1 Introduction and problem formulation

In this paper, we will consider the nonlinear elliptic equation

with integral boundary conditions

and Dirichlet boundary conditions at the points of the remaining two sides of the rectangle Ω

The boundary value problems for elliptic equations with nonlocal conditions as some elementary generalization of classical boundary value problems were formulated in [5, 8]. On the other hand, many papers were published on the nonlocal boundary value problems for various types of equations as mathematical models of some physical phenomena related to plasma physics, heat transfer, thermoelasticity, chemical diffusion, underground water flow, biochemistry, population dynamics, etc. (see, e.g., [12] and references therein).

Both facts motivated active investigation of numerical methods for elliptic equations with nonlocal conditions. As a consequence, the numerical methods for linear and nonlinear elliptic equations with various nonlocal conditions were intensively investigated during past two decades. The numerical methods for linear two-dimensional elliptic equation with Bitsadze–Samarskii and multipoints nonlocal conditions were investigated in the papers [13, 28, 32].

The existence and uniqueness of a solution in the corresponding Sobolev spaces for a multidimensional elliptic equation with two integral boundary conditions

were proved in [4]. The radial basis function collocation technique for the solution of such problem was presented in [18]. The convergence of finite difference method for the two-dimensional elliptic equation with condition (5) was proved in [7].

The nonlocal boundary conditions (2), (3) for the first time were formulated in [10,11] for the parabolic equation with application to thermoelasticity and thermodynamics.

The integral conditions in the form of (2), (3) are used by many authors considering nonlocal problems, also for differential equations of other types: for hyperbolic equations [16], equations with fractional derivatives [17], problems with complex parameter in the equation or nonlocal condition [23].

Also there are few papers in which problem (1)(2)(3)(4) is considered with only one nonlocal condition, usually under presumption α(x) = 0 or β(x) = 0. Thus, in the paper [2] the following problem

was investigated. A is self-adjoint positive define multidimensional elliptic operator.

The second order of the accuracy difference scheme for the approximate solution of this nonlocal boundary value problem is presented under the assumption

In many papers for the elliptic equation the multipoint nonlocal condition (usually one, not two)

instead of two integral conditions (2), (3), where 0 < ξ1 < ξ2 < < ξm < 1, is used. This condition sometimes is called as generalized condition of Bitsadze–Samarskii because it generalizes in some sense the simplest Bitsadze–Samarskii condition u(1, y) = αu(ξ, y), 0 < ξ < 1.

Essentially, condition (7) is a discrete analogue of condition (3). Considering difference schemes for the elliptic equation with nonlocal condition (7), many authors use the same assumptions for the coefficients αk as for the functions α(x), β(x) in integral conditions (2) and (3).

In the paper [1] the well-posedness of the second order of accuracy difference scheme for the elliptic equation is established under the assumption

Analogous result for an inverse problem of multidimensional elliptic equation with nonlocal condition (7) is proved in [3] under the assumption

In [6] the slightly modified assumption

which is sometimes more effective than other analogous assumptions, is used.

One more assumption

is presented in [14]. This assumption could be interpreted in a following way. If all the coefficients are nonnegative (αk ≥ 0), then (11) and (8) coincide. But if a part of the coefficients are negative (αk < 0), there is no limitation for them, and the sum of positive coefficients must not exceed one. With this assumption, the convergence of the difference scheme of the second order of accuracy is proved.

In the paper [29], several iterative methods for the solution of the system of difference equations approximating problem (1)–(4) are presented when

Also we note that assumptions (6), (12) for the functions α(x) and ρ(λ) and assumptions (8)(9)(10)(11) for the coefficients αk are only sufficient, but not the necessary conditions for the statements proved in the papers would be right.

The convergence of the difference schemes is a very important issue for any differential equation with nonlocal conditions. However, in the case of nonlinear elliptic equations, there is one more important task. It is necessary to consider how to solve the system of nonlinear difference equations. Usually, some iterative methods must be applied for this purpose. However, there are comparably not many papers in which the iterative methods for the difference systems with nonlocal conditions were investigated (see, e.g., [22, 29] and references therein).

In [25] the Poisson equation with the variable α(x) and β(x) expressions in conditions (2), (3) is solved by the Peaceman–Rachford alternating direction method. The functions α(x) and β(x) are selected in a way that all the eigenvalues of respective difference operators would be positive. That guaranties the convergence of the method.

Nonlinear elliptic equation (1) with the nonlocal conditions (2), (3) in which α(x) = const, β(x) = const is solved in [26] by alternating direction method.

At the present time, it is not clear, where the assumptions, under which the convergence of difference schemes can be proved, are sufficient also for the convergence of some iterative methods. As far as authors know, that problem has not been investigated earlier. It is important to emphasize that in most cases of elliptic equations with nonlocal conditions in the form of (2), (3) the matrix of difference problem under some assumption to functions α(x) and β(x) has the properties appropriate for the M-matrices [22, 29].

Based on this property, it is possible to prove the convergence of many iterative methods [22].

Further, during some last years, the stability and convergence of difference schemes for some parabolic and elliptic equations with nonlocal conditions were investigated using the properties of the M-matrices [9, 15, 27]. We note that applications of the M-matrices for the elliptic and parabolic equations with Dirichlet boundary conditions have been described by Varga [30].

So, it is possible to state that the M-matrices theory is the appropriate methodology for other theoretical aspects of difference scheme in the case of nonlocal conditions, in particular, conditions in the form (2), (3).

The main aim of the present paper is to investigate when the matrix of the difference problem approximating problem (1)(2)(4) is an M-matrix with the weaker or more general assumptions for the coefficients α(x) and β(x) in comparison with the known assumptions (9)(10)(11)(12). In the paper, using the properties of M-matrices, the new algorithm for the solution of the system of difference equations is provided.

The structure of the paper is following. In Section 2 the difference problem is formulated and rewritten in matrix form. The corresponding difference eigenvalue problem is considered in Section 3. Taking into apart, in this section the conditions, under which for all eigenvalues of difference operator the condition Re λ > 0 is satisfied, are obtained. In Section 4 the iterative methods for the solution of the system of nonlinear difference equations with nonlocal conditions are investigated. Results of numerical experiment supplementing the theoretical investigation are provided in Section 5. In Section 6 the generalizations and conclusions are formulated.

2 A difference problem

The boundary value problem with nonlocal conditions (1)(2)(3)(4) is solved by the finite difference method. By this aim we provide that the unique, sufficiently smooth solution of this problem exists.

We define the finite difference operators

where h = 1/N ; N —integer. The difference problem approximating differential problem (1)(2)(3)(4) is as follows:

We assume the following hypotheses are right:

(H1) 0 ≤ m0 ≤ ∂f/∂u ≤ m1 for all the values (x, y) ∈ Ω and u;

(H2) Functions α(x) and β(x) are nonnegative and bounded: 0 ≤ α(x) ≤ M1, 0 ≤ β(x) ≤ M2;

(H3) The grid step h is sufficiently small, i.e., hM1 ≤ 1/2, hM2 ≤ 1/2.

We write down the system of difference equations (13)–(16) in the matrix form. For this purpose, for every fixed value j = 1, 2, . . . , N - 1, we express from conditions (14), (15) as two equations system the unknowns u0j and uNj via other unknowns uij, i = 1, 2, . . . , N − 1:

where

Lemma 1. If hypotheses (H2) and (H3) are true, then α˜i ≥ 0, β˜i ≥ 0.

Proof. First, it will be observed that D ≠ 0. Indeed, if (H2) and (H3) are true, then

So, formulas (18) always have sense. Further, α˜i ≥ αi(1−hβN /2)/D ≥ 0. Analogously, β˜i ≥ βi(1 − hα0/2)/D ≥ 0.

Remark 1. If the conditions of Lemma 1 are fulfilled, then

We put expressions (17) into difference equations (13) as i = 1 and i = N - 1. Then we get

So, the system of finite difference equation (13)–(16) in which there are (N + 1) (N - 1) unknowns uij, i = 0, 1, . . . , N , j = 1, 2, . . . , N - 1, is written in other equivalent form. This is system (19) in which there are (N 1)2 unknowns uij, i, j = 1, 2, . . . , N 1, and explicit formulas (17) for unknowns u0j, uNj, j = 1, 2, . . . , N - 1.

We rewrite the system of difference equations (19) in matrix form

where A is matrix of the order (N 1)2, f (u) and ϕ are vectors of the order (N 1)2. Vector f (u) is formed from the components fij(uij), and vector ϕ—from the values of µ˜1j, µ˜2j, µ3i and µ4i. We do not express the value of ϕ because it will not be used further.

Matrix A is formed as follows:

A = Λ − C,

where Λ is the square matrix of order (N − 1)2 corresponding to a difference operator

x Importar tabla

−δ2 − δ2 in the rectangular domain with the Dirichlet-type homogeneous boundary value conditions. Matrix C consists of multipliers hα˜i or hβ˜i of unknowns uij in the equations of system (19). More exactly, C is a block matrix

where

The number of blocks of matrix C and order of matrix C1 are N - 1.

Matrix A = Λ - C, taking into account the expressions of Λ and C, could be written also in other form:

where I is an identity matrix of order (N − 1),

Now we consider the eigenvalue problem

for the matrix A that is necessary for the further investigation of the systems of nonlinear equations (19) or (20).

3 An eigenvalue problem of matrix A

We start from the difference eigenvalue problem

We take this problem in matrix form. With this purpose, we express u0j and uNj from conditions (26), (27):

where α˜i and β˜i are defined by formulas (18). Putting these expressions of u0j and uNj into equation (25) as i = 1 or i = N 1, we get (24), where A is defined by formula (22). So, we get an important conclusion:

Corollary 1. Difference eigenvalue problem (25)(26)(27)(28) is equivalent to the eigenvalue problem for the matrix A.

Using the Fourier method, we separate variables in (25)–(28)

uij = viwj, i, j = 0, 1, . . . , N.

Putting the expression of these unknowns into (25)–(28), analogously as in [24], we obtain two one-dimensional problems:

and

For the eigenvalues λ of problem (25)–(28), the following equality is true:

For the corresponding eigenvectors, we have

where i and j are the numbers of vectors coordinates, and k and l are numbers of eigenvectors.

The solution of eigenvalue problem (31) is known [19]:

We rewrite problem (30) in matrix form analogously for problem (25)–(28). We express from nonlocal conditions of problem (30)

where α˜i and β˜i are defined, as earlier, by formulas (18).

Substituting these expressions into first equation of (30) as i = 1 and i = N - 1, we get

where Λx is defined by formula (23), v = vi , i = 1, 2, . . . , N - 1.

Together with eigenvalue problem (30) or (33), we consider another problem as α(x) = γ1, β(x) = γ2:

This problem could be written in matrix form as follows:

where

and

We define auxiliary block-tridiagonal matrix

where Λ(0) is defined by formula (35). It is evident that A(0) is the matrix of difference eigenvalue problem (25)(26)(27)(28) with α(x) = γ1, β(x) = γ2.

Now we need a few properties of M-matrices.

Definition 1. (See [31].) A real square matrix A = akl , k, l = 1, 2, . . . , n, with akl ≤ 0 for all k ≠ l is called an M-matrix if A−1 is nonsingular and all elements of A−1 are nonnegative.

It follows from the definition that akk > 0. We will denote A > 0 (A ≥ 0) if akl > 0 (akl ≥ 0). Additionally, A < B if akl < bkl. We will use similar notation for vectors also.

If A is such that akl ≤ 0, k = / l, then the following statements are equivalent [31]:

(S1) A−1 exists and A−1 ≥ 0;

(S2) The real parts of all the eigenvalues of the matrix A are positive: Re λ(A) > 0.

We will prove some properties of the matrices A and Λx defined by formulas (22), (23).

Lemma 2. If hypotheses (H2) and (H3) are true and

then matrix Λx defined by formula (23) is an M-matrix.

Proof. We take the difference eigenvalue problem (34) and its matrix form. In this problem, we define

We evaluate α˜ and β˜ from (36). According to (H3), (37) and (38), we get

In this way the diagonal elements of matrix Λ(0) are positive, and nondiagonal elements— nonpositive. It has been proved [20, 21] that all the eigenvalues of the difference problem (34) are positive if and only if γ1 + γ2 < 2. So, according to Definition 1 and statements (S1), (S2), matrix Λx(0) is an M-matrix.

Further, we compare the elements of the matrices Λx and Λx(0). From (18), according to hypothesis (H2), we get

Now it follows from (36) that α˜ ≥ α˜i. Analogously, β˜ ≥ β˜i.

We denote elements of matrix Λ(0) as b(0) and elements of matrix Λx as bij . So, we have:

It follows from these three properties that Λx also is an M-matrix.

Remark 2. Condition (37) is only sufficient, but not necessary one for the matrix Λx to be an M-matrix. This condition could be weakened with the concrete expressions α(x) and β(x). But in the general case, not concreting the expressions α(x) and β(x), is not possible to improve this condition. Indeed, e.g., when α(x) = M1, β(x) = M2, then Λx is an M-matrix if and only if M1 ≥ 0, M2 ≥ 0, M1 + M2 < 2. And this is very close to condition (37).

We will obtain the conditions, which should be satisfied by the functions α(x) and β(x) that the eigenvalues of two-dimensional problem (25)–(28) satisfy the inequality

Re λkl > 0.

As far as the authors know, such formulation of the problem with nonlocal conditions has not been considered before.

Lemma 3. If h < 2/(γ1 + γ2), then number γ0 > 2 exists, such that the inequality λ(A(0)) > 0 is satisfied for all 0 ≤ γ1 + γ2 < γ0.

Proof of lemma is the same as for Lemma 3 in the paper [22] in which the case of γ2 = 0, 0 ≤ γ1 < γ0 was investigated. There it was proved that

where

Theorem 1. If hypotheses (H2) and (H3) and inequality

where γ0 is defined by formula (39), are true, then matrix A defined by (22) is an M-matrix.

Proof. Like in the proof of Lemma 2, γ1 and γ2 are defined by formulas (38). Then according to Lemma 3, we get λkl(A(0)) > 0 as condition (40) is satisfied. So, matrix A(0) = {a(0)} is an M-matrix because

Now we compare the elements of matrices A = {aij} and A(0). As for the elements of matrices Λx = {bij } and Λx(0) = {bij (0)}, inequalities (i), (ii) are true, then it follows from the definitions of A and A(0) that aii ≥ a(0) > 0, 0 ≥ aij ≥ a(0), I ≠ j. Whereas A(0) is an M-matrix, it follows that A is also an M-matrix.

Remark 3. Assumption (40) is only sufficient but not necessary condition for the matrix A to be an M-matrix. If

M1 + M2 + hM1M2 = γ0,

then λ(A(0)) = 0 in the case (38). In this case, we could not exploit the method used in the proof of Theorem 1 for the investigation that the matrix A is an M-matrix or not. In more detail, this case will be investigated in Section 5.1 using the numerical experiment.

It is possible to reformulate the statement of Theorem 1. Precisely, the following statement is right.

Corollary 2. If hypothesis (H2) and condition (40) are satisfied and h is sufficiently small, i.e., hM1 ≤ 1/2, hM2 ≤ 1/2, then for all eigenvalues λkl of two-dimensional eigenvalue problem (25)–(28), the following inequality is true:

4 Iterative methods

We solve the system of nonlinear difference equations (20)

Au + f (u) = ϕ,

where A = Λ C, by iterative methods. According to Theorem 1, matrix A of this system defined by formula (22) is an M-matrix. We remind that for the nonlinear function f (u), hypothesis (H1) is satisfied.

Lemma 4. If conditions in Theorem 1 and hypothesis (H1) hold, then the unique solution of system of equations (20) exists.

Proof. Statement of lemma follows from [22].

Now we can present the main result of our paper.

In paper [22], under conditions α(x) ≥ 0, 1 α(x) dx ≤ ρ < 1, β(x) = 0, several implicit iterative methods for system (20) were investigated:

where m1 is constant from (H1).

The main assumption for the iterative method to converge is following: matrix A is an M-matrix. Then along with the every new regular splitting of the matrix A = M - N, M−1 ≥ 0, N ≥ 0, we get a new iterative method (41)(42)(43).

From the Theorem 1 the statement follows:

Theorem 2. If conditions of Theorem 1 and hypothesis (H1) hold, then every of methods (41)–(43) for system (21) converges.

Remark 4. In the paper [22] the convergence of iterative methods (41)–(43) is proven under the assumptions

With these assumptions, matrix A is diagonally predominant. Meanwhile, with assumption (40), neither the symmetricity nor diagonal dominance are not necessary.

Each of these iterative methods is characterized by specific interpretation. To obtain un+1 when the values of un are known, we need to solve an additional simpler system of equations. The inner iteration is usually used for this. In method (41) the values un+1 are obtained by solving the system of nonlinear difference equation with Dirichlet boundary condition. In method (42) the system of linear equation with nonlocal condition is solved. And using method (43), on every step of iteration the system of linear equation with Dirichlet boundary condition is solved. To obtain un+1 when un is known, in all three cases of iterative methods, it is necessary to solve the system of difference equations corresponding to two-dimensional elliptic equation.

Further, we consider one more iterative method in each step of which it is needed to solve only the system of linear tridiagonal equations with nonlocal conditions.

With this aim, we write matrix A defined by formula (22) in the form

A = M − N ,

where M and N are block-diagonal and block-tridiagonal matrices, respectively:

We note that M is an M-matrix, N ≥ 0.

Now for the system of equations (20), we write down the following iterative method:

The convergence of method (44) is proved likewise as of iterative methods (41)(42)(43) in [22], on the basis of the properties of M-matrices.

Iterative method (44) like methods (41), (42) and (43) is implicit one. The main advantage of method (44) is that on every step of the iteration, we have to solve not a twodimensional, but one-dimensional problem. We write method (44) by the coordinate form such as system (13)(14)(15)(16):

5 Numerical results

In this section, we present some results of numerical experiment. The main goal of the numerical experiment is to illustrate the theoretical results obtained in the paper. Our numerical experiment consists of two parts.

5.1 Conditions under which the matrix of difference problem is an M-matrix

This is the first part of numerical experiment. In Lemma 2 and Theorem 1 the sufficient conditions (37) and (40) for the matrix Λx and respectively A were formulated under which these matrices are the M-matrices. Condition (40) differs from the ones used by other authors investigating difference schemes for two-dimensional differential equations with nonlocal boundary conditions of the same form (see (6), (8)(9)(10)(11)(12)).

To explain the effectiveness of assumption (40), we carried out the numerical experiment with fixed values of h and some different expressions of α(x) and β(x) directly computing all the eigenvalues of matrix Λx.

To obtain the eigenvalues of matrix A, we used formula (32). For chosen expressions α(x) and β(x), we varied coefficients M1 and M2 in these expressions checking when property Re λ(Λ

019

is satisfied. In this way, using the numerical experiment, we obtain the area on the coordinate plane (M1, M2), where the inequality Re λ(A) > 0 is satisfied for all the eigenvalues of A. We compared the results obtained with the theoretical results presented in Theorem 1.

We chose five following different functions α(x) and β(x).

Example 1. α(x) = M1(x + 1)/2, β(x) = M2(1 − x/2);

Example 2. α(x) = M1 sin πx, β(x) = M2(1 − sin πx);

Example 3. α(x) = M1(2x − 1)2, β(x) = 4M2x(1 − x);

Example 4. α(x) = M1, β(x) = M2x;

Example 5. α(x) = M1x, β(x) = M2(1 − x).

In all examples, M1 ≥ 0, M2 ≥ 0. So, it gets

0 ≤ α(x) ≤ M1, 0 ≤ β(x) ≤ M2.

In each of Figs. 1(a)–1(e), we present the results of numerical experiment for corresponding example. The continuous line passing through the points (M1 = 3.42, M2 = 0) and (M1 = 0, M2 = 3.42) is the graph of the curve

M1 + M2 + hM1M2 = 3.42.

The pointwise line consists of the points (M1, M2), where Re λ(A) = 0 according to results of the numerical experiment. Thus, according to Theorem 1, in the points of the area bounded by the straight lines M1 = 0, M2 = 0 and continuous curve, the inequalities α(x) ≥ 0, β(x) ≥ 0 and Re λ(A) > 0 are satisfied. It means that the matrix A is an M-matrix according to Theorem 1. In other words, if the point ( M1, M2 ), where M1 ≥ 0, M2 ≥ 0, is placed below of the continuous curve or belongs to it, then condition (40) is satisfied.

Analogously, if the point (M1, M2 ), where M1 ≥ 0, M2 ≥ 0, is placed below the pointwise curve, then the inequalities α(x) ≥ 0, β(x) ≥ 0 and Re λ(A) > 0 are satisfied

according to the results of the numerical experiment. It means that the matrix A in this (bigger) area is an M-matrix according to the results of the numerical experiment.

The following conclusions could be formulated from results of the numerical experiment.

If the functions α(x) and β(x) change sufficiently slow in the interval x ϵ (0, 1), i.e., when the variations of these functions are relatively insignificant, then the sufficient condition (40) for A to be an M-matrix is close to be the necessary condition. In other words, the pointwise curve in this case is close to the continuous curve. This conclusion is illustrated by the results of numerical experiment provided in Figs. 1(a) and 1(d) for Examples 1 and 4. In Example 1 the variation of functions α(x) and β(x) is half of size of these functions in Examples 2, 3, 5.

If functions α(x) and β(x) sufficiently differ from constants, assumption (40) remains correct, but it could be noticeably improved. From Figs. 1(b), 1(c), 1(e) it could be seen that matrix A is an M-matrix (Re λ(A) > 0) with less restricted assumption for M1 and M2 in comparison with the statement of Theorem 1.

5.2 Iterative methods

In the second part of the numerical experiment the system of nonlinear difference equations (13)(14)(15)(16) was solved by iterative method (45).

The values of M1 and M2 in the expressions of functions α(x) and β(x) were chosen from the first part of numerical experiment. More precisely, the theory (Theorem 1) does not guarantee that with the chosen functions α(x) and β(x), matrix A is an M-matrix. However, according to the results of numerical experiment, it distinguishes by such property. Thus, iterative method (44) must converge, what is illustrated by the results of computations.

We consider problem (1)(2)(3)(4) with

These expressions were chosen in a way that the function

u(x, y) = ex+y−1

would be the exact solution of this problem. We admit that functions α(x) and β(x) are chosen in the same way as in Example 5.

Numerical results are presented in Tables 1,2,3.

In all Tables 1,2,3 the errors ϵ = max(i,j) uij - u∗(xi, yj) of the approximate solution uij are presented, where u∗(xi, yj) is the solution of the differential problem.

The results provided in Table 1 show the dependence of the error from α(x) and β(x) when M1 and M2 satisfy the condition M1 + M2 = 3.42.

The iterative method converges, the error is of the order O(h2). We admit that the error is decreasing lawfully when M1 is increasing (correspondingly, M2 is decreasing). In other words, inequality M1 > M2 influences smaller error in comparison with the case of M1 < M2.

In Table 2 the values of the errors ϵ are provided for some qualitative different values of (M1, M2). Namely, if M1 = M2 = 0.1 then nonlocal conditions (2), (3) are close to Dirichlet conditions. If M1 = 2, M2 = 1.42, the matrix A is an M-matrix according to Theorem 1. And in the case of M1 = 0.42, M2 = 10.1, condition (40) of Theorem 1 is not satisfied, but the matrix A is an M-matrix according to the results of the numerical experiment. Again, we admit that the error is of the order O(h2), i.e., the error is proportional to the error of approximation.

In Table 3 the results on the dependence of the error on the values M1 and M2 are provided when M1 = M2. This error is steadily, although slowly increasing when the value M1 = M2 is increasing. This could be explained by the fact that the error of trapezoid formula increases when M1 = M2 increases.

We could also formulate one more subtle conclusion. When the point (M1, M2), M1 = M2 is placed above of the pointwise line, matrix A is no longer an M-matrix according to the results of numerical experiment. Therefore, condition (40) is not satisfied. However, the iterative method still converges for some value M1 = M2 a little bit bigger than 3.42. It could be explained because the convergence of the iterative method (44) is assured by the matrix M + m1I is being an M-matrix.

6 Remarks and generalization

As it was mentioned, one of the most important aims of this paper was to obtain weaker restrictions for the coefficients α(x) and β(x) in formulas (2), (3) for the matrix A, defined by formula (35) of the system of difference equations (25)-(28), to be an M-matrix. From Theorem 1 and the results of numerical experiment it follows such qualitative conclusions. The matrix of the system of difference equations (25)(26)(27)(28) could be an M-matrix for sufficiently wide class of functions α(x) ≥ 0 and β(x) ≥ 0.

We would like to highlight the role of the M-matrices in the theoretical investigation of finite difference method for the differential equations with nonlocal conditions.

According to the theory of M-matrices, the stability [15] and convergence [9] of finite difference schemes for parabolic equation were investigated. Convergence of finite difference method for elliptic equations was considered in [27], and iterative methods for the solution of the system of difference equations were considered in [22,29]. These were the first results of application of the M-matrices for the investigation of difference schemes for differential equations with nonlocal conditions. The present paper is the continuation of the mentioned investigations.

Therefore, it is possible to assert that application of the M-matrices for the investigation of difference methods is a promising methodology in the numerical analysis for the boundary value problems with nonlocal conditions, separately taken, conditions (2), (3).

References

1 A. Ashyralyev, E. Ozturk, On Bitsadze–Samarskii type nonlocal boundary value problems for elliptic differential and difference equations: Well–posedness, Appl. Math. Comput., 219: 1093–1107, 2012, https://doi.org/10.1016/j.amc.2012.07.016

2 A. Ashyralyev, E. Ozturk, On a difference scheme of second order of accuracy for the Bitsadze– Samarskii type nonlocal boundary–value problem, Bound. Value Probl., 2014(14):1–19, 2014, https://doi.org/10.1186/1687-2770-2014-14

3 C. Ashyralyyev, G. Akyuz, M. Dedeturk, Approximate solution for on inverse problem of multidimentional elliptic equation with multipoint nonlocal and Neumann boundary conditions, Electron. J. Differ. Equ., 2017(197):1–16, 2017.

4 G. Avalishvili, M. Avalishvili, D. Gordeziani, On a nonlocal problem with integral boundary conditions for a multidimensional elliptic equation, Appl. Math. Lett., 24:566–571, 2011, https://doi.org/10.1016/j.aml.2010.11.014

5 R.W. Beals, Nonlocal elliptic boundary value problems, Bull. Am. Math. Soc., 70:693–696, 1964, https://doi.org/10.1090/s0002-9904-1964-11166-6

6 G.K. Berikelashvili, On the convergence rate of the finite-difference solution of a nonlocal boundary value problem for a second-order elliptic equation, Differ. Equations, 39(7):945– 953, 2003, https://doi.org/10.1023/b:dieq.0000009190.72873.ed

7 G.K. Berikelashvili, N. Khomeriki, On the convergence rate of a difference solution of the Poisson equation with fully nonlocal constraints, Nonlinear Anal. Model. Control, 19(3):367– 381, 2014, https://doi.org/10.15388/na.2014.3.4

8 A.V. Bitsadze, A. A. Samarskii, Some elementary generalizations of linear elliptic boundary value problems, Dokl. Akad. Nauk SSSR, 185:739–740, 1969 (in Russian).

9 R. Cˇ iupaila, M. Sapagovas, K. Pupalaige˙, M-matrices and convergence of finite difference scheme for parabolic equation with an integral boundary condition, Math. Model. Anal., 25(2):167–183, 2020, https://doi.org/10.3846/mma.2020.8023

10 W.A. Day, Extences of a property of solutions of the heat equation subject to linear thermoelasticity and other theories, Q. Appl. Math., 40:319–330, 1982, https://doi.org/10.1090/qam/678203

11 W.A. Day, A decreasing property of solutions of a parabolic equation with applications to thermoelasticity and other theories, Q. Appl. Math., 41:468–475, 1983, https://doi.org/10.1090/qam/693879

12 M. Dehghan, Efficient techniques for the second-order parabolic equation subject to nonlocal specifications, Appl. Numer. Math., 52:39–62, 2005, https://doi.org/10.1016/j.apnum.2004.02.002

13 D.G. Gordeziani, Solution methods for a class of nonlocal boundary value problems, Tbilisi, 1981 (in Russian).

14 V.A. Il’in, E.I. Moiseev, Two-dimensional nonlocal boundary value problem for Poisson’s operator in differential and difference variants, Mat. Model., .:139–156, 1990 (in Russian).

15 K. Jakube˙liene˙, R. Cˇ iupaila, M. Sapagovas, Semi–implicit difference scheme for a two– dimensional parabolic equation with an integral boundary condition, Math. Model. Anal., 22(5):617–633, 2017, https://doi.org/10.3846/13926292.2017.1342709

16 J. Martin-Vaquero, A.H. Encinas, A. Queiruga-Dios, V. Gayoso-Martinez, A. Martin del Rey, Numerical schemes for general Klein–Gordon equations with Dirichlet and nonlocal boundary conditions, Nonlinear Anal. Model. Control, 23(1):50–62, 2018, https://doi.org/10.15388/na.2018.1.5

17 J. Ren, C. Zhai, Nonlocal q-fractional boundary value problem with Stieltjes integral conditions, Nonlinear Anal. Model. Control, 24(4):582–602, 2019, https://doi.org/10.15388/na.2019.4.6

18 S. Sajavicˇius, Radial basis function method for a multidimensional linear elliptic equation with nonlocal boundary conditions, Comput. Math. Appl., 67:1407–1420, 2014, https://doi.org/10.1016/j.camwa.2014.01.014

19 A.A. Samarskii, The Theory of Difference Schemes, CRC Press, New York, 2001, https://doi.org/10.1201/9780203908518

20 M. Sapagovas, On stability of finite-difference schemes for one-dimensional parabolic equations subject to integral conditions, Zh. Obchysl. Prykl. Mat., 92:70–90, 2005.

21 M. Sapagovas, On the stability of a finite-difference scheme for nonlocal parabolic boundaryvalue problems, Lith. Math. J., 48(3):339–356, 2008, https://doi.org/10.1007/s10986-008-9017-5

22 M. Sapagovas, V. Griškoniene˙, O. Štikoniene˙, Application of M-matrices for solution of a nonlinear elliptic equation with an integral condition, Nonlinear. Anal. Model. Control, 22(4):489–504, 2017, https://doi.org/10.15388/na.2017.4.5

23 M. Sapagovas, T. Meškauskas, F. Ivanauskas, Influence of complex coefficients on the stability of difference scheme for parabolic equations with nonlocal conditions, Appl. Math. Comput., 332:228–240, 2018, https://doi.org/10.1016/j.amc.2018.03.072

24 M. Sapagovas, J. Novickij, A. Štikonas, Stability analysis of a weighted difference scheme for two-dimensional hyperbolic equations with integral conditions, Electron. J. Differ. Equ., 2019(04):1–13, 2019.

25 M. Sapagovas, A. Štikonas, O. Štikoniene˙, Alternating direction method for the Poisson equation with variable weight coefficients in an integral conditions, Differ. Equ., 47(8):1176– 1187, 2011, https://doi.org/10.1134/s0012266111080118

26 M. Sapagovas, O. Štikoniene˙, Alternating–direction method for a mildly nonlinear elliptic equation with nonlocal integral conditions, Nonlinear Anal. Model. Control, 16(2):220–230, 2011, https://doi.org/10.15388/na.16.2.14107

27 M. Sapagovas, O. Štikoniene˙, K. Jakube˙liene˙, R. Cˇ iupaila, Finite difference method for boundary value problem for nonlinear elliptic equation with nonlocal conditions, Bound. Value Probl., 2019(94):1–16, 2019, https://doi.org/10.1186/s13661-019-1202-4

28 Siraj-ul-Islam, I. Aziz, M. Ahmad, Numerical solution of two-dimensional elleptic PDEs with nonlocal boundary conditions, Comput. Math. Appl., 69:180–205, 2015, https://doi.org/10.1016/j.camwa.2014.12.003O

29 Štikoniene˙, M. Sapagovas, R. Cˇ iupaila, On iterative methods for some elliptic equations with nonlocal conditions, Nonlinear. Anal. Model. Control, 19(3):517–535, 2014, https://doi.org/10.15388/na.2014.3.14

30 R.S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962.

31 V. Voevodin, Y. Kuznecov, Matrices and Computations, Nauka, Moscow, 1984 (in Russian).

32 E.A. Volkov, A.A. Dosiyev, S.C. Buranay, On the solution of a nonlocal problem, Comput. Math. Appl., 66(3):330–338, 2013, https://doi.org/10.1016/j.camwa.2013.05.010