* The differential-symbol method of constructing the quasi-polynomial solutions of two-point problem

The solvability of the problemwith local nonhomogeneous two-point in time conditions for a homogeneous PDEof the secondorder in time and infinite order in spatial variable in the casewhen the set of zeroes of the characteristic determinant is not empty and does not coincide with C is investigated. The existence of a solution of the problem in which the right-hand sides of the two-point conditions are quasi-polynomials is proved. We propose the differential-symbol method of constructing the solutions of the problem.


Introduction
Many physical, economic, demographic and medical-biological processes can be described by models with multipoint in time conditions for PDEs and systems of them. The problems with such conditions have a simple interpretation of process observations at different points in time. For example, the two-point problem describes the process of oscillation of an infinite string when the profile of the string is given at two moments of time t = 0 and t = h > 0. Unlike the Cauchy problem for equation (1) with zero initial conditions which has only a trivial solution, the two-point problem for equation (1) with zero two-point conditions has nontrivial solutions of the form U(t, x) = sin πt h cos πx ah .
Let us notice that the initial conditions, namely the conditions in which U(0, x) and ∂U ∂t (0, x) are given at the one moment of time, are idealized. If a profile of a string for a oscillatory process is given at one moment Zinovii Nytrebych: Lviv Polytechnic National University, Lviv, Ukraine; E-mail: znytrebych@gmail.com *Corresponding Author: Oksana Malanchuk: Danylo Halytsky Lviv National Medical University, Lviv, Ukraine; E-mail: Oksana.Malan@gmail.com of time and the speed of change of the profile is given at another moment of time (although very close), then we obtain the problem for equation (1) with two-point conditions Obviously, the two-point in time conditions (5) and (6) are a partial case of the following conditions where A 1 , A 2 , B 1 , B 2 ∈ C. Conditions (5) are obtained from (7) for Therefore, the study of the solvability of the problem for a PDE with multipoint in time conditions is important for constructing a general theory of boundary-value problems and for practical problems.
Let us note the problems with multipoint in time conditions for PDEs are a generalization of problems with n-point conditions for ODEs which are known in the literature as Valle-Poussin problems [1][2][3]. The formulation and the first important results, which regard the solving of n-point problems for PDEs, are obtained in [4]. More general results of the solvability of multipoint problems in bounded domains for PDEs and systems of them are obtained in many works (see, for example, [5][6][7][8] and the bibliographies in them). Papers [9][10][11] are devoted to the establishment of the classes of uniqueness of the solution of the problems with multipoint in time conditions in unbounded domains.
In recent years the multipoint in time conditions become more general and get a new content. In particular, in conditions (7) instead of constants A 1 , A 2 , B 1 , and B 2 arbitrary differential polynomials of spatial variables can be considered. Similar conditions are present in works [12,13].
Many works of scientists [14][15][16] are devoted to constructing the polynomial and quasi-polynomial solutions for the equations and boundary-value problems for them. Let us note that function (4), which is a nontrivial solution to the homogeneous problem (1), (3) is quasi-polynomial, which can be written in the form U(t, x) = Ae αt (e βx + e −βx ) − Ae −αt (e βx + e −βx ), In unbounded domain of variables t ∈ (T 0 , T 1 ), x ∈ R s (s ∈ N), the differential-symbol method turned out to be convenient for solving the problem of the form in which A 1 ( ∂ ∂x ), ..., An( ∂ ∂x ) are differential polynomials in spatial variables, l 1 , ..., ln are linear functionals in the variable t, −∞ ≤ T 0 < T 1 ≤ ∞. The ideas behind this method are described in [17]. This differentialsymbol method allows establishing the classes of entire functions as classes of existence and uniqueness of the solution of problem (8), (9) in the form of actions of the differential expressions, whose symbols are the right-hand sides of conditions (9) and the right-hand side of equation (8), onto some functions of the parameter λ and the vector-parameter ν and after the actions the parameters are set to zero. It means the solution of the problem can be found according to this method in the following form where G(t, x, λ, ν), T 1 (t, x, ν), . . . , Tn(t, x, ν) are functions which are especially constructed for each specific problem, O = (0, . . . , 0). The problems for PDEs with conditions in time variables by using the differentialsymbol method were studied in [18][19][20][21]. Symbol calculus which is inherent for differential-symbol method we can see, for example, in papers [22,23]. In particular, for finding the definite integral We note that in ( [17], p. 26) the formula (10) is presented in the following form The present work continues the research of [24][25][26][27] and is devoted to the generalized problem (1), (7). The question of existence of solutions of the problem for the equation of the second order in time and generally infinite order in spatial variable with two-point conditions (7) in which A 1 , A 2 , B 1 , B 2 are operator coefficients in the variable x is being studied.

Problem statement
In the domain (t, x) ∈ R 2 , we investigate the solvability of the following two-point problem where )︀ are the differential expressions of finite or infinite orders, symbols of which are the in variables t and x which satisfies equation (11) in domain R 2 and conditions (12) in R.
Let us write down corresponding ODE for the PDE (11) where by {T 0 (t, ν), T 1 (t, ν)} we denote the normal at the point t = 0 fundamental system of solutions to the ODE (13).
Let us consider the determinant of the form The solvability of problem (11), (12) depends on the values of the determinant ∆ (ν). This determinant is called the characteristic determinant of problem (11), (12).
For a subset M of the space C we consider the following classes of quasi-polynomials: -K M is the class of quasi-polynomials of the form In this work we study the case M ≠ ∅ and M ≠ C, and the right-hand sides φ 0 (x) and φ 1 (x) of the twopoint conditions (12) can belong to K M . Note that the cases M = ∅ and M = C are investigated in [21,24,27]. Moreover, in [21] the existence of the unique solution of problem (11), (12) in the class K C,C∖M is proved when φ 0 (x) and φ 1 (x) belong to K C∖M and the following formula is proposed for solving problem (11), (12): wherẽ︀ T 0 (t, ν) ,̃︀ T 1 (t, ν) are functions of the form︀ It should be noted that, in formulas (16), the substituting ν = 0 and ν = α j is done after the action of the differential expressions by the parameter ν.
The functions̃︀ T 0 (t, ν) ,̃︀ T 1 (t, ν) are solutions of the ODE (13) and satisfy the conditions where δ kj is the Kronecker delta.
Also, we note that the action of the differential expression φ (︀ ∂ ∂ν )︀ , symbol of which is the quasi-polynomial φ(x) of the form (15), in formula (16) we understand as follows: Obviously, if φ 0 (x) and φ 1 (x) belong to K C∖M , then we do not get zero ∆(ν). This article is devoted to the case when at least one of the functions φ 0 (x) or φ 1 (x) belongs to K M . We will prove that a solution of problem (11), (12) exists in the class K C,M for φ 0 , φ 1 ∈ K M . This solution is not unique in K C,M , since nontrivial solutions of equation (11) which satisfy the homogeneous conditions l k U(t, x) = 0, k ∈ {0, 1} , exist in this class [24].
In the case φ 0 , φ 1 ∈ K M , the question regarding the solvability of problem (11), (12) was not investigated. So, in this work we propose the differential-symbol method for constructing the solutions of problem (11), (12), if φ 0 (x) or φ 1 (x) belongs to K M .
Theorem 3.1. Let the right-hand sides φ 0 (x) and φ 1 (x) of conditions (12) belong to K M and have the form (18). Then the solution of problem (11), (12) in the class of quasi-polynomials K C,M exists and it can be written in the form where Proof. It should be noted, that in the formula (20) the differential expression Q j ( ∂ ∂ν ) is obtained from the polynomial Q j (x) by substituting x with ∂ ∂ν . First we note that the functions (20) are entire functions. Moreover, these functions are solutions to equation (11). Really, for j ∈ {0, 1} we have In this chain of equalities we used the fact that the functions̃︀ T 0 (t, ν),̃︀ T 1 (t, ν) are solutions to the ODE (13) and the following equality Let us now prove that the function which is defined by equality (19) satisfies the two-point conditions (11). Really, for k ∈ {0, 1} we obtain In these equalities we use the fact that ( d dν ) s ∆ n j +1 (ν)  (11), (12) for each term of the polynomial and the sum of found solutions will also be a solution to problem (11), (12).
For problem (21), (22) the functions̃︀ T 0 (t, ν),̃︀ T 1 (t, ν) have the following form︀ Then we can write the functions ρ 0 (t, x, ν), ρ 1 (t, x, ν) down as By formula (19), the solution of problem (21), (22) can be found as This solution is not unique in the class K C,M , since in [24] it is proved that the two-point problem for equation (21) with homogeneous conditions has nontrivial solutions of the form where c 1 , c 2 , c 3 , c 4 are arbitrary complex constants.
The solution to problem (25), (26) we can find by formula (24): where the functions̃︀ T 0 (t, ν),̃︀ T 1 (t, ν), ρ 0 (t, x, ν) are following:︀ Let us note that the found solution of problem (25), (26) is not unique in the class K C,C , since, for example, the functions U 1 (t, x) = c 1 e −x , U 2 (t, x) = c 2 (1 − t − x)e −x , c 1 , c 2 ∈ C, are solutions to the corresponding homogeneous problem and the sum of each of these solutions with the found solution U(t, x) is also a solution to problem (25), (26).