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International Journal of Partial Differential Equations
Volume 2013 (2013), Article ID 157947, 7 pages
Boundary Value Problems for the Classical and Mixed Integrodifferential Equations with Riemann-Liouville Operators
1National University of Uzbekistan, Tashkent, Uzbekistan
2Urgench State University, Urgench, Uzbekistan
Received 27 April 2013; Accepted 30 July 2013
Academic Editor: Michael Grinfeld
Copyright © 2013 B. Islomov and U. I. Baltaeva. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
By method of integral equations, unique solvability is proved for the solution of boundary value problems of loaded third-order integrodifferential equations with Riemann-Liouville operators.
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 because of 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 1920–1930 by Tricomi  and Gellerstedt . The works of M. A. Lavrent’ev, A. V. Bitsadze, F. I. Frankl, M. Protter, and C. Morawetz have had a great impact on this theory, where outstanding theoretical results were obtained and pointed out important practical values. Bibliography of the main fundamental results on this direction can be found, among others, in the monographs of Bitsadze , Bers , Salakhitdinov and Urinov , and Nakhushev .
In the recent years, in connection with intensive research on problems of optimal control of the agroeconomical 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 “loaded equations.” Such equations were investigated for the first time in works of N. N. Nazarov and N. Kochin. However, they did not use the term “loaded equation.” For the first time, the term has been used in works of Nakhushev , where the most general definition of a loaded equation is given and various loaded equations are classified in detail, for example, loaded differential, integral, integrodifferential, functional equations and so forth, and numerous applications are described [6, 8].
Basic questions of the theory of boundary value problems for partial differential equations are the same for the boundary value problems for the loaded differential equations. However, existence of the loaded operator does not always make it possible to apply directly the known theory of boundary value problems.
Works of Nakhushev, M. Kh. Shkhankov, A. B. Borodin, V. M. Kaziev, A. Kh. Attaev, C. C. Pomraning, E. W. Larsen, V. A. Eleev, M. T. Dzhenaliev, J. Wiener, B. Islomov and D. M. Kuriazov, K. U. Khubiev, and M. I. Ramazanov et al. are devoted to loaded second-order partial differential equations.
It should be noted that boundary value problems for loaded equations of a hyperbolic, parabolic-hyperbolic, and elliptic-hyperbolic types of the third order are not well understood. We indicate only the works of V. A. Eleev, Islomov, D. M. Kur’yazov, and A. V. Dzarakhokhov.
The present paper is devoted to formulation and investigation of the analogue of the Cauchy-Goursat problem for the loaded equation of a hyperbolic type and a boundary value problem for a loaded equation of a mixed parabolic hyperbolic type where . Assume that and coefficients , , are given real parameters, and .
2. Analogue of the Cauchy-Goursat Problem for a Loaded Equation of the Hyperbolic Type
Let be a domain bounded at by the characteristics of (1) and the segment of the axis .
Let us consider the following analogue of the Cauchy-Goursat problem for the loaded equation (1) in the domain .
Problem A. Find a solution to (1), which is regular in the domain continuous in , has continuous derivatives , , up to , and satisfies the boundary value conditions where is an inner normal and , , are real-valued functions.
Theorem 1. If and then there exists a unique solution to the problem in the domain .
Proof of Theorem 1. An important role in proving Theorem 1 is played by the following.
Lemma 2. Any regular solution to (1) is represented in the formwhere is the solution of the equationand —is the solution of the following ordinary differential equation:
Proof of Lemma 2. Let , represented by Formula (7), be the solution of (1). Then, substituting satisfies (1).
Then, vice versa, let be a regular solution to (1) and let be a certain solution equation
Let us prove the validity of relation (7). Manifestly, the function is a solution to (1), where is a solution to (8) and the function is a partial solution to (1). Hence, (1) entails the validity of representation (7); that is, .
It follows from the latter representation that . Then, (11) provides and the function satisfies (8).
Lemma 2 is proved.
Invoking that the function satisfies (8), we can assume without loss of generality that when studying problem A.
Let us solve the Cauchy problem for (9) with conditions (15) with respect to .
The solution to the Cauchy problem for (9) with conditions (15) has the form where
The last equality with respect to designation and after some transformation becomes where
And with recurring index implied summation. Solving the next equation with respect to  and Dirichlet’s formula we have where
By virtue of representation (7), problem A is reduced to problem A* of finding a solution of (8), which is regular in the domain conditions where is defined by (20).
Similarly to [9, 10], we can write out the solution to (8) with conditions (22) by means of the general representation in view of (6) and : where is the Riemann-Hadamard function  and, is the modified Bessel function .
Assuming that in (23) and invoking (20), we obtain the following functional correlation, transferred from the domain onto : where , , are are the modified Bessel functions .
We can assume without loss of generality that when and when .(1)Let , and then for any function applying the Dirichlet permutation integration formula from (25) we have (2)Let , and then taking account of conversion  into (25) we get
Hence, we conclude that the integral equations (28), (29) with respect to (6),  always have a solution, which is unique .
Thus, it is proved that problem A is uniquely solvable. Theorem 1 is proved.
3. Investigation of Problem C for (2)
Formulation of Problem C for (2). Let be a domain bounded by the segments , , , and of the straight lines , , , and , respectively, when . is a characteristic triangle bounded by the segment of the axis and two characteristics of (2) for .
Let us introduce the following notation:
Problem C. Find the function , possessing the following properties:(1); (2) is continuous up to;(3) is a regular solution of (2) in the domains and ;(4)the sewing conditions are satisfied on ;(5) satisfies the boundary value conditions where is the inner normal and , , , , and are given functions.
Theorem 3. If and then there exists a unique solution to the problem C in the domain .
Proof of Theorem 3. The following theorem holds.
Lemma 4. Any regular solution of (2) (when ) is represented in the formwhere is a solution to the equation is a solution of the following ordinary differential equation:
The lemma is proved similarly to Lemma 2.
Invoking that the function satisfies (37), we can subordinate the function to the conditions
Solution of the Cauchy problem for (38) with the conditions (39) can be represented in the form (20).
By virtue of representation (36), (2) and the boundary value conditions (33), in view of (39), are reduced to the form (37):
Derivation of Basic Functional Relations. As it is known from problem A, the solution to (37) with the boundary value conditions (41), (42), and is given by the formula (23).
Assuming that in (23), in view of (39), and we obtain the functional relation, transferred from the domain onto : where where can be represented in the form (26).
Denoting from (44) and using the inversion formula for such equations  in view of (35) and (45), we obtain with respect to in the form
Due to the property of Problem C and in view of (43), (44), we obtain  from (37) in , tending . Here is an unknown constant to be defined.
The equality (50) is a second functional relation between and , transferred from the domain to .
Existence of Solution to Problem C. Solving (50) with respect to with the conditions we have
Omitting the function , in (49) and (52), in view of the sewing condition, we obtain an integral equation with a shift with respect to : where
Assuming that we write (53) in the form
Hence, (56) is an integral Volterra equation of the second kind, which is unconditionally and uniquely solvable in class . Thus, the solution of (56) has the from where is the resolvent of the kernel .
In view of (55) and the Dirichlet formula, (57) has where
Hence, we conclude that (58) always has a solution that is unique and can be represented in the form  where is the resolvent of the kernel .
Hence, by virtue of the condition , are determined uniquely. Upon determining , we find the functions and from (49) and (20).
Thus, the solution of problem C in the domain in view of (20) and (23) is determined uniquely according to the formula (36), and in the domain we arrive to the problem for an nonloaded equation of the third order . Thus, the solution of problem B in the domains and can be constructed from (36) in view of (20), (23), and Problem .
Thus, problem C is uniquely solvable. Theorem 3 is proved.
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