Boundary Value Problems for the Classical and Mixed 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 [1] and Gellerstedt [2]. 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 [3], Bers [4], Salakhitdinov and Urinov [5], and Nakhushev [6]. 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 [7], 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


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 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 [1] and Gellerstedt [2].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 [3], Bers [4], Salakhitdinov and Urinov [5], and Nakhushev [6].
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 [7], 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 where (3) 3 (0, 1), ,  are given real parameters, and  > 0.

Analogue of the Cauchy-Goursat Problem for a Loaded Equation of the Hyperbolic Type
Let  ⊂  2 be a domain bounded at  < 0 by the characteristics of (1) and the segment  of the axis  = 0. Let us consider the following analogue of the Cauchy-Goursat problem for the loaded equation (1) where  is an inner normal and ](),  1 (),  2 () are realvalued functions.
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 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  = 1, 2, . . .,  implied summation.Solving the next equation with respect to [7] 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).
Thus, it is proved that problem A is uniquely solvable.Theorem 1 is proved.
then there exists a unique solution to the problem C in the domain Ω.
Proof of Theorem 3. The following theorem holds.
Lemma 4. Any regular solution of (2) where (, ) 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).
Thus, the solution of problem C in the domain Ω 2 in view of (20) and ( 23) is determined uniquely according to the formula (36), and in the domain Ω 1 we arrive to the problem for an nonloaded equation of the third order [9].Thus, the solution of problem B in the domains Ω 1 and Ω 2 can be constructed from (36) in view of (20), (23), and Problem Γ 11 [9].
Thus, problem C is uniquely solvable.Theorem 3 is proved.

2 International
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   (  −   − ) −   ∑ =1   ()    0  (, 0) = 0 (1) Journal of Partial Differential Equations and a boundary value problem for a loaded equation of a mixed parabolic hyperbolic type