Assanova A.T., Sabalahova A.P., Baygulova N.Z.

Institute of Mathematics, Ministry of Education and Science Republic of Kazakhstan, Almaty, Kazakhstan, M.Auezov South Kazakhstan State University, Shymkent, Kazakhstan

 

Solvability conditions and the algorithms for finding a solution to a nonlocal boundary value problem for a class of hybrid systems

 

Considered a nonlocal boundary value problem for a system of composite type in a rectangle

     ,                                 (1)

with the conditions

,             ,                                          (2)

  ,             ,                                          (3)

,             ,                                          (4)

where the functions , , , ,  continuous on , , function  continuously differentiable on , functions , ,  continuous on , functions , ,  continuously differentiable on , and and matching condition about the data is performed: .

Hybrid systems often arise in the mathematical modeling of numerous processes in the automotive, aerospace, robotics, power et al. [1-4]. Hybrid systems can be combined continuous and discrete equations of different types according to the described application. In addition, the linearization of nonlinear problems for partial differential equations and systems of partial differential equations often leads to approximating linear problems for hybrid systems corresponding to the form [5-6].

In this paper we study nonlocal boundary value problem for a hybrid system of a special type. The system of equations (1) consists of a first order differential equations and partial differential equations of hyperbolic type second order, which are interconnected through the desired functions. For the first order differential equation is given undivided two-point condition (2), and for a hyperbolic equation with mixed derivatives are given boundary condition (3), non-local condition (4).

Hybrid systems of the form (1) are found in the study of wave processes in different environments, in theory of the population, in the theory of adsorbed mixtures in metal cleaning technology et al. [7]. Interest in hybrid systems consisting of different types of equations, related both to their great practical significance, and in the non-classical nature of the resulting problems.

Materials and methods. Solution of a nonlocal boundary value problem for the hybrid system (1)-(4) is a pair of functions , where ,  continuous on  functions, have continuous partial derivatives , , ,  on , satisfy the system of equations (1) and the boundary conditions (2), (3) and (4).

The purpose of the work is study questions of existence, uniqueness of the solution of a nonlocal boundary value problem for the hybrid system (1)-(4), as well as the construction of algorithms for finding an approximate solution. The main method used to address these issues is a method of introducing functional parameters [8]. This method was developed in the works of one of the authors for solving nonlocal boundary value problems for hyperbolic equations with mixed derivatives. Have been proposed algorithms for finding approximate solutions and the conditions for the unique solvability of nonlocal boundary value problems [9-16].

The problem under consideration with the new unknown functions reduces to a family of two-point boundary value problems for partial differential equations of the first order and integral relations. Algorithms for finding a solution to obtain an equivalent problem. The conditions of existence of a unique solution of a nonlocal boundary value problem for the hybrid system (1) - (4) in terms of the coefficients of the equations and boundary functions.

Reduction to an equivalent problem by the introduction of functional parameters. Let , . By introducing these new features in a nonlocal boundary value problem (1)-(4) go to the equivalent problem

         ,                             (5)

                  ,             ,                                      (6)

,       ,    (7)

  ,     ,         ,         (8)

where, , , .

Here, the condition (3) with respect to the values of function  on line , taken into account in the integral relations (8).

Problem (5)-(8) consists of a system of differential equations depending on a parameter , (5) with the boundary conditions (6), (7) and integral relations (8).

Solution of the problem (5)-(8) is the four functions , where continuous on  functions , , ,  satisfy the system of equations (5), boundary conditions (6)-(7) and the function  associated integral relations (8) with the functions , .

If a pair of functions  is a solution of problem (1)-(4), the four functions , where , , are a solution of problem (5)-(8). Conversely, if the four functions are a solution of problem (5)-(8), a pair of functions is a solution of the original problem (1)-(4).

For fixed , , problem (5)-(7) is a family of two-point boundary value problems for systems of partial differential equations of the first order with respect to functions , , where the variable  is a parameter and is continuously changed in the interval .

If we know the function , , from the family of two-point boundary value problems (5)-(7) can be defined solution - a pair of functions , and if the function , of integral equations (8) through it you can find functions   .

Since the unknowns are functions , , and functions , , applies an iterative process. To determine the sequence of approximate solutions of the problem (5)-(8) - four functions  proposed the following algorithm:

0-step.  1) Counting ,  from the family of two-point boundary value problems for systems of partial differential equations of the first order (5)-(7), we find , , ; 2) Of integral relations (8) by ,  we find , , .

1-step. 1) Counting , , from the family of two-point boundary value problems for systems of partial differential equations (5)-(7), we find  , , ; 2) Of integral relations (8) by , , we find ,  , .

-step. 1) Counting , , from the family of two-point boundary value problems for systems of equations (5)-(7), we find , ,  ; 2) Of integral relations (8) by ,  , we find , , ,

.

Now become an important issue questions the feasibility and the convergence of the constructed algorithm for finding an approximate solution of problem   (5)-(8). Approval of the following theorem gives answers to these questions under certain assumptions about the source data.

Theorem 1. Let   and  

 for all . Then the sequence of quadruples ,   determined by the algorithm converges uniformly to four functions  - the only solution of the problem (5) - (8) for all .

Proof. We use 0-step algorithm built.

Consider a family of two-point boundary value problems

,                                            (9)

                                   ,             ,              (10)

           ,                               (11)

,      .      (12)

When the conditions of the theorem, ie, if the functions ,  different from zero for all , then solving (9), (10) and (11), (12) will have the following form

,

.

The initial approximation of functions, determined from the integral relations

  ,      for all .

Continuing the process, to -step of the algorithm we obtain

                  ,

where   , .

-th approximation of functions ,  determined from the integral relations  ,      for all .

Let , , , ,  ,  . 

Then the difference of successive approximations we have the estimates

,                 (13)

,                  (14)

,                    (15)

,                   (16)

.

Using (15), (16) and the differential equation relatively , from (13) and (14) we obtain

,                 (17)

,                              (18)

,                                           (19)

where ,  

      .

We introduce the notation  

,

.

Then from (15)-(19) follows the basic inequality

,    .                     (20)

The estimate (20) follows a uniform relatively  convergence of sequences , ,  by  to functions , , , respectively. Inequalities (15) and (16) provide a uniform relatively  convergence of sequences ,  by  to functions , , respectively.

Thus, there exists a solution (5)-(8).

Let us prove the uniqueness of the solution. Assume the contrary. Suppose there are two solutions of the problem (5)-(8) - four functions   and .

Similarly, the estimates (15) - (19) we obtain

,                               (21)

,                              (22)

,                 (23)

,                              (24)

.                                            (25)

Let    .

Then, similar to the estimate (20), we obtain the basic inequality

.                         (26)

Applying the inequality Bellman-Gronwall estimate (26) we find that . It follows that , , . Then from inequalities (21), (22) follows , . The uniqueness of the solution is proved. Theorem 1 is proved.

The equivalence of problems (1)-(4) and (5)-(8) follows

Theorem 2. Let     and 

for all . Then the nonlocal boundary value problem for the hybrid system (1)-(4) has a unique solution .

Studied nonlocal boundary value problem for a class of hybrid systems consisting of ordinary differential equations and hyperbolic equations with mixed derivative. By introducing new unknown functions considered problem is reduced to an equivalent problem, consisting of families of two-point boundary value problems for systems of differential equations and integral equations. An algorithm for finding an approximate solution of the resulting problem. We prove the convergence of the proposed algorithm to solve the problem. Coefficient set conditions for the existence of a unique solution of a nonlocal boundary value problem for the hybrid system.

 

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