γ1 (x),
0 < x < 1,
uy (x, +0) = β1 (x) uy (x, −0) + α2 (x) uy (x, −0) + γ2 (x),
0 < x < 1,
where α1 , γ1 ∈ C 3 , α2 , β1 , γ2 ∈ C 2 are given functions, and α1 β1 6= 0 for 0 < x < 1.
For problems 2.1 and 2.2, in this case there exist also unique solutions.
Acknowledgements. The authors gratefully acknowledge Professor Ingo Witt for
his advice and suggestions.
References
[1] Baltaeva, U. I.; Solvability of the analogs of the problem Tricomi for the mixed type loaded
equations with parabolic-hyperbolic operators, Boundary Value Problems, 2014:211 (2014),
pp. 1-12.
[2] Berdyshev, A. S.; Cabada, A.; Karimov, E. T.; On a non-local boundary problem for a
parabolic-hyperbolic equation involving a Riemann-Liouville fractional differential operator,
Nonlinear Analysis, 75 (2012), pp. 3268-3273.
[3] Chen, S. X.; Mixed type equations in gas dynamics, Quart.Appl. Math. LVXIII (3) (2010),
pp. 487-511.
[4] Djuraev, T. D.; Boundary Value Problems for Equations of Mixed and Mixed-Composite
Types, Fan, Tashkent, 1979, 240 p.
[5] Djuraev, T. D.; Sopuev, A.; Mamajonov, M.; Boundary value problems for the problems for
the parabolic-hyperbolic type equations, Fan, Tashkent, 1986.
[6] Eleev, V. A.; Boundary value problems for mixed loaded equations second and third orders,
Vol. 30, No. 2(1994), pp. 230-237.
[7] Eleev, V. A.; Dzarakhokhov, A. V.; On a nonlocal boundary value problem for a third-order
loaded equation. Vladikavkaz. Mat. Zh. 6 (2004), no. 3, pp. 36-46.
[8] Gel‘fand, I. M.; Some questions of analysis and differential equations, Uspekhi Mat. Nauk
Ser. 3 (87), Vol. 14, (1959), pp. 3-19.
[9] Islomov, B.; Baltaeva, U.I.; The boundary value problems for the loaded differential equations hyperbolic and parabolic-hyperbolic types third order, Ufa mathematical journal, Vol. 3,
3(2011), pp. 15-25.
[10] Kneser, A.; Belastete Integralgleihungen Rendiconti del Circolo Mathematiko di Palermo
38(1914), pp. 169-197.
[11] Lichtenstein, L.; Vorlesungen uber einege Klassen nichtlinear Integral gleichungen und Integral differential gleihungen nebst. Anwendungen. Berlin: Springer, 1931.
[12] Nakhushev, A. M.; On the Darboux problem for a degenerate loaded integral differential
equation of the second order, Differential equations, -Vol. 12, No. 1(1976), pp. 103–108.
[13] Nakhushev, A .M.; Equations of mathematical biology, Vishaya shkola, Moscow, 1995, 302 p.
[14] Pulkin, S. P.; Integral representation of the solution of the Cauchy-Goursat, Scientific notes
Kuibyshev KPI physical and mathematical sciences, Vol.29 (1959), pp. 25-40.
[15] Rassias, J. M.; Mixed type partial differential equations with initial and boundary values in
fluid mechanics, Int.J. Appl.Math. Stat. 13(J08)(2008), pp. 77-107.
[16] Sabitov, K. B.; Boundary value problem for a third-order equation of mixed type in a rectangular domain, Differential equations, 49, No. 2(2013), pp. 187-197.
[17] Salakhitdinov, M. S.; Equation of a mixed composite type, Fan, Tashkent, 1974, 156 p.
[18] Salakhitdinov, M. S.; Urinov, A. K.; Boundary value problems for equations of mixed type
with a spectral parameter, Fan, Tashkent, 1997.
[19] Vodakhova, V. A., Guchaeva, Z. K.; Nonlokal problem for the loaded equation of the third
order with multiple characteristics, Advances in current natural sciences, no 7(2014), pp.
90-92.
[20] Wiener, J.; Debnath, L.; Partial differential equations with piecewise constant delay, Internat.
J. Math. and Math. Scz. 14 (1991), pp. 485-496.
10
B. ISLOMOV, U. I. BALTAEVA
EJDE-2015/221
Bozor Islomov
National University of Uzbekistan, 100174, Tashkent, Uzbekistan
E-mail address: islomovbozor@yandex.ru
http://math.nuu.uz/en/islomov-bozor-islomovich
Umida Ismoilovna Baltaeva
Mathematical Institute University of Göttingen, 37073 Göttingen, Germany
E-mail address: umida baltayeva@mail.ru