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# A fourth-order accurate difference Dirichlet problem for the approximate solution of Laplace’s equation with integral boundary condition

*Advances in Difference Equations*
**volume 2019**, Article number: 340 (2019)

## Abstract

A new constructive method for the finite-difference solution of the Laplace equation with the integral boundary condition is proposed and justified. In this method, the approximate solution of the given problem is defined as a sequence of 9-point solutions of the local Dirichlet problems. It is proved that when the exact solution \(u(x,y)\) belongs to the Hölder calsses \(C^{4,\lambda }\), \(0<\lambda <1\), on the closed solution domain, the uniform estimate of the error of the approximate solution is of order \(O(h^{4})\), where *h* is the mesh step. Numerical experiments are given to support analysis made.

## 1 Introduction

Different finite-difference problems as approximations of the nonlocal problems with integral boundary condition have been studied by many authors (see [1,2,3,4,5] and references given therein). They all were basically focusing on the following difficulties related to the existence of a quadrature approximation of the integral condition on the side of the domain where nonlocal condition was given: (i) finding an approximate solution by solving the obtained system of equations which are non-band matrices, (ii) determining the rate of convergence of the approximate solution by appropriate smoothness conditions on the given data. In [1], a system of finite difference equations for the Poisson problem has been studied for the spectrum of the matrix to apply an iterative method. Moreover, the author obtained some conditions, under which this system has a unique solution. In [2] and [3], for the error of approximate solution, the order of estimation of \(O(h^{2})\) in the difference \(W_{2}^{1}\) metric is obtained, where *h* is the mesh step. In [4], the radial basis function collocation technique is used to find an approximate solution of an elliptic equation with nonlocal integral boundary condition. In [5], a finite-difference approximation for the problem with integral boundary conditions is constructed by reducing the given problem to the problem with nonlocal conditions containing derivatives. The authors proved that when the fourth-order partial derivatives of the exact solution are continuous on the closed solution domain, the uniform estimate of order \(O(h^{2} \vert \ln h \vert )\) is obtained for the error of the approximate solution.

In this paper, we propose and justify a new constructive method to solve a system of nonlocal 9-point finite-difference problem for the Laplace equation with the integral boundary condition. The solution of this nonlocal difference problem is defined as a solution of the 9-point Dirichlet problem by constructing approximate values of the solution on the side where the integral condition was given. Therefore, the approximate solution is obtained by solving a system with 9 diagonal matrices, for the realization of which many fast algorithms have been proposed (see [6, 7]). Moreover, the uniform estimate of the error of approximate solution is of order \(O(h^{4})\) when the given boundary functions on the sides belong to the Hölder classes \(C^{4,\lambda }\), \(0<\lambda <1\). Finally, numerical experiments are demonstrated to support the theoretical results.

The proposed method with the 5-point scheme was announced in [8].

Other nonlocal boundary value problems are stated and developed in numerous papers (see [9,10,11,12,13,14,15,16,17,18,19,20] and references therein).

## 2 Nonlocal boundary value problem

Let

be an open rectangle, \(\gamma ^{m}\), \(m=1,2,3,4\), be its sides including the endpoints, numbered in the clockwise direction, beginning with the side lying on the *y*-axis, and let \(\gamma =\bigcup_{m=1}^{4}\gamma ^{m}\) be the boundary of *R* and \(\overline{R}=R\cup \gamma \). Let \(C^{0}\) denote the linear space of continuous functions of one variable *x* on the interval \([ 0,a ] \) of the *x*-axis, and vanishing at the points \(x=0\) and \(x=a\). For a function \(f\in C^{0}\), we define the norm

It is clear that the space \(C^{0}\) with this norm is complete.

Consider the following nonlocal boundary value problem:

where \(\Delta \equiv \partial ^{2}/\partial x^{2}+\partial ^{2}/\partial x^{2}\) is the Laplacian, \(\tau =\tau (x)\) and \(\mu =\mu (x)\) are given functions which belong to \(C^{0}\), and *α* is a given constant which satisfies the following inequality:

## 3 Nonlocal finite-difference problem and its reduction to the Dirichlet problem

We define a square mesh with size \(h=\frac{a}{N}=\frac{b}{ M^{\ast }}\), where \(N,M^{\ast }>2\) are integers, constructed with the lines \(x,y=h,2h,\ldots \) . Let \(D_{h}\) be the set of nodes of this square grid and let \(R_{h}=R\cap D_{h}\), \(\overline{R}_{h}=\overline{R}\cap D_{h}\). We put \(\gamma _{h}^{m}=\gamma ^{m}\cap D_{h}\), \(m=1,2,3,4\), and \(\gamma _{h}=\bigcup_{m=1}^{4}\gamma _{h}^{m}\).

Let

be the set of points divided by the step size *h* on \([ 0,a ]\).

Let \(C_{h}^{0}\) be the linear space of grid functions defined on \([0,a]_{h}\) that vanish at \(x=0\) and \(x=a\). The norm of a function \(f_{h}\in C_{h}^{0}\) is defined as

We introduce the operator \(B_{h}\) by

For the approximate solution of the nonlocal problem (1)–(2), we consider a solution of the following system of difference equations (see [1]):

where equation (5) is obtained by approximating the integral in (2) and using Simpson’s rule with \(\rho _{1}=\rho _{M}= \frac{h}{3}\), \(\rho _{j}=\frac{h}{3} ( 3+ ( -1 ) ^{j} ) \) for \(j=2,3,\ldots,M-1\), \(\eta _{j}=\xi +(j-1)h\), \(j=1,2,\ldots,M\), \(h=\frac{a}{N}\), \((M-1)h+\xi =b\), \(\mu _{h}\) is the trace of *μ* on \(\gamma _{h}^{4}\), and \(\frac{\xi }{h}\) is an integer.

We reduce a solution of the nonlocal differential problem to the solution of the local Dirichlet problem.

Let \(v_{h}\) be the solution of the finite-difference Dirichlet problem

and we put

where \(\tau _{h}\) is the trace of *τ* on \(\gamma _{h}^{2}\).

Let \(w_{h}\) be a solution of the following finite difference Dirichlet problem:

where \(\widetilde{f}_{h}\in C_{h}^{0}\), is an arbitrary function.

We define a linear operator \(B_{i}^{h}\) from \(C_{h}^{0}\) to \(C_{h}^{0}\) as follows:

where \(w_{h}\) is the solution of problem (8).

Let

We have

Since \(w_{h}^{\ast }=Bw_{h}^{\ast }\) on \(R_{h}\), from (9)–(10) and by a comparison theorem (see [21, Chap. 4]), we have

and then for the norm of operator \(B_{i}^{h}\), we get

Let

where \(\widetilde{\varphi }_{k,h}(x)\) is the function from (7).

In the view of inequality (3), we have

Inequalities (12) and (14) yield

### Lemma 1

*A solution of the finite difference problem* (4)*–*(5) *can be represented as*

*where*
\(v_{h}\)
*is the solution of problem* (6) *and*
\(w_{h}\)
*is the solution of problem* (8) *with*
\(\widetilde{f}_{h}\)
*being a solution of the following nonlinear equation*:

### Proof

According to (4), (6), and (8), relation (16) holds on \(R_{h}\) and the boundary sides \(\gamma _{h}^{m}\), \(m=1,2,3\).

From (13) and (17), it follows that

Relying on (7) and (9), we have

By virtue of (6) and (8), we obtain

Due to (5), this shows that relation (16) is also satisfied on \(\gamma _{h}^{4}\). □

Thus, the unknown function on \(\gamma _{h}^{4}\) in problem (8) is a solution of the nonlinear equation (17).

### Theorem 2

*There exists a unique solution*
\(\widetilde{f}_{h}\)
*of the nonlinear equation* (17).

### Proof

Consider the following sequences in \(C_{h}^{0}\):

From this, for the positive integers *m* and *n* with \(m>n\), we get

Applying inequality (11), we reach

where *q* is defined by (15). In a similar way, from (19) we obtain

which shows that sequences (18) are Cauchy. Since \(C_{h}^{0}\) is complete, there are limits

Using (20) and taking the limit of (18) as \(n\rightarrow \infty \), we have

We multiply both sides of equation (21) by \(\alpha \rho _{i}\) and sum over \(i=1,2,\ldots,M\) to get

In view of relations (17) and (22), we obtain a solution of the nonlinear equation (17) as

To show the uniqueness, let \(\widetilde{f}_{h,p}\in C_{h}^{0}\), \(p=1,2\), be two functions satisfying relation (17). Then, we obtain the following inequality:

where \(0< q<1\) is defined by (15). Hence \(\widetilde{f}_{h,1}= \widetilde{f}_{h,2}\). □

## 4 Convergence of the finite-difference problem

We say that \(F\in C^{k,\lambda }(E)\), if *F* has *k*th derivatives on *E* satisfying Hölder condition with exponent *λ*. We assume that \(\tau (x)\) and \(\mu (x)\) in (1) and (2) are from \(C^{4,\lambda }\), \(0<\lambda <1\), on \(\gamma ^{2}\) and \(\gamma ^{4}\), respectively, and \(\tau ^{ ( 2m ) } ( 0 ) = \tau ^{ ( 2m ) } ( a ) =0\), \(\mu ^{ ( 2m ) } ( 0 ) =\mu ^{ ( 2m ) } ( a ) =0\), \(m=0,1,2\). By using the *n*th iteration \(\widetilde{\psi }_{i,h}^{n}\), \(n\geq 1\) of (18), we define the function

Hence, for the approximate solution of the nonlocal problem (1)–(2), we define the following difference problem:

### Theorem 3

*The following estimate holds*:

*where*
\(\widetilde{u}_{h}^{n}\)
*is a solution of problem* (24)*–*(25), *u*
*is the exact solution of nonlocal boundary value problem* (1)*–*(2), \(c_{1}\)
*and*
\(c^{\ast }\)
*are constants independent of*
*h*, \(q_{0}\)
*is defined by* (14), *and*
\(q_{1}=1-\frac{\xi }{b}\).

### Proof

Let *U* be the exact solution of the system of the following problem:

Let *V* be a solution of the Dirichlet problem

and denote by

where \(\eta _{k}=\xi +(k-1)h\), \(k=1,2,\ldots,M\). We define the function

Consider the Dirichlet problem

where *f* is an unknown function from \(C^{0}\). The linear operator \(B_{i}:C^{0}\rightarrow C^{0}\) is defined as

Then following inequality holds for the norm \(\vert B_{i} \vert \):

By analogy with the results in [18], it is shown that a solution *U* of problem (27)–(28) can be represented as \(U=V+W\) where *V* and *W* are the solutions of problem (29) and (32), respectively, when *f* is defined by

Here the functions \(\psi _{1},\psi _{2},\ldots,\psi _{M}\) are from \(C^{0}\), and are defined as the solutions of the nonlinear equations

Therefore, the nonlocal problem (27)–(28) is reduced to the following Dirichlet problem:

where *f* is defined by (33). The solution \(\psi _{i}\), \(i=1,2,\ldots,M\), of system (34) is found as a limit of the infinite sequence of functions \(\{ \psi _{i}^{n} \} _{n _{=0}}^{\infty }\) in \(C^{0}\) defined by

Since \(\tau (x)\) in (29) belongs to \(C^{4,\lambda } ( \gamma ^{2} ) \) and \(\tau ^{ ( 2m ) } ( 0 ) =\tau ^{ ( 2m ) } ( a ) =0\), \(m=0,1,2\), it follows from [22] that

where \(v_{h}\) is a solution of problem (6), \(V_{h}\) is the trace of the solution of (29) on \(\overline{R}_{h}\) and \(c_{2}\) is a constant independent of *h*. Let \(\varphi _{h}\), \(\psi _{i,h}\), and \(\psi _{i,h}^{n}\) be the trace of *φ*, \(\psi _{i}\), and \(\psi _{i}^{n}\) on \([ 0,a ] _{h}\), respectively, and let \(( B_{i} ( F ) ) _{h}\) be the trace of \(B_{i} ( F ) \) on \([ 0,a ] _{h}\) for any function \(F\in C^{4,\lambda } [ 0,a ] \). By (7), (13), (30), (31), and (38), we obtain

where \(c_{3}\) is a constant independent of *h*. By using (18) and (37), we have, for all \(i=1,2,\ldots,M\),

Applying (11) and (39), it follows that

where \(c_{4}\) is a constant independent of *h*. Similar to inequality (38), we have

where \(c_{5}\) is a constant independent of *h*. From the relations (40)–(42), we have

where \(c_{6}\) is a constant independent of *h*. For \(n\geq 2\), we have

Then,

By analogy with (54) in [20], it follows that

where \(c_{7}\) is a constant independent of *h*. From (45), we find that

where \(c_{8}= \vert \alpha \vert ( b-\xi ) c _{7}\). In the view of (11), (14), (42), (44), and (46), we have

where \(q_{0}\) is defined by (14) and \(c_{9}\) is a constant independent of *h*. By virtue of (43) and (47), we obtain

where \(c_{10}\) and \(c_{11}\) are constants independent of *h*. According to (37), it follows that

where *φ* is defined by (31). From (49) and (50), we have

where \(q_{1}=1-\frac{\xi }{b}\). Moreover, for any \(m=1,2,\ldots \) , we obtain

Since

by taking the limit as \(m\rightarrow \infty \), from (51) and (52), it follows that

Let \(U_{h}(x,y)\) be the solution of the system of grid equations

which approximates problem (35)–(36) when \(f_{h}\) is the trace of *f* on \([ 0,a ] _{h}\). Since *τ*, *μ*, *φ*, and \(\psi _{i}\), \(i=1,2,\ldots,M\), belong to \(C^{4,\lambda }\), \(0< \lambda <1\), on the interval \(0\leq x\leq 1\), and the \((2m)\)th order derivatives vanish at the endpoints for \(m=0,1,2\) (see [20]), by [22], we have

where *U* is the solution of problem (35)–(36) and \(c_{12}\) is a constant independent of *h*. In view of inequalities (39) and (54), we obtain

where \(q_{0}\) is defined by (14) and \(c_{13}\) is a constant independent of *h*. By the grid maximum principle and from (58), we have

where \(\widetilde{u}_{h}^{n}\) is the solution of problem (24)–(25) and \(U_{h}\) is the solution of problem (55)–(56). According to estimates (57) and (59), the following inequality holds:

where *U* is the solution of problem (35)–(36) and \(c_{14}\) is a constant independent of *h*.

Using the estimate (60) and by the maximum principle for the Laplace equation with the truncation error of Simpson’s rule, which is order of \(O(h^{4})\), we obtain the final estimate

where *u* is the solution of problem (1)–(2), \(c_{1}\) is a constant independent of *h*, and \(c^{\ast }= \Vert \varphi \Vert _{C^{0}}+ \Vert \mu \Vert _{C^{0}}\). □

### Remark 4

In (61), the right-hand side is of order \(O (h^{4} )\), when

From (62) it follows that

where \([ a ] \) is the integer part of *a*.

## 5 Numerical experiments

Let

### Problem 1

### Problem 2

The exact solutions of Problems 1 and 2 are unknown. The approximate values of Problems 1 and 2 on the line \(y=0\) obtained by the proposed method are given in Tables 1 and 2, respectively. According to repeated digits, for the decreasing mesh steps \(h=\frac{1}{16},\frac{1}{32}, \frac{1}{64},\frac{1}{128} \), it follows that the maximum error on this line decreases as \(O ( h^{4} ) \). To obtain these results, 14 iterations are run for the construction of \(\widetilde{f}_{h}^{n}\) with the successive error which is less than 10^{−16}.

### Problem 3

where \(u=e^{\pi y}\sin \pi x\) is the exact solution, \(\mu (x)= [ 1+\frac{\alpha }{\pi } ( 1-e^{2\pi } ) ] \sin \pi x\).

In Table 3 for Problem 3, the maximum error for each step \(h= \frac{1}{2^{k}}\), \(k=4,5,6,7\) and the reduction orders are given. From the third column it follows that the convergence order is \(O ( h ^{4} ) \).

In Tables 4, 5, and 6, the results of the CPU times (in seconds), when solving Problems 1, 2, and 3, respectively, are given. In columns 2 and 3, the CPU times for the realization of the proposed approaches by the discrete Fourier method and by the Gauss–Seidel method are given. For the construction of the local function \(\widetilde{f}_{h}^{n}\) for Problems 1 and 2, just 14 iterations are used. Problem 3 needs 11 iterations. In column 4, the Gauss–Seidel method is used to solve the given problems without reducing to the Dirichlet problem. From these results it follows that the discrete Fourier method, which cannot be used on the problem without reducing to the Dirichlet problem, is faster than others. The third and fourth columns show that for the method which is applicable for both approaches (as Gauss–Seidel), the CPU times with reducing are less than the CPU times without reducing to the Dirichlet problem.

As it follows from Tables 4–6, the CPU times for Problems 1 and 3 in Tables 4 and 6 are less than those for Problem 2 in Table 5. This takes place because of low smoothness of the boundary function in Problem 2.

## 6 Conclusion

A new constructive method for the approximate solution of the nonlocal boundary value for Laplace’s equation with integral boundary condition is given. In the proposed method, the system of finite-difference equations is defined as the 9-point solution of the Dirichlet problem by constructing the function on the side of the rectangle where the nonlocal boundary condition was given. This function is defined by using the *n*th term of the convergent simplest fixed point iteration (18) for the solution of the nonlinear system of (21). A uniform estimate for the error of the approximate solution of the nonlocal problem by using the *n*th term for \(n=\max \{ [ ( \ln h^{4}(1-q_{1}) ) /\ln q_{1} ] ,1 \} \) is of order \(O ( h^{4} ) \), where *h* is the step size.

The proposed method gives an opportunity to solve nonlocal problems by using different fast algorithms constructed for the local Dirichlet problem by many authors (see [6] and the references therein).

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Dosiyev, A., Reis, R. A fourth-order accurate difference Dirichlet problem for the approximate solution of Laplace’s equation with integral boundary condition.
*Adv Differ Equ* **2019**, 340 (2019). https://doi.org/10.1186/s13662-019-2282-2

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DOI: https://doi.org/10.1186/s13662-019-2282-2

### Keywords

- Finite difference method
- Nonlocal integral boundary condition
- Laplace’s equation
- Uniform error