1. Introduction
Since the fifties of the last century, researchers have been intensively studying local and nonlocal problems for partial differential equations in simply connected regions. Currently, the range of such tasks is expanding in various directions. Among them, problems with displacement occupy a special place. This is explained by the fact that such problems cover various correct local boundary value problems, and many problems are reduced to them, for example, biological synergetics, genetics, immunology, transonic gas dynamics and thermal physics.
One of the most important classes of unloaded partial differential equations are second-order equations parabolo - hyperbolic and elliptic - hyperbolic types. An analogue of the Tricomi problem in a doubly connected domain for an equation of mixed type with one line of degeneracy was first posed and studied in A.V. Bitsadze [1, p.30], and for the Lavrentiev–Bitsadze equation - in the works of M. S. Salakhitdinov and A. K. Urinov [2].
Boundary value problems in classical domains for a loaded equation of hyperbolic type were studied in A.M.Nakhushev [3], V.M.Kaziev [4], A.Kh.Attaev [5], and the work of M.T.Dzhenaliev [6], M.Kh.Shkhankova [7] studied the local and nonlocal problem for a loaded equation of parabolic type. A.V. Borodin [8] studied the Dirichlet problem for a loaded equation of elliptic type. In the works of K.U.Khubiev [9], B.Islomov and D.M.Kuryazov [10], M.I.Ramazanov [11], analogues of the Tricomi and Gellerstedt problem for loaded hyperbolic-parabolic type equations were studied, and the Dirichlet problem for loaded equation with the Lavrentiev–Bitsadze operator in a rectangular domain were studied in the works of K.B. Sabitova and E.P. Melisheva [12-13]. Boundary value problems for an equation of hyperbolic type with a fractional derivative were also studied in works[30-33].
Local and non-local problems in simply connected domains for loaded equations of elliptic -hyperbolic and parabolic - hyperbolic types, when the loaded part contains a trace or derivative of the desired function, have been little studied. We note the works of B.I. Islomov and D.M. Kuryazov [14], B.I. Islomov and U.I. Boltaeva [15], [16], B.I. Islomov and Zh.A. Kholbekov [17], K.B. Sabitov [18], R.R. Ashurova and S.Z. Zhamalova [19], Yu.K. Sabitova [20], V.A. Eleeva [21]. This is due, first of all, to the lack of representation of a general solution for such equations; on the other hand, such problems are reduced to little-studied integral equations with a shift.
Currently, the range of such tasks is expanding in various special areas. However, boundary value problems for loaded equations of mixed type with an integral operator of fractional order in doubly connected domains have still not been sufficiently studied. Note that local and nonlocal boundary value problems for a loaded equation of elliptic -hyperbolic type in a doubly connected domain were studied in [22-25], in which the loaded part contains a differential operator or a trace of the desired function. The works [26-27] outline a technique for formulating correct boundary value problems with displacement for loaded second-order linear hyperbolic equations in special domains. It is shown that the correctness of such boundary value problems is significantly influenced by the loaded part of the equation under consideration.
Based on this, the present work is devoted to the formulation and study of a nonlocal boundary value problem for a loaded equation of parabolic - hyperbolic type in a special domain.
2. Statement of the problem
Let
is the area bounded by the segments
direct
,
at
.
is area limited by axis
segments
and at
characteristics
equations
emerging from points
and
intersecting at points
and
,
In equation (
1)
is given real numbers, and
Let us introduce the following notation:
Through
and
,
respectively, denote the characteristic triangle and
and quadrilaterals
,
.
characteristic intersection points
with a characteristic coming from the point
.
Task .Find a function that has the following properties:
1)
2)
and satisfies equation (
1) in the areas
and
3)
satisfies the boundary conditions:
where are
is the given functions, and
3. Investigation of the problem
For the study of the problem, the following tasks play an important role .
Task .Find a function that has the following properties:
1)
2)
and satisfies equation (
1) in the areas
and
3)
satisfies boundary conditions (
4) and (
5).
Theorem 1.
If conditions (2), (7)-(10) and
then Ω there is a unique regular solution to the problem in the region
Proof of Theorem 1 Solution of the Cauchy problem with conditions
for equation (
1) in the region
has the form:
Substituting (
13) at
in (
5) taking into account
we have
Differentiating (
14) with respect to
x, we obtain a functional relationship between
and
brought from the area
to
:
Similarly,
using the solution to the Cauchy problem (
13) with initial data (
12) for equation (
1) in the domain
taking into account (
3) and (
5), we obtain the functional relationship between
and
brought from the area
to
:
Consequently, as in [28 , pp. 40-47], due to conditions 1) - 2) of the problem
, passing to the limit at
in equation (
1) taking into account (
12), we obtain a functional relationship between
and
brought from the region
to
:
where
unknown is a constant to be determined.
Having excluded
from relations (
15), (
16) and (
17), after some calculations we obtain an integral equation for
:
where
By virtue of (
2) and (
9) from (
19) - (
22) it follows that
Thus, by virtue of (
23), (
24) and (
25), equation (
18) is a Volterra integral equation of the second kind. According to Volterra’s theory of integral equations [29, pp. 36-51], we conclude that the integral equation (
18) is uniquely solvable in the class
and its solution is given by the formula
where is
the resolvent of the kernel
Integrating (
26) from
x to 1 at
(from
to
x at
) taking into account
we have
Now putting in (
27) and (
28) respectively
and
taking into account
we find the unknown constant
:
By virtue of (
2), (
11), from (
19) and (
20) it follows that
Consequently, the resolvent of the kernel
is also negative, i.e.
This means that the denominator of formula (
29) for any
does not vanish, i.e.
.
By virtue of (
23) - (
25) from (
27) and (
28) taking into account (
29) we conclude that
Putting (
30) into (
15) and (
16) taking into account (
9), (
30), we define the function
from class
.
Thus, the solution to the problem
can be restored in the domain
as a solution to the first boundary value problem for equation (
1) [28, p. 99], and in the domains
as a solution to the Cauchy problem for equation (
1) (see (13)).
Therefore, the taskis uniquely solvable.
Theorem 1 is proven.
Now to restore the solution to the problem
in the regions
and
solve Goursat problems for equation (
1).
Task. Find function with the following properties:
1)
2)
is twice continuously differentiable solution of equation (
1) in region
3)
satisfies boundary conditions (
6) and
where
are determined from
here
are
are known functions and they are determined respectively from (
15), (
16), and (
27)-(
29).
belongs to the class
and
.
Task. Find a function with the following properties:
1) and the function can go to infinity of order less than one at the ends of the interval I;
2)
is twice continuously differentiable solution of equation (
1) in region
3)
satisfies the boundary conditions
where
are determined from
and belongs to the class
and
Here
is known functions and they are determined from (
6), (
12), (
27), (
28) and (
29) respectively.
Theorem 2.
Theorem 2. If conditions (2), (10), (30), (32) are met, then the solution to the problem
exists and is unique in the domain.
Proof of Theorem 2 The general solution to the equation (
1) in the domain
has the form [30, p. 77]:
where
are arbitrary twice continuously differentiable functions, and the function
is determined from (
27), (
28) and (
39).
Substituting (
35) into (
6) and (
31) , we find
Taking into account the properties of the functions
and
from (
36) it follows that the solution to the problem
exists, is unique and belongs to the class
So the task is uniquely solvable.
Theorem 2 is proven.
Theorem 3.
Theorem 3. If conditions (2), (34) are satisfied, then in the domain the solution to the problem
exists and is unique.
Proof of Theorem 3 To prove Theorem 3, the following problems play an important role :
Task Find in the region
a solution
to equation (
1) satisfying conditions (
33) and
where
is a given function, and
Let’s explore the problem. The general solution to the equation
in the region
has the form [15], [30, pp. 135-137]:
where
are arbitrary twice continuously differentiable functions, and the function
is determined from (
27), (
28) and (
29).
Then, substituting (
39) into (
33) and (
37), we find a solution to the problem
for
equation (
1) in the areas
and
:
and
respectively.
By virtue of (
30), (
34), (
38) from (
40) and (
41), we conclude that the solution to the problem
exists, is unique, and belongs to the class
.
Now let’s move on to exploring the problem . Differentiating (
40) and (
41) with respect to
x, then tending
x to zero, we obtain the functional relationship between
and
, brought from the region
and
to
I:
where
Eliminating the function from (
42)
taking (
43) into account, we obtain
Integrating (
45) from
up
y to taking into account
we have
By virtue of (
30), (
34) from (
46) it follows that
. Consequently, after defining the function,
the problem
is reduced to the study of problems
and
, where
is determined by formula (
46).
Hence we conclude that the task is uniquely solvable.
Theorem 3 has been proven.
This completes the study of the problem
for equation (
1).
4. Conclusions
This work is devoted to the formulation and study of a nonlocal boundary value problem for a loaded equation of parabolic-hyperbolic type in a special domain, as well as related results on the existence and behavior of solutions to the problem with integral gluing conditions. Thus, we can study various boundary value problems for loaded equations of generalized parabolic and hyperbolic types with fractional operators.
Author Contributions
Conceptualization, I.B. and A.A.; methodology, Y.O.; investigation, I.B. and A.A.; data curation, Y.O.; writing—original draft preparation, I.B.; writing—review and editing, I.B.; supervision, A.A.; project administration, A.A.; funding acquisition, Y.O. All authors have read and agreed to the published version of the manuscript.
Funding
Please add: This research was funded by the Ministry of Science and Higher Education of the Russian Federation within the framework of the state assignment No. 075-03-2022-010 dated 14 January 2022 (Additional agreement 075-03-2022-010/10 dated 09 November 2022, Additional agreement 075-03-2023-004/4 dated 22 May 2023).
Data Availability Statement
No new data were created or analyzed in this study. Data sharing is not applicable to this article.
Acknowledgments
The authors thank Professor N.I. Vatin for his expertise and assistance in writing the manuscript.
Conflicts of Interest
The authors declare no conflict of interest.
References
- Bitsadze, A.V. On the problem of equations of mixed type. Tr. MIAN USSR. 1953, 41, 3–59. https://www.mathnet.ru/rus/tm1177 (in Russian). [Google Scholar]
- Salakhitdinov M. S.; Urinov A. K. Nonlocal boundary value problem in a doubly connected domain for an equation of mixed type with nonsmooth lines of degeneracy. Dokl. Academy of Sciences of the USSR. 1988, 299(1), 63–66. https://www.mathnet.ru/rus/dan7748 (in Russian).
- Nakhushev A.M. Loaded equations and their applications. Differential equations 1983, 19(1), 86–94. https://www.mathnet.ru/rus/de4747 (in Russian).
- Kaziev V.M. On the Darboux problem for one loaded second-order integro-differential equation. Differential equations 1978, 14(1), 181–184. https://www.mathnet.ru/rus/de3294 (in Russian).
- Attaev A.Kh. Boundary value problems for the loaded wave equation. Differential equations: Abstract. report regional interuniversity seminar. May 20-25, 1984 Kuibishev. 1984, 9–10. https://mathematics-vestnik.ksu.kz/apart/2017-86-2/1.pdf (in Russian).
- Dzhenaliev N.T. On a boundary value problem for a linear loaded parabolic equation with nonlocal and boundary conditions. Differential equations 1991, 27(10), 1925–1927. https://www.mathnet.ru/rus/de7631 (in Russian).
- Shkhanukov M.Kh. Difference method for solving one loaded equation of parabolic type. Differential equations 1977, 13(1), 163-167. https://www.mathnet.ru/rus/de2979 (in Russian).
- Borodin A.B. About one estimate for elliptic equations and its application to loaded equations. Differential equations 1977, 13(1), 17–22. https://www.mathnet.ru/rus/de2959 (in Russian).
- Khubiev K.U. Gellerstedt problem for a loaded equation of mixed type with data on non-parallel characteristics. Dokl. Adyg. inter.academic.sciences. 2008, 1, 1–4. https://www.mathnet.ru/rus/mmkz925 (in Russian).
- Islomov B.; Kuryazov D.M. Boundary value problems for a mixed loaded equation of third order of parabolic-hyperbolic type. DUzbek mathematical journal. 2000, 2, 29–35. (in Russian).
- Ramazanov M.I. On a nonlocal problem for a loaded hyperbolic-elliptic type in a rectangular domain. Siberian mathematical journal. 2002, 2(4), 75–81. (in Russian).
- Sabitov K.B.; Melisheva E.P. The dirichlet problem for a loaded mixed-type equation in a rectangular domain. Russian Mathematics 2013, 57(7), 53–65. [CrossRef]
- Melisheva E.P. The Dirichlet problem for the loaded Lavrentiev–Bitsadze equation. Vestn. SamSU. Natural science ser. 2010, 6(80), 39–47. https://www.mathnet.ru/rus/vsgu192 (in Russian).
- Islomov B.; Kuryazov D.M. On a boundary value problem for a loaded second-order equation. Doclady ANRUz. 1996, 1(2), 3–6. (in Russian).
- Islomov B.; Baltaeva U.I. Boundary value problems for loaded differential equations of hyperbolic and mixed types of third order. Ufa Mathematical Journal. 2011, 3(3), 15–25. https://www.mathnet.ru/rus/ufa100.
- Islomov B.I.; Kholbekov Zh. On a nonlocal boundary-value problem for a loaded parabolic-hyperbolic equation with three lines of degeneracy. Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta, Seriya Fiziko-Matematicheskie Nauki 2021, 25(3), 407–422. [CrossRef]
- Sabitov K.B. Initial-boundary problem for a parabolic-hyperbolic equation with loaded terms. Russian Mathematics 2021, 59(6), 23–33. [CrossRef]
- Dzhamalov S.Z.; Ashurov R.R. On a nonlocal boundary-value problem for second kind second-order mixed type loaded equation in a rectangle. Uzbek Mathematical Journal 2018, 3, 63–72. [CrossRef]
- Sabitova Yu.K. Dirichlet problem for the Lavrentiev–Bitsadze equation with loaded terms. Russian Mathematics. 2018, 9, 42–58. http://kpfu.ru/science/nauchnye-izdaniya/ivrm/.
- Sabitova Yu.K. On some boundary value problems for mixed loaded equations of the second and third order. Differential equations. 1994, 30(2), 230–237. https://www.mathnet.ru/rus/de8291.
- Abdullayev O.Kh. About a method of research of the non-local problem for the loaded mixed type equation in double-connected domain. Bulletin KRASEC. Phys. & Math. Sci. 2014, 9(2), 3–12. [CrossRef]
- Abdullayev O.Kh. Boundary value problem for a loaded equation of elliptic-hyperbolic type in a doubly connected domain. Bulletin KRASEC. Phys. & Math. Sci. 2014, 1(8), 33–48. [CrossRef]
- Islomov B.I.; Abdullaev O.Kh. A boundary value problem of the type of the Bitsadze problem for a third-order equation of elliptic-hyperbolic type in a doubly connected domain. Reports of the Adyghe (Circassian) International Academy of Sciences 2004, 7(1), 42–46.[in Russian].
- Abdullayev O.Kh. Nonlocal problem for a loaded equation of mixed type with an integral operator. Russian mathematics. 2016, 20(2), 220–240. [CrossRef]
- Islomov B.I.; Yunusov O.M. A boundary value problem of the type of the Bitsadze problem for a third-order equation of elliptic-hyperbolic type in a doubly connected domain. Doclady ANRUz. 2016, 4, 8–12.[in Russian].
- Islomov B.I.; Yunusov O.M. A boundary value problem of the type of the Bitsadze problem for a third-order equation of elliptic-hyperbolic type in a doubly connected domain. Doclady ANRUz. 2016, 4, 8–12.[in Russian].
- Juraev T.D. Boundary value problems for equations of mixed and mixed-composite type, Publisher: Fan, Tashkent, Uzbekistan, 1979; 240 p.[in Russian].
- Mikhlin S.G. Lectures on linear integral equations, Publisher: Fizmatgiz, Moscow, Russia, 1959; 232 p. https://djvu.online/file/pSYEQg6jU8PRq.
- Tikhinov A.N.; Samarsky A.A. Equations of mathematical physics, Publisher: Nauka, Moscow, Russia, 1977; 736 p. https://djvu.online/file/XY0Zgytes78pt.
- Fedorov V.E., Turov M.M., Kien Bui Trong. A class of quasilinear equations with Riemann–Liouville derivatives and bounded operators. Axioms. 2022, 11(3), 96. [CrossRef]
- Kirianova L.V. The Boundary Value Problem with Stationary Inhomogeneities for a Hyperbolic-Type Equation with a Fractional Derivative. Axioms. 2022, 11, 207. [CrossRef]
- Beybalaev, V.D., Aliverdiev A.A., Hristov J. Transient Heat Conduction in a Semi-Infinite Domain with a Memory Effect: Analytical Solutions with a Robin Boundary Condition. Fractal and Fractional. 2023, 7(10), 770. [CrossRef]
- Fedorov V.E., Plekhanova M.V., Melekhina D.V. Nonlinear Inverse Problems for Equations with Dzhrbashyan–Nersesyan Derivatives. Fractal and Fractional. 2023, 7(6), 464. [CrossRef]
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