Correct and Stable Algorithm for Numerical Solving Nonlocal Heat Conduction Problems with Not Strongly Regular Boundary Conditions

Abstract

For a nonlocal initial-boundary value problem for a one-dimensional heat equation with not strongly regular boundary conditions of general type, an approximate difference scheme with weights is constructed. A correct and stable algorithm for the numerical solving of the difference problem is proposed. It is proven that the difference scheme with weights is stable and its solution converges to the exact solution of the differential problem in the grid $L_2^h$-norm. Stability conditions are established. An estimate of the numerical solution with respect to the initial data and the right-hand side of the difference problem is given.

1. Introduction

The problems considered in the paper belong to the class of nonlocal boundary/initial-boundary value problems for differential equations. In such problems, nonsolution and/or derivative values on fixed parts of the boundary are specified, but linear combinations of these values on different parts of the boundary. The problems with nonlocal boundary conditions have continuously been the focus of research due to their wide application in the mathematical modeling of processes and phenomena associated with the diffusion of particles in a turbulent plasma, heat transfer for a given total change in the amount of heat, thermal stability, chemical diffusion, groundwater flow, and pollution processes in rivers and seas, medium oscillation, biochemistry, dynamics of biological populations, etc.; a large number of publications are also devoted to nonlocal inverse problems (see, e.g., [1,2,3,4,5,6,7,8,9], and references therein). For example, nonlocal conditions turn out to be preferable to local ones, when it is difficult to separately measure data on fixed parts of the boundary or control the process in a closed contour with an ideal/nonideal contact at the ends of the contour.

The general statement of a wide class of nonlocal problems for partial differential equations was first formulated in [10] (see also [11]). The separation of regular boundary conditions from general nonlocal boundary conditions made it possible to form a class of boundary value problems whose spatially differential operators have complete systems of root functions [12,13,14]. However, such a system, as it turns out, does not always form a basis. Restricting this class to strongly regular boundary conditions leads to an almost self-adjointed spatially differential operator with two series of eigenvalues of different asymptotics. This guarantees the existence of a Riesz basis [15,16,17]. To construct solutions to boundary/initial-boundary value problems with strongly regular boundary conditions, one can already use the theory for self-adjointed operators (or their perturbations) and well-known methods, in particular, the method of separation of variables. The remaining boundary conditions from the class of regular ones are called not strongly regular boundary conditions. Sturm-type conditions are a special case of strongly regular conditions ([13], p. 70). Particular cases of not strongly regular conditions are periodic and antiperiodic boundary conditions ([13], p. 71) and Samarskii–Ionkin conditions [1].

In problems with not strongly regular boundary conditions, spatial differential operators are, as a rule, non-self-adjointed and have two series of eigenvalues with the same asymptotics. This prevents eigenfunctions from forming the basis. Moreover, eigenvalues can be multiples, there can be an infinite number of associated functions, they are not uniquely defined, and, as a result, the Riesz basis either exists or does not (depending on the choice of associated functions). Examples are known when it is possible to construct systems consisting of combinations of root functions that form the Riesz basis, and, accordingly, write out solutions to boundary value problems [18,19]. However, when the boundary conditions are not strongly regular, the available methods and theories developed for problems with self-adjointed spatially differential operators cannot generally be used. Therefore, the development of new and/or modernization (adaptation) of currently known methods for solving these problems is required. Exceptions, and perhaps the only ones, are boundary-value problems with periodic and antiperiodic boundary conditions. In these problems, spatially differential operators (with real coefficients) are self-adjointed. This guarantees the existence of orthonormal basis ([13], p. 32).

In this paper, we consider an initial-boundary value problem for a one-dimensional heat equation with not strongly regular boundary conditions of general type. The finite difference method is used for the numerical solving the problem. We construct a nonlocal difference scheme with weights, which approximates the original differential problem. To establish the stability of the scheme, we adapt the algorithm from [20], developed for differential problems with not strongly regular boundary conditions, for the difference scheme with weights. Necessary and sufficient conditions for the stability of the nonlocal difference scheme in $L_2^h$-norm is obtained, and this condition turns out to be the same as for local difference schemes. The stability and convergence of the solution of the nonlocal difference scheme to the exact solution of the nonlocal differential problem in $L_2^h$-norm are proven.

2. The Problem Statement in a Differential Form

Consider $\Omega =\left\{(x,t):0<x<1,0<t<T\right\}$ the initial-boundary value problem for the one-dimensional heat equation

$$\frac{\partial u(x,t)}{\partial t}=L_{x}u(x,t)+f(x,t),0<x<1,0<t<T,$$

(1)

with the initial condition

$$u(x,0)=g\left(x\right),0\le x\le 1,$$

(2)

and boundary conditions of the form

$$\left\{\begin{array}{c}{a}_{11}{\displaystyle \frac{\partial u(0,t)}{\partial x}}+b_{11}{\displaystyle \frac{\partial u(1,t)}{\partial x}}+a_{10}u(0,t)+b_{10}u(1,t)=0,\\ a_{21}{\displaystyle \frac{\partial u(0,t)}{\partial x}}+b_{21}{\displaystyle \frac{\partial u(1,t)}{\partial x}}+a_{20}u(0,t)+b_{20}u(1,t)=0.\end{array}\right.$$

(3)

Here, $L_{x}u(x,t)={\displaystyle \frac{{\partial }^{2}u(x,t)}{\partial x^2}}$, the coefficients $a_{ij}$, $b_{ij}$$(i=1,2;j=0,1)$ are real numbers, and $g\left(x\right)$, $f(x,t)$ are given real functions. Moreover, the agreement conditions are met,

$$\left\{\begin{array}{c}{a}_{11}{g}^{\prime }\left(0\right)+b_{11}{g}^{\prime }\left(1\right)+a_{10}g\left(0\right)+b_{10}g\left(1\right)=0,\\ a_{21}{g}^{\prime }\left(0\right)+b_{21}{g}^{\prime }\left(1\right)+a_{20}g\left(0\right)+b_{20}g\left(1\right)=0.\end{array}\right.$$

In the general case, conditions (3) are asymmetrical, so the operator $L_x$ is non-self-adjointed. More precisely, when solving problem (1)–(3) by the method of separation of variables, there arises a spectral problem for the non-self-adjointed spatial operator L is given by the differential expression $Lv\left(x\right)=-v^{{}^{″}}\left(x\right)$, with nonlocal boundary conditions of the form

$$\left\{\begin{array}{c}{a}_{11}{v}^{{}^{\prime }}\left(0\right)+b_{11}{v}^{{}^{\prime }}\left(1\right)+a_{10}v\left(0\right)+b_{10}v\left(1\right)=0,\\ a_{21}{v}^{{}^{\prime }}\left(0\right)+b_{21}{v}^{{}^{\prime }}\left(1\right)+a_{20}v\left(0\right)+b_{20}v\left(1\right)=0.\end{array}\right.$$

(4)

Let us indicate the classification of the boundary conditions (4) and, respectively, conditions (3) according to [13]. The boundary conditions (4) (3) are called regular (by Birkhoff [21]) ([13], p. 66) (see also [22]), if one of the following three relations holds:

$$\begin{array}{c}{a}_{11}{b}_{21}-a_{21}{b}_{11}\ne 0,\\ a_{11}{b}_{21}-a_{21}{b}_{11}=0,\left|a_{11}{|+|b}_{11}\right|>0,a_{11}{b}_{20}+b_{11}{a}_{20}\ne 0,\\ a_{11}=b_{11}=a_{21}=b_{21}=0,a_{10}{b}_{20}-a_{20}{b}_{10}\ne 0.\end{array}$$

(5)

The regular boundary conditions are strongly regular in the first and third cases, and in the second case under the additional condition

$$a_{11}{a}_{20}+b_{11}{b}_{20}\ne \pm {(a}_{11}{b}_{20}+b_{11}{a}_{20}).$$

(6)

Any not strongly regular boundary conditions (3) according to ([20], Lemma 1) can be reduced to the form

$$\left\{\begin{array}{c}{a}_1{\displaystyle \frac{\partial u(0,t)}{\partial x}}+b_1{\displaystyle \frac{\partial u(1,t)}{\partial x}}+a_{0}u(0,t)+b_{0}u(1,t)=0,\left|a_1{|+|b}_1\right|>0,\\ c_{0}u(0,t)+d_{0}u(1,t)=0\end{array}\right.$$

(7)

of one of the following four types

$$\begin{array}{c}I-II.a_1\pm b_1=0,c_0\mp d_0\ne 0;\\ III-IV.c_0\pm d_0=0,a_1\mp b_1\ne 0.\end{array}$$

(8)

The model (1)–(2), (7) describes the nonstationary process of heat transfer in a homogeneous bar of the length 1. The ends of the bar match, but this contact may not be ideal. In this model, $u(x,t)$ is the temperature at the point x of the bar at time t, Laplacian $L_{x}u(x,t)={\displaystyle \frac{{\partial }^{2}u(x,t)}{\partial x^2}}$ specifies the rate of change of heat flux and $\frac{\partial u(x,t)}{\partial t}$ is the rate of temperature change over time, $f(x,t)$ characterizes the intensity of the sources (sinks) due to external influences, $g\left(x\right)$ is the initial temperature distribution of $u(x,t)$ in the bar, $t=0$ is an initial time point, and $t=T$ is a final one.

The classical solution of problems (1)–(2), (7) exists and is unique ([20], Theorem 2). This solution $u(x,t)$ is a continuous function in the closed domain $\overline{\Omega }$, and it belongs to the class $C^{2,1}\left(\overline{\Omega }\right)$, and satisfies (1) in $\Omega$, condition (2) at the initial moment of time and conditions (7) on the boundary.

3. The Problem Statement in a Difference Form. Construction of a Difference Scheme

Note that the correctness of the differential problem (1)–(2), (7) does not guarantee that its solution can be given in an analytical form or as a convergent series, or that the problem can be solved by the method of separation of variables. Therefore, we apply the numerical method for obtaining an approximate solution to the problem. If the method is correct and stable, then the approximate solution converges to the solution of the original differential problem in a certain sense, for example, in the chosen norm.

Problem (1)–(2), (7) is solved by the finite difference method ([23], p. 50). To do this, in $\overline{\Omega }$ we introduce a uniform grid ${\overline{\omega }}_{h\tau }={\overline{\omega }}_h\times {\overline{\omega }}_{\tau }$, where ${\overline{\omega }}_h=\left\{x_i=ih,i=\overline{0,N}\right\}$, ${\overline{\omega }}_{\tau }=\left\{t_n=n\tau ,n=\overline{0,M}\right\}$ and instead of the function $u(x,t)$ of the continuous arguments x, t we pass to the grid function $u(x_i,t_n)$ given at nodes of the grid ${\overline{\omega }}_{h\tau }$. Let us construct a discrete analogue of the differential problem, that is, a one-parameter family of difference schemes depending on the real parameter $\sigma$. As usual, we consider $0\le \sigma \le 1$. Denote by $u(x_i,t_n)=u_i^n\equiv u_i$, $y(x_i,t_n)=y_i^n\equiv y_i$ values of exact and approximate (numerical) solutions of problem (1)–(2), (7) at the node $(x_i,t_n)$, respectively. Denote $y\equiv y^n={(y_0^n,\dots ,y_N^n)}^{{}^{\prime }}$. For the differential equation, we write down a difference equation approximating it ([23], pp. 301–303) as

$$y_t=\Lambda y^{\left(\sigma \right)}+F,$$

where $y_t=(y^{n+1}-y^n)/\tau$, $\Lambda y=y_{\overline{x}x}$, ${(\Lambda y)}_i=y_{\overline{x}x,i}=(y_{i-1}-2y_i+y_{i+1})/h^2$, $y^{\left(\sigma \right)}=\sigma y^{n+1}+(1-\sigma )y^n$, $i=\overline{1,N-1}$, $n=0,1,\dots ,M$, F is a grid function approximating the right-hand side of the equation; let $F=f^{\left(\sigma \right)}$. We assume that the initial condition is accurate approximated: $y_i^0=g\left(x_i\right)$, $i=\overline{0,N}$.

It is known that the difference operator $\Lambda$ approximates the differential operator $L_x$ with order $O\left(h^2\right)$ ([23], pp. 58–59). Let us approximate the boundary conditions with the order of approximation not lower than in the main equation, i.e., $O\left(h^2\right)$. Obviously, the second boundary condition is accurate approximated:

$$c_0{y}_0+d_0{y}_N=0,n=1,2,\dots ,M.$$

To approximate the derivatives ${\displaystyle \frac{\partial u(0,t)}{\partial x}}$ and ${\displaystyle \frac{\partial u(1,t)}{\partial x}}$ in the first boundary condition, we use Equation (1):

$$\begin{array}{c}{\displaystyle \frac{\partial u(0,t)}{\partial x}}=y_{x,0}-0.5h\left(y_{t,0}-f_0\right)+O\left(h^2\right),\\ {\displaystyle \frac{\partial u(1,t)}{\partial x}}=y_{\overline{x},N}+0.5h\left(y_{t,N}-f_N\right)+O\left(h^2\right),\end{array}$$

where $y_{x,0}=\left(y_1-y_0\right)/h$, $y_{\overline{x},N}=(y_N-y_{N-1})/h$. Then the first boundary condition is written in the difference form

$$a_1\left(y_{x,0}-0.5h{y}_{t,0}+0.5h{f}_0\right)+b_1\left(y_{\overline{x},N}+0.5h{y}_{t,N}-0.5h{f}_N\right)+a_0{y}_0+b_0{y}_N=0.$$

The difference scheme consists of a difference equation, difference initial and boundary conditions and approximates the differential problem (1)–(2), (7).

Let us write the scheme in a two-layer form ([23], p. 397). Because the coefficients $c_0$, $d_0$ are not simultaneously equal to zero, then for definiteness we choose $c_0\ne 0$. Let us make the change in the first difference boundary condition

$$y_0=\beta y_N,\beta =-d_0/c_0.$$

As a result, the boundary condition takes the form

$$b{y}_{t,N}-\frac{2}{h}\left(a_1{y}_{x,0}+b_1{y}_{\overline{x},N}+\eta y_N\right)=a_1{f}_0-b_1{f}_N,$$

where $b=\beta a_1-b_1$, $\eta =\beta a_0+b_0$.

Denote by $B=diag(1,1,\dots ,1,b)$ the diagonal matrix of order N. Then the two-layer scheme for problem (1)–(2), (7) is written (for each $n=1,2,\dots )$ as follows:

$$\left\{\begin{array}{c}B{y}_t=\Lambda y^{\left(\sigma \right)}+F,\\ {\left(\Lambda y\right)}_i=y_{\overline{x}x,i},i=\overline{1,N-1},\\ {\left(\Lambda y\right)}_N=2h^{-1}\left(a_1{y}_{x,0}+b_1{y}_{\overline{x},N}+\eta y_N\right),\\ c_0{y}_0+d_0{y}_N=0,\\ y_i^0=g\left(x_i\right),i=\overline{0,N}.\end{array}\right.$$

(9)

As F, we choose the vector function

$$F_i=\left\{\begin{array}{c}{f}_i^{\left(\sigma \right)},i=\overline{1,N-1},\\ {(a_1{f}_0-b_1{f}_N)}^{\left(\sigma \right)},i=N.\end{array}\right.$$

4. Approximation Order of the Difference Scheme

Let us estimate the approximation order of the difference scheme (9). We introduce into consideration a grid function $z=y-u$, that is, the error of the solution of the difference scheme. To estimate the approximation error of difference scheme (9) on the solution u, we assume that the solution $u(x,t)$ belongs to the class $C^{l,k}\left(\overline{\Omega }\right)$ with necessary number of derivatives l and k.

First of all, define the approximation order of the difference equation in (9). To do this, we substitute $y=z+u$ into the equation assuming u as a given function. As a result, we obtain the equation for z,

$$z_t=\Lambda z^{\left(\sigma \right)}+\psi ,$$

where $\psi =\Lambda u^{\left(\sigma \right)}-u_t+\phi$ is the approximation error of the difference equation on the solution $u\left(x_i,t_n\right)$, $i=1,2,\dots ,N-1$, $n=1,2,\dots ,M$, $\phi$ is the approximation of the right-hand side of the equation, not necessarily accurate. In this case, $\phi =f^{\left(\sigma \right)}$.

We represent $u^{\left(\sigma \right)}$ as $u^{\left(\sigma \right)}=u+\tau \sigma u_t$. Substituting the expression $u_t={\displaystyle \frac{\partial u}{\partial t}}+{\displaystyle \frac{\tau }{2}}{\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}+O\left({\tau }^2\right)$ into $u^{\left(\sigma \right)}$, we have $u^{\left(\sigma \right)}=u+\tau \sigma {\displaystyle \frac{\partial u}{\partial t}}+O\left({\tau }^2\right)$. Moreover, to obtain the estimate, we need the following relations:

$$\begin{array}{c}\Lambda u=L_{x}u+{\displaystyle \frac{h^2}{12}}{L}_x^{2}u+O\left(h^4\right),L_{x}u+f={\displaystyle \frac{\partial u}{\partial t}},L_x^{2}u=L_x{\displaystyle \frac{\partial u}{\partial t}}-L_{x}f,\\ {\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}=L_x{\displaystyle \frac{\partial u}{\partial t}}+f_t,\tau h^2\le ({\tau }^2+h^4)/2.\end{array}$$

Here $L_x^{2}u={\displaystyle \frac{{\partial }^{4}u}{\partial x^4}}$. We get

$$\begin{array}{c}\psi =\Lambda u^{\left(\sigma \right)}-u_t+f^{\left(\sigma \right)}=L_x{u}^{\left(\sigma \right)}+{\displaystyle \frac{h^2}{12}}{L}_x^2{u}^{\left(\sigma \right)}+f^{\left(\sigma \right)}+O\left(h^4\right)-{\displaystyle \frac{\partial u}{\partial t}}-{\displaystyle \frac{\tau }{2}}{\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}+O\left({\tau }^2\right)\\ =\left((\sigma -0.5)\tau +{\displaystyle \frac{h^2}{12}}\right)L_x{\displaystyle \frac{\partial u}{\partial t}}+\left((\sigma -0.5)\tau {\displaystyle \frac{\partial f}{\partial t}}-{\displaystyle \frac{h^2}{12}}{L}_{x}f\right)+O\left({\tau }^2+h^4\right).\end{array}$$

It is obvious that the order of approximation remains unchanged from the replacement ${\displaystyle \frac{\partial f}{\partial t}}\approx f_t$, $L_{x}f\approx \Lambda f$. This shows that the difference equation generates an approximation of order $O\left(\tau +h^2\right)$ for $\sigma \ne 0.5$ and $u\in C^{4,2}\left(\overline{\Omega }\right)$, and $O\left({\tau }^2+h^2\right)$ for $\sigma =0.5$ and $u\in C^{4,3}\left(\overline{\Omega }\right)$. In this case, $\phi =f^{\left(\sigma \right)}$. The approximation order can be increased to $O\left({\tau }^2+h^4\right)$ by equating the coefficient at $L_x{\displaystyle \frac{\partial u}{\partial t}}$ to zero, finding

$$\sigma ={\sigma }_{*}=0.5-\frac{h^2}{12\tau }.$$

For an unchanged approximation order, when $\sigma ={\sigma }_{*}$, a proper choice of the function $\phi$ must be chosen as $\phi =f+{\displaystyle \frac{\tau }{2}}{f}_t+{\displaystyle \frac{h^2}{12}}\Lambda f$ and $u\in C^{6,3}\left(\overline{\Omega }\right)$. We can also take $\phi$ in the form ${\phi }^n=f^{n+1/2}+{\displaystyle \frac{h^2}{12}}\Lambda f^{n+1/2}$ ([23], p. 305).

Let us now estimate the approximation order of the first boundary condition

$$b{y}_{t,N}-\frac{2}{h}{\left(a_1{y}_{x,0}+b_1{y}_{\overline{x,}N}+\eta y_N\right)}^{\left(\sigma \right)}={\left(a_1{f}_0-b_1{f}_N\right)}^{\left(\sigma \right)}.$$

To do this, we rewrite it in the form

$$0.5h\left(a_1{y}_{t,0}-b_1{y}_{t,N}\right)-{\left(a_1{y}_{x,0}+b_1{y}_{\overline{x,}N}+a_0{y}_0+b_0{y}_N\right)}^{\left(\sigma \right)}=0.5h{\left(a_1{f}_0-b_1{f}_N\right)}^{\left(\sigma \right)}.$$

Substituting $y=z+u$ into the boundary condition and assuming u as a given function, we obtain a condition for z:

$$0.5h\left(a_1{z}_{t,0}-b_1{z}_{t,N}\right)-{\left(a_1{z}_{x,0}+b_1{z}_{\overline{x,}N}+a_0{z}_0+b_0{z}_N\right)}^{\left(\sigma \right)}={\psi }_1,$$

where ${\psi }_1={\left(a_1{u}_{x,0}+b_1{u}_{\overline{x,}N}+a_0{u}_0+b_0{u}_N\right)}^{\left(\sigma \right)}-0.5h\left(a_1{u}_{t,0}-b_1{u}_{t,N}\right)+0.5h{\phi }_1$ is the approximation error of the difference boundary condition on the solution u, ${\phi }_1$ is the right-hand side of the boundary condition obtained as a result of the approximation. In this case, ${\phi }_1={\left(a_1{f}_0-b_1{f}_N\right)}^{\left(\sigma \right)}$. We further use the relations given above for the main equation, as well as the expansion of the function u into Taylor series. We carry out some transformations and simplify the expression for ${\psi }_1$. As a result, we get

$$\begin{array}{c}{\psi }_1={\left(a_1{u}_{x,0}+b_1{u}_{\overline{x,}N}+a_0{u}_0+b_0{u}_N\right)}^{\left(\sigma \right)}-0.5h\left(a_1{u}_{t,0}-b_1{u}_{t,N}\right)+{\phi }_1\\ ={\left(a_1{\left({\displaystyle \frac{\partial u}{\partial x}}\right)}_0+0.5h{a}_1{\left({\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}\right)}_0+a_0{u}_0+b_1{\left({\displaystyle \frac{\partial u}{\partial x}}\right)}_N-0.5h{b}_1{\left({\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}\right)}_N+b_0{u}_N\right)}^{\left(\sigma \right)}\\ -0.5h\left(a_1{\left({\displaystyle \frac{\partial u}{\partial t}}\right)}_0+0.5a_1\tau {\left({\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}\right)}_0-b_1{\left({\displaystyle \frac{\partial u}{\partial t}}\right)}_N-0.5b_1\tau {\left({\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}\right)}_N\right)\\ +{0.5h\left(a_1{f}_0-b_1{f}_N\right)}^{\left(\sigma \right)}=0.5\tau h{a}_1(\sigma -0.5){\left(L_x{\displaystyle \frac{\partial u}{\partial t}}\right)}_0-0.5\tau h{b}_1(\sigma -0.5){\left(L_x{\displaystyle \frac{\partial u}{\partial t}}\right)}_N\\ +0.5h\left(a_1{\left(L_{x}u\right)}_0-b_1{\left(L_{x}u\right)}_N-a_1{\left({\displaystyle \frac{\partial u}{\partial t}}\right)}_0+b_1{\left({\displaystyle \frac{\partial u}{\partial t}}\right)}_N\right)-{\left(0.5\right)}^2\tau h{a}_1{\left({\displaystyle \frac{\partial f}{\partial t}}\right)}_0\\ +{\left(0.5\right)}^2\tau h{b}_1{\left({\displaystyle \frac{\partial f}{\partial t}}\right)}_N+0.5h{a}_1{f}_0+0.5\tau h\sigma a_1{\left({\displaystyle \frac{\partial f}{\partial t}}\right)}_0-0.5h{b}_1{f}_N-0.5\tau h\sigma b_1{\left({\displaystyle \frac{\partial f}{\partial t}}\right)}_N\\ +O\left({\tau }^2+h^2\right).\end{array}$$

Finally, we have the following expression for ${\psi }_1$:

$$\begin{array}{c}{\psi }_1=0.5\tau h(\sigma -0.5)\left(a_1{\left(L_x{\displaystyle \frac{\partial u}{\partial t}}\right)}_0-b_1{\left(L_x{\displaystyle \frac{\partial u}{\partial t}}\right)}_N+a_1{\left({\displaystyle \frac{\partial f}{\partial t}}\right)}_0-b_1{\left({\displaystyle \frac{\partial f}{\partial t}}\right)}_N\right)\\ +O\left({\tau }^2+h^2\right).\end{array}$$

Taking into account the obvious inequality $\tau h\le ({\tau }^2+h^2)/2$, we obtain the estimate

$${\psi }_1=O\left({\tau }^2+h^2\right).$$

This shows that the difference boundary condition approximates the original differential boundary condition with order $O\left({\tau }^2+h^2\right)$. In this case, ${\phi }_1={\left(a_1{f}_0-b_1{f}_N\right)}^{\left(\sigma \right)}$. As ${\phi }_1$ we can also choose the function ${\phi }_1=0.5\tau \left(a_1{f}_{t,0}-b_1{f}_{t,N}\right)+\left(a_1{f}_0-b_1{f}_N\right)$ or, combining similar terms, we get a different form ${\phi }_1=a_1{f}_0^{n+1/2}-b_1{f}_N^{n+1/2}$. The approximation order of the boundary condition can be increased to $O\left({\tau }^2+h^4\right)$ for the parameter value $\sigma ={\sigma }_{*}=0.5-{\displaystyle \frac{h^2}{12\tau }}$. Here, we use the result obtained in ([23], p. 321) for approximation of the boundary conditions of the third kind. It is easy to show that the difference boundary condition in this case is

$$\begin{array}{c}0.5h\left(\left(a_1-{\displaystyle \frac{h}{3}}{a}_0\right)y_{t,0}-\left(b_1+{\displaystyle \frac{h}{3}}{b}_0\right)y_{t,N}\right)-{\left(a_1{y}_{x,0}+b_1{y}_{\overline{x,}N}+a_0{y}_0+b_0{y}_N\right)}^{\left(\sigma \right)}\\ =0.5h{\phi }_2^{\left(\sigma \right)},\end{array}$$

where

$$\begin{array}{c}{\phi }_2={\displaystyle \frac{h^2}{6}}\left(a_1{\overline{f}}_0^{{}^{\prime }}+b_1{\overline{f}}_N^{{}^{\prime }}+a_0{\overline{f}}_0+b_0{\overline{f}}_N\right)\\ +{\displaystyle \frac{h^2}{2}}\left(\left(a_1-{\displaystyle \frac{h}{3}}{a}_0\right)\left({\overline{f}}_0+{\displaystyle \frac{h^2}{12}}{\overline{f}}_0^{{}^{″}}\right)+\left(b_1+{\displaystyle \frac{h}{3}}{b}_0\right)\left({\overline{f}}_N+{\displaystyle \frac{h^2}{12}}{\overline{f}}_N^{{}^{″}}\right)\right)\end{array}$$

and ${\overline{v}}_i=v_i^{n+1/2}$ is the value of the function $v(x,t)$ at the point $(x_i,t_{n+1/2})$, $f^{{}^{\prime }}={\displaystyle \frac{\partial f}{\partial x}}$, $f^{{}^{″}}={\displaystyle \frac{{\partial }^{2}f}{\partial x^2}}$. Then, in scheme (9) for $\sigma ={\sigma }_{*}$ the right-hand side is a function of the form

$$F_i=\left\{\begin{array}{c}{f}_i^{\left(\sigma \right)},i=\overline{1,N-1},\\ {\phi }_2^{\left(\sigma \right)},i=0;N,\end{array}\right.$$

$b=\beta \left(a_1-{\displaystyle \frac{h}{3}}{a}_0\right)-\left(b_1+{\displaystyle \frac{h}{3}}{b}_0\right)$ and $\eta$ remains the same.

Scheme (9) for $\sigma ={\sigma }_{*}$ and with appropriate choice of the function F is usually termed a higher-accuracy scheme. Respectively, we obtain an explicit scheme for $\sigma =0$, a pure implicit scheme for $\sigma =1$ and a symmetric difference scheme (scheme of Crank–Nicolson) for $\sigma =0.5$.

5. Algorithm for Reducing the Solving Nonlocal Difference Problem to the Successive Solving Two Local Difference Problems

The convergence of the solution of scheme (9) to the exact solution of problem (1)–(2), (7) follows from its approximation and stability with respect to initial data and the right-hand side ([23], p. 313). But in the general case, the problem of stability of the difference schemes with not strongly regular boundary conditions has not yet been completely solved. Some special cases of the difference schemes and the methods developed for proving the stability of these schemes can be found in the literature. As a rule, the stability of schemes is established in spaces with specially constructed grid energy norms, either in the form of operator inequalities which are difficult to check, or hard constraints are set on the parameters of the difference scheme; the stability of schemes is established by using computer simulation as well [24,25,26,27]. For implicit difference schemes the solution on the upper $(n+1)$ time layer can be obtained by one of the variants of the sweep method (in [23] it is termed as elimination method), if the conditions for applicability of the sweep method and stability of the scheme with respect to initial data coincide ([28], p. 72). However, the matrix of the system of linear algebraic equations obtained on the $(n+1)$ time layer is either ill-conditioned or there is no diagonal dominance in such a matrix ([28], p. 81). Checking the correctness of the algorithm of the sweep method in this case is another problem. Sufficient conditions which are easily checked for the applicability of the sweep method are introduced in ([28], p. 83) but these conditions are not always met even for problems with strongly regular boundary conditions. These difficulties complicate the solving of difference problems.

We apply an approach for finding a solution of the nonlocal difference scheme (9), in which we avoid proving the stability of the scheme and the solvability of the corresponding system of linear algebraic equations. For this, we use the results from [20]. In [20], the authors propose an algorithm for solving differential problems with not strongly regular boundary conditions that do not use the basis properties of the system of root functions. The algorithm is based on reducing the solving nonlocal differential problem to the successive solving two local differential problems with Sturm-type boundary conditions with respect to a spatial variable.

For numerical solving differential problems with not strongly regular boundary conditions, we adapt the algorithm from [20] for difference schemes approximating differential problems. If correctness and stability of the algorithm are fulfilled, then the study of the correctness of nonlocal difference schemes, in particular, their stability and convergence, is replaced by the same study, but for local difference schemes.

In difference scheme (9), we represent the solution y on each time layer n as the sum of two functions:

$$y=Q+S,$$

where $Q_i=(y_i+y_{N-i})/2$ is an even part of $y_i$, and $S_i=(y_i-y_{N-i})/2$ is an odd part of $y_i$. From the definition of the functions Q and S, we obtain their properties:

$$\begin{array}{c}{Q}_i=Q_{N-i},Q_{x,i}=-Q_{\overline{x},N-i},Q_{\overline{x}x,i}=Q_{\overline{x}x,N-i},\dots ,\\ S_i={-S}_{N-i},S_{x,i}=S_{\overline{x},N-i},S_{\overline{x}x,i}={-S}_{\overline{x}x,N-i},\dots ,i=0,\dots ,N.\end{array}$$

Let us show that at the point $\left(x_i,t_n\right)$, $i=1,\dots ,N-1$, $n=1,2,\dots ,M$, Q, and S satisfy equations similar to the difference equation of scheme (9). The difference is only on the right-hand sides of the equation. To do this, we write a system of two equations

$$\left\{\begin{array}{c}{y}_{t,i}^n-\sigma y_{\overline{x}x,i}^{n+1}-(1-\sigma )y_{\overline{x}x,i}^n=F_i,\\ y_{t,N-i}^n-\sigma y_{\overline{x}x,N-i}^{n+1}-(1-\sigma )y_{\overline{x}x,N-i}^n=F_{N-i},\end{array}\right.i=\overline{1,N-1}.$$

Multiply both equations by $0.5$; when adding the equations, we get an equation for Q and when subtracting one equation from another, we get an equation for S:

$$\begin{array}{c}{Q}_t-\Lambda Q^{\left(\sigma \right)}=\Phi ,Q^0\mathrm{is}\mathrm{given},\\ S_t-\Lambda S^{\left(\sigma \right)}=\Psi ,S^0\mathrm{is}\mathrm{given}.\end{array}$$

Here ${\Phi }_i=(F_i+F_{N-i})/2$, ${\Psi }_i=(F_i-F_{N-i})/2$, $i=1,\dots ,N-1$.

Substitute $y=Q+S$ into the first boundary condition

$$0.5h\left(a_1{y}_{t,0}-b_1{y}_{t,N}\right)-{\left(a_1{y}_{x,0}+b_1{y}_{\overline{x,}N}+a_0{y}_0+b_0{y}_N\right)}^{\left(\sigma \right)}=0.5h{\left(a_1{f}_0-b_1{f}_N\right)}^{\left(\sigma \right)}.$$

Then we get

$$\begin{array}{c}{0.5ha}_1\left(Q_{t,0}+S_{t,0}\right)-0.5h{b}_1\left(Q_{t,N}+S_{t,N}\right)\\ {-\left(a_1\left(Q_{x,0}+S_{x,0}\right)+b_1\left(Q_{\overline{x,}N}+S_{\overline{x,}N}\right)+a_0{Q}_0{+a}_0{S}_0+b_0{Q}_N+b_0{S}_N\right)}^{\left(\sigma \right)}\\ =0.5h{\left(a_1{f}_0-b_1{f}_N\right)}^{\left(\sigma \right)}.\end{array}$$

By using the properties of the functions Q, S, we finally obtain the following form of the boundary conditions at the point $(x_0,t_n)$:

$$\left\{\begin{array}{c}(a_1-b_1)Q_{t,0}+(a_1+b_1)S_{t,0}-2h^{-1}((a_1-b_1)Q_{x,0}+(a_1+b_1)S_{x,0}\\ +(a_0+b_0)Q_0+(a_0-b_0)S_0{)}^{\left(\sigma \right)}={\left(a_1{f}_0-b_1{f}_N\right)}^{\left(\sigma \right)},\\ (c_0+d_0)Q_0+(c_0-d_0)S_0=0.\end{array}\right.$$

At the point $(x_N,t_n)$, the conditions are as follows:

$$\left\{\begin{array}{c}{(a}_1-b_1)Q_{t,N}-{(a}_1+b_1)S_{t,N}{-2h^{-1}(-(a}_1-b_1)Q_{\overline{x,}N}+{(a}_1+b_1)S_{\overline{x,}N}\\ +{(a}_0+b_0)Q_N-{(a}_0-b_0){S_N)}^{\left(\sigma \right)}={\left(a_1{f}_0-b_1{f}_N\right)}^{\left(\sigma \right)},\\ (c_0+d_0)Q_N-(c_0-d_0)S_N=0.\end{array}\right.$$

Recall that the coefficients of the boundary conditions can only be of $I-IV$ types (see (8)). Consider the difference schemes with all types of conditions separately.

I. Let $a_1+b_1=0$, then $c_0-d_0\ne 0$ and $a_1-b_1\ne 0$. If we find $S_0$ from the second boundary condition and substitute this expression into the first boundary condition, then we obtain separate boundary conditions for the functions Q, S at the point $(x_0,t_n)$:

$$\left\{\begin{array}{c}{Q}_{t,0}-2h^{-1}{\left(Q_{x,0}-\mu Q_0\right)}^{\left(\sigma \right)}={\left(a_1-b_1\right)}^{-1}{\left(a_1{f}_0-b_1{f}_N\right)}^{\left(\sigma \right)},\\ S_0=-\frac{c_0+d_0}{c_0-d_0}{Q}_0,\end{array}\right.$$

where $\mu =\frac{(c_0+d_0)(a_0-b_0)}{c_0-d_0}-\frac{a_0+b_0}{a_1-b_1}.$ Similarly we have the boundary conditions for the functions Q, S at the point $(x_N,t_n)$:

$$\left\{\begin{array}{c}{Q}_{t,N}-2h^{-1}{\left(-Q_{\overline{x,}N}-\mu Q_N\right)}^{\left(\sigma \right)}={\left(a_1-b_1\right)}^{-1}{\left(a_1{f}_0-b_1{f}_N\right)}^{\left(\sigma \right)},\\ S_N=\frac{c_0+d_0}{c_0-d_0}{Q}_N.\end{array}\right.$$

The appropriate difference problems for the functions Q, S are written in the form

$$\left\{\begin{array}{c}{Q}_t-{\Lambda }_1{Q}^{\left(\sigma \right)}=\Phi ,Q^0\mathrm{is}\mathrm{given},\\ {\left({\Lambda }_{1}Q\right)}_i\equiv {\left(\Lambda Q\right)}_i=Q_{\overline{x}x,i},i=\overline{1,N-1},\\ {\left({\Lambda }_{1}Q\right)}_0=2h^{-1}\left(Q_{x,0}-\mu Q_0\right),\\ {\left({\Lambda }_{1}Q\right)}_N=-2h^{-1}\left(Q_{\overline{x,}N}+\mu Q_N\right);\end{array}\right.$$

(10)

$$\left\{\begin{array}{c}{S}_t-\Lambda S^{\left(\sigma \right)}=\Psi ,S^0\mathrm{is}\mathrm{given},\\ {\left(\Lambda S\right)}_i=S_{\overline{x}x,i},i=\overline{1,N-1},\\ S_0=\nu Q_0,S_N=-\nu Q_N,\end{array}\right.$$

(11)

where $\nu =-\frac{c_0+d_0}{c_0-d_0}$, ${\Phi }_0={\Phi }_N={\left(a_1-b_1\right)}^{-1}{\left(a_1{f}_0-b_1{f}_N\right)}^{\left(\sigma \right)}$. Obviously, first one needs to solve problem (10) and determine the function Q, and then solve the inhomogeneous Dirichlet problem (11) and determine the function S, where the values $Q_0$ and $Q_N$ of the function Q are included in the right-hand side of the boundary conditions.

$II$. Let $a_1-b_1=0$, then $c_0+d_0\ne 0$, and $a_1+b_1\ne 0$. If we find $Q_0$ from the second boundary condition and substitute this expression into the first boundary condition, then we obtain difference problems for the functions Q, S, similar to (10), (11):

$$\left\{\begin{array}{c}{S}_t-{\Lambda }_2{S}^{\left(\sigma \right)}=\Psi ,S^0\mathrm{is}\mathrm{given},\\ {\left({\Lambda }_{2}S\right)}_i\equiv {\left(\Lambda S\right)}_i=S_{\overline{x}x,i},i=\overline{1,N-1},\\ {\left({\Lambda }_{2}S\right)}_0=2h^{-1}\left(S_{x,0}-{\mu }_1{S}_0\right),\\ {\left({\Lambda }_{2}S\right)}_N=-2h^{-1}\left(S_{\overline{x,}N}+{\mu }_1{S}_N\right);\end{array}\right.$$

(12)

$$\left\{\begin{array}{c}{Q}_t-\Lambda Q^{\left(\sigma \right)}=\Phi ,Q^0\mathrm{is}\mathrm{given},\\ {\left(\Lambda Q\right)}_i=Q_{\overline{x}x,i},i=\overline{1,N-1},\\ Q_0={\nu }_1{S}_0,Q_N=-{\nu }_1{S}_N,\end{array}\right.$$

(13)

where ${\mu }_1=\frac{(c_0-d_0)(a_0+b_0)}{c_0+d_0}-\frac{a_0-b_0}{a_1+b_1}$, ${\nu }_1=-\frac{c_0-d_0}{c_0+d_0}$, ${\Psi }_0=-{\Psi }_N={\left(a_1+b_1\right)}^{-1}{\left(a_1{f}_0-b_1{f}_N\right)}^{\left(\sigma \right)}$.

$III$. Let $c_0+d_0=0$, then $a_1-b_1\ne 0$ and $c_0-d_0\ne 0$. From the second boundary condition, we get $S_0=S_N=0$, which we substitute into the first boundary condition. The appropriate difference problems for S and Q are written in the form

$$\left\{\begin{array}{c}{S}_t-\Lambda S^{\left(\sigma \right)}=\Psi ,S^0\mathrm{is}\mathrm{given},\\ {\left(\Lambda S\right)}_i=S_{\overline{x}x,i},i=\overline{1,N-1},\\ S_0=S_N=0;\end{array}\right.$$

(14)

$$\left\{\begin{array}{c}{Q}_t-{\Lambda }_3{Q}^{\left(\sigma \right)}=\Phi ,Q^0\mathrm{is}\mathrm{given},\\ {\left({\Lambda }_{3}Q\right)}_i\equiv {\left(\Lambda Q\right)}_i=Q_{\overline{x}x,i},i=\overline{1,N-1},\\ {\left({\Lambda }_{3}Q\right)}_0=2h^{-1}\left(Q_{x,0}-{\mu }_2{Q}_0\right),\\ {\left({\Lambda }_{3}Q\right)}_N=-2h^{-1}\left(Q_{\overline{x,}N}+{\mu }_2{Q}_N\right),\end{array}\right.$$

(15)

where ${\mu }_2=-\frac{a_0+b_0}{a_1-b_1}$, ${\Phi }_0={\Phi }_N={\left(a_1-b_1\right)}^{-1}{\left(a_1{f}_0-b_1{f}_N\right)}^{\left(\sigma \right)}+2h^{-1}\frac{a_1+b_1}{a_1-b_1}{S}_{x,0}^{\left(\sigma \right)}$.

$IV$. Let $c_0-d_0=0$, then $a_1+b_1\ne 0$, and $c_0+d_0\ne 0$. From the second boundary condition, we get $Q_0=Q_N=0$ and appropriate difference problems are written in the form

$$\left\{\begin{array}{c}{Q}_t-\Lambda Q^{\left(\sigma \right)}=\Phi ,Q^0\mathrm{is}\mathrm{given},\\ {\left(\Lambda Q\right)}_i=Q_{\overline{x}x,i},i=\overline{1,N-1},\\ Q_0=Q_N=0;\end{array}\right.$$

(16)

$$\left\{\begin{array}{c}{S}_t-{\Lambda }_4{S}^{\left(\sigma \right)}=\Psi ,S^0\mathrm{is}\mathrm{given},\\ {\left({\Lambda }_{4}S\right)}_i\equiv {\left(\Lambda S\right)}_i=S_{\overline{x}x,i},i=\overline{1,N-1},\\ {\left({\Lambda }_{4}S\right)}_0=2h^{-1}\left(S_{x,0}-{\mu }_3{S}_0\right),\\ {\left({\Lambda }_{4}S\right)}_N=-2h^{-1}\left(S_{\overline{x,}N}+{\mu }_3{S}_N\right),\end{array}\right.$$

(17)

where ${\mu }_3=-\frac{a_0-b_0}{a_1+b_1}$, ${\Psi }_0={\left(a_1+b_1\right)}^{-1}{\left(a_1{f}_0-b_1{f}_N\right)}^{\left(\sigma \right)}+2h^{-1}\frac{a_1-b_1}{a_1+b_1}{Q}_{x,0}^{\left(\sigma \right)}$, ${\Psi }_N=$$-{\left(a_1+b_1\right)}^{-1}{\left(a_1{f}_0-b_1{f}_N\right)}^{\left(\sigma \right)}+2h^{-1}\frac{a_1-b_1}{a_1+b_1}{Q}_{\overline{x,}N}^{\left(\sigma \right)}$.

Thus, in each of the four types (8) the solving nonlocal difference problem with not strongly regular conditions is reduced to the successive solving two local difference problems with Sturm-type boundary conditions.

6. Proof of the Stability of Non-Local Difference Scheme

In the space of the grid functions v we introduce the scalar product and the norm

$$(v,w)=\sum _1^{N-1}h{v}_i{w}_i+0.5h(v_0{w}_0+v_N{w}_N),\parallel v\parallel ={\parallel v\parallel }_{L_2^h}=\sqrt{(v,v)}.$$

(18)

The grid analogue of the norm can be chosen in C:

$$\parallel v\parallel ={\parallel v\parallel }_C=\underset{0\le i\le N}{max}\left|v_i\right|.$$

To investigate the stability of local difference schemes (10)–(17) in one of the norms for any $\sigma \in [0,1]$, one can choose the method of energy inequalities, or the method of separation of variables, or the maximum principle ([23], p. 323). Recall that under natural assumptions the convergence of the difference scheme for linear problems follows from its approximation and stability ([23], p. 313). To find the solutions Q and S on the upper $(n+1)$ time layer, if the scheme is implicit, one can use the sweep method, which is stable and correct for boundary value problems for the heat equation with boundary conditions of the first, second, and third kind ([23], p. 323).

To justify the applicability of the algorithm from [20], adapted for difference schemes, it is necessary to derive estimates of the stability of the schemes with respect to the initial data and the right-hand side of scheme (9), i.e., with respect to the functions g and F. From the estimates for Q and S, and by using the well-known triangle inequality $\parallel y\parallel \le \parallel Q\parallel +\parallel S\parallel$, we derive an estimate of the stability of the solution y and, as a result, its convergence to the solution u of the differential problem (1)–(2), (7).

As a sample, we consider schemes of type $IV$ for which the following conditions are met: $a_1+b_1\ne 0$, $c_0-d_0=0$. To prove the stability of the schemes and the algorithm itself, we slightly modify the representation of the solution y. Note that, without loss of generality, we can assume $c_0=d_0=1$ and $a_1+b_1=1$. Denote $a_1=c$, $b_1=1-c$. Let us introduce the functions: Q, as before, is an even part of the solution y, and W is a function of the form

$$W=y-(1-(1-2c)(2x-1))Q,x\in {\overline{\omega }}_h.$$

Though the function W is not an odd part of the solution y, this function is odd. Indeed, W can be represented as $W=S-(1-2c)(2x-1)Q$, where S, as before, is the odd part of the solution y, and the term $-(1-2c)(2x-1)Q$) is an odd function on ${\omega }_h$. Once the functions W and Q are defined, the solution y is found by the formula

$$y=W+(1-(1-2c)(2x-1))Q,x\in {\overline{\omega }}_h.$$

Let us construct difference schemes for the functions Q and W. Clearly, the difference equation for Q remains unchanged:

$$Q_t-\Lambda Q^{\left(\sigma \right)}=\Phi ,Q^0\mathrm{is}\mathrm{given}.$$

Denote $r\left(x\right)=-(1-(1-2c\left)\right(2x-1\left)\right)$, $x\in {\overline{\omega }}_h$. Then, $r_0=-2(1-c)$, $r_N=-2c$, $r_x=r_{\overline{x}}=2(1-2c)$, $r_{\overline{x}x}=0$. In addition, we need the estimate

$$\underset{x\in {\overline{\omega }}_h}{max}\left|r\left(x\right)\right|=max|r_0|,|r_N|\le 2+\left|r_N\right|=A.$$

Applying formulas of difference differentiation, we obtain

$$\Lambda {\left(rQ\right)}_i={\left(rQ\right)}_{\overline{x}x,i}=r_{x,i}{Q}_{\overline{x},i}+r_{\overline{x},i}{Q}_{x,i-1}+r_i{Q}_{\overline{x}x,i}=4(1-2c)Q_{\overline{x},i}+r_i{(\Lambda Q)}_i,i=\overline{1,N-1}.$$

Then,

$$W_t-\Lambda W^{\left(\sigma \right)}\equiv {(y+rQ)}_t-\Lambda {(y+rQ)}^{\left(\sigma \right)}=F+r\left(Q_t-\Lambda Q^{\left(\sigma \right)}\right)-4(1-2c)Q_{\overline{x}}.$$

Finally, we obtain the following difference equation for W

$$W_t-\Lambda W^{\left(\sigma \right)}=R,W^0\mathrm{is}\mathrm{given},$$

where $R=F+r\Phi -4(1-2c)Q_{\overline{x}}$, $W^0=g\left(x\right)+r{Q}^0$. Substitute $y=W-rQ$ into the second boundary condition $y_0+y_N=0$. As a result, we get

$$W_0-r_0{Q}_0+W_N-r_N{Q}_N=-(r_0+r_N)Q_0=0,$$

which implies that $Q_0=Q_N=0$. Thus, the difference scheme for Q still has the form (16). Let us define the boundary condition for W at $x_0=0$. To do this, we substitute $y=W-rQ$ into the first boundary condition

$$\begin{array}{c}0.5h(c{y}_{t,0}-(1-c)y_{t,N})-{(c{y}_{x,0}+(1-c)y_{\overline{x,}N}+a_0{y}_0+b_0{y}_N)}^{\left(\sigma \right)}\\ =0.5h{(c{f}_0-(1-c)f_N)}^{\left(\sigma \right)},\end{array}$$

where $Q_0=Q_N=0$. We get

$$\begin{array}{c}0.5h(c{W}_{t,0}-(1-c)W_{t,N})-{\left(c{(W-rQ)}_{x,0}+(1-c){(W-rQ)}_{\overline{x,}N}+a_0{W}_0+b_0{W}_N\right)}^{\left(\sigma \right)}\\ =0.5h{(c{f}_0-(1-c)f_N)}^{\left(\sigma \right)}\end{array}$$

or, taking into account that $c{\left(rQ\right)}_{x,0}+(1-c){\left(rQ\right)}_{\overline{x,}N}=2h(1-2c)Q_{x,0}$, we obtain

$$W_{t,0}-\frac{2}{h}{(W_{x,0}-{\mu }_4{W}_0)}^{\left(\sigma \right)}=R_0,$$

where ${\mu }_4=b_0-a_0$, $R_0={(c{f}_0-(1-c)f_N)}^{\left(\sigma \right)}-4(1-2c)Q_{x,0}^{\left(\sigma \right)}$. Thus, the difference schemes for Q and W are written as follows:

$$\left\{\begin{array}{c}{Q}_t-\Lambda Q^{\left(\sigma \right)}=\Phi ,Q^0\mathrm{is}\mathrm{given},\\ {\left(\Lambda Q\right)}_i=Q_{\overline{x}x,i},i=\overline{1,N-1},\\ Q_0=Q_N=0;\end{array}\right.$$

(19)

$$\left\{\begin{array}{c}{W}_t-{\Lambda }_5{W}^{\left(\sigma \right)}=R,W^0\mathrm{is}\mathrm{given},\\ {\left({\Lambda }_{5}W\right)}_i\equiv {\left(\Lambda W\right)}_i=W_{\overline{x}x,i},i=\overline{1,N-1},\\ {\left({\Lambda }_{5}W\right)}_0=2h^{-1}\left(W_{x,0}-{\mu }_4{W}_0\right),\\ {\left({\Lambda }_{5}W\right)}_N=-2h^{-1}\left(W_{\overline{x},N}+{\mu }_4{W}_N\right),\end{array}\right.$$

(20)

where R is a function of the form

$$R=\left\{\begin{array}{c}{F}_i+r_i{\Phi }_i-4(1-2c)Q_{\overline{x},i}^{\left(\sigma \right)},i=\overline{1,N-1},\\ {\left(c{f}_0-(1-c)f_N\right)}^{\left(\sigma \right)}-4(1-2c)Q_{x,0}^{\left(\sigma \right)},i=0,\\ -{\left(c{f}_0-(1-c)f_N\right)}^{\left(\sigma \right)}-4(1-2c)Q_{\overline{x},N}^{\left(\sigma \right)},i=N.\end{array}\right.$$

Scheme (19) is called uniformly stable, if for its solution the estimate

$$\parallel Q^{n+1}\parallel \le \parallel Q^n\parallel$$

is valid ([23], p. 392), which implies the estimate with respect to the initial data and the right-hand side

$$\parallel Q^{n+1}\parallel \le \parallel Q^0\parallel +\sum _{j=0}^n\tau \parallel {\Phi }^j\parallel .$$

For difference scheme (20), there are similar estimates with the values $\parallel W^0\parallel$, ${\sum }_{j=0}^n\tau \parallel R^j\parallel$.

According to ([23], p. 313) schemes (19), (20) are uniformly stable for $\sigma \ge 0.5-{\displaystyle \frac{h^2}{4\tau }}$. Taking into account the representations $Q_i=(y_i+y_{N-i})/2$ and ${\Phi }_i=(F_i+F_{N-i})/2$, $F_i=f_i^{\left(\sigma \right)}$, $i=\overline{1,N-1}$, and the conditions $Q_0=Q_N=0$, we get the estimates

$$\parallel Q^{n+1}\parallel \le \parallel Q^n\parallel ,\parallel Q^{n+1}\parallel \le \parallel g\parallel +\sum _{j=0}^n\tau \parallel F^j\parallel ,$$

i.e., scheme (19) is stable with respect to g and F. In addition, there are a priori estimates

$$\parallel Q_{\overline{x}}^{n+1}\left]\right|\le \parallel Q_{\overline{x}}^n\left]\right|,\parallel Q_{\overline{x}}^{n+1}\left]\right|\le \parallel Q_{\overline{x}}^0\left]\right|,\parallel v\left]\right|={\left(\sum _{j=1}^{N}h{v}_i^2\right)}^{1/2},$$

which mean the uniform stability of scheme (19) in the norm $\parallel Q_{\overline{x}}\left]\right|$, being a grid analogue of the norm in $\stackrel{\circ }{W_2^1}$ ([23], p. 319). The following estimate is also obvious:

$$\parallel Q_{\overline{x}}^{\left(\sigma \right)}{\left]\right|}^2\le \parallel Q_{\overline{x}}^0{\left]\right|}^2=\sum _{i=1}^{N}h{\left(Q_{\overline{x},i}^0\right)}^2\le 0.5\sum _{i=1}^{N}h\left({\left(y_{\overline{x},i}^0\right)}^2+{\left(y_{\overline{x},N-i}^0\right)}^2\right)\le \sum _{i=1}^{N}h{\left(y_{\overline{x},i}^0\right)}^2\le \parallel g_{\overline{x}}{\left]\right|}^2.$$

To estimate $\parallel W^{n+1}\parallel$, as the norm $\parallel R\parallel$ we take the norm which is concordance with $\parallel W\parallel$ ([23], p. 394) (in this case $\parallel R\parallel ={\parallel R\parallel }_{L_2^h}$) and $g\in W_2^1$. As a result, we get

$$\begin{array}{c}\parallel W^0\parallel \le \parallel g\parallel +\underset{x\in {\overline{\omega }}_h}{max}\left|r\left(x\right)\right|\parallel g\parallel =(1+A)\parallel g\parallel ,\\ \parallel R\parallel \le \parallel F\parallel +\underset{x\in {\overline{\omega }}_h}{max}\left|r\left(x\right)\right|\parallel F\parallel +4|1-2c|\parallel g_{\overline{x}}\left]\right|\le (1+A)\parallel F\parallel +4|1-2c|\parallel g_{\overline{x}}\left]\right|,\\ \parallel W^{n+1}\parallel \le \parallel W^0\parallel +\sum _{j=0}^n\tau \parallel R^j\parallel \le (1+A)\parallel g\parallel +(1+A)\sum _{j=0}^n\tau \parallel F^j\parallel +4|1-2c|\parallel g_{\overline{x}}\left]\right|.\end{array}$$

For the solution y we get the estimate

$$\begin{array}{c}\parallel y^{n+1}\parallel \le \parallel W^{n+1}\parallel +\underset{x\in {\overline{\omega }}_h}{max}\left|r\left(x\right)\right|\parallel Q^{n+1}\parallel \\ \le (1+A)\parallel g\parallel +(1+A)\sum _{j=0}^n\tau \parallel F^j\parallel +4|1-2c|\parallel g_{\overline{x}}\left]\right|+A\left(\parallel g\parallel +\sum _{j=0}^n\tau \parallel F^j\parallel \right).\end{array}$$

Denote $M=max(1+2A,4|1-2c\left|\right)$. Finally, we derive the estimate

$$\parallel y^{n+1}\parallel \le M\left(\parallel g\parallel +\parallel g_{\overline{x}}\left]\right|+\sum _{j=0}^n\tau \parallel F^j\parallel \right).$$

(21)

Similarly, one can show the stability of difference scheme (9) for boundary conditions of $I-III$ types.

Let us introduce the following definition of the stability of difference scheme (9).

Definition 1.

A difference scheme (9) is said to be stable if its solution y admits the estimate (21), where $M>0$ is a constant independent on h, τ both and $g\in W_2^1(0,1)$.

Taking into account the above estimates, we arrive at the following statements.

Theorem 1.

The algorithm for reducing the solving scheme (9) with not strongly regular boundary conditions to the successive solving of two difference schemes with Sturm-type boundary conditions with respect to a spatial variable is correct and stable.

Theorem 2.

The difference scheme (9) is stable in $L_2^h$-norm for $\sigma \ge 0.5-\frac{h^2}{4\tau }$ and its solution y admits the estimate (21).

Consequently, we have considered all types of boundary value problems (1)–(2), (7) with not strongly regular boundary conditions and found their numerical solutions by using the algorithm from [20] adapted for difference schemes. The numerical solution y of scheme (9) is stable both to errors in the initial data and the right-hand side and to round-off errors on each time layer. Thus, difference scheme (9) is stable, approximates the original differential problem (1)–(2), (7), and its solution y converges to the exact solution u of the original problem (1)–(2), (7) in the grid $L_2^h$-norm. It is worth noting that the condition $\sigma \ge 0.5-\frac{h^2}{4\tau }$ is necessary and sufficient for solving the difference problems with local boundary conditions. Thus, based on the present study, we can state that the condition on $\sigma$ is also necessary and sufficient for the stability of the difference schemes with non-local boundary conditions (not strongly regular ones).

7. Examples

Let us present examples of well-known difference schemes with not strongly regular boundary conditions. For these difference schemes, the stability and convergence have already been investigated by other authors. Although solutions of the difference schemes are obtained in various norms and under different restrictions on the parameters of the schemes, according to the algorithm described in Section 5, all schemes are stable and converge in $L_2^h$-norm under the condition: $\sigma \ge 0.5-\frac{h^2}{4\tau }$.

The differential problem has the form

$$\left\{\begin{array}{c}{u}_t(x,t)=u_{xx}(x,t),0<x<1,0<t\le T,\\ u(x,0)=g\left(x\right),0\le x\le 1,t=0,\\ \gamma u_x(0,t)-u_x(1,t)-\alpha u(1,t)=0,\\ u(0,t)-du(1,t)=0.\end{array}\right.$$

(22)

Here $\alpha$, $\gamma$, d are real parameters, $g\left(x\right)$ is a given real function. The approximating difference scheme for this problem is written in the form:

$$\left\{\begin{array}{c}{y}_t=\Lambda y^{\left(\sigma \right)},\\ {(\Lambda y)}_i=y_{\overline{x}x,i},i=\overline{1,N-1},\\ {(\Lambda y)}_N=2h^{-1}(\gamma y_{x,0}-y_{\overline{x},N}-\alpha y_N),\\ y_i^0=g\left(x_i\right),i=\overline{0,N};y_0-d{y}_N=0.\end{array}\right.$$

(23)

Example 1.

Consider scheme (23) with boundary conditions of $I-II$ types: $a_1\pm b_1=0$, $c_0\mp d_0\ne 0$. Assume that $d=0$. In this case, the scheme has the form

$$\left\{\begin{array}{c}{y}_t=\Lambda y^{\left(\sigma \right)},\\ {(\Lambda y)}_i=y_{\overline{x}x,i},i=\overline{1,N-1},\\ {(\Lambda y)}_N=2h^{-1}(\gamma y_{x,0}-y_{\overline{x},N}-\alpha y_N),\\ y_i^0=g\left(x_i\right),i=\overline{0,N};y_0=0.\end{array}\right.$$

(24)

Depending on the values of the parameters $\alpha$ and $\gamma$ in (24), we have the following problems previously studied by other authors:

(1)

$\alpha =0$, $\gamma =1$ is the Samarskii–Ionkin problem [29,30,31,32];

(2)

$\alpha \ne 0$, $\gamma =1$ is the Samarskii–Ionkin problem with perturbation [19,31]; and

(3)

$\alpha =0$, $\gamma \ne 0;1$ is the Samarskii–Ionkin problem with a parameter [25].

For $\left|\gamma \right|\ne 1$ and $\alpha$ is arbitrary, we obtain problems with strongly regular boundary conditions according to the second condition from (5) and (6).

For $\left|\gamma \right|=1$ and $\alpha$ is arbitrary, we have schemes with not strongly regular boundary conditions.

In the case (1) $c_0=1$, $d_0=0$, $a_1=-b_1=1$, $a_0=b_0=0$. The difference scheme is rewritten as follows:

$$\left\{\begin{array}{c}{y}_t=\Lambda y^{\left(\sigma \right)},\\ {(\Lambda y)}_i=y_{\overline{x}x,i},i=\overline{1,N-1},\\ {(\Lambda y)}_N=2h^{-1}(\gamma y_{x,0}-y_{\overline{x},N}),\\ y_i^0=g\left(x_i\right),i=\overline{0,N};y_0=0.\end{array}\right.$$

(25)

The uniform stability of scheme (25) with respect to the initial data and the right-hand side is proven in [33]. The energy norm is constructed for which the scheme is stable, and the equivalence constants of this norm and the $L_2^h$-norm independent of $h>0$ are determined. In this case, the condition $\sigma \ge 0.5-\frac{h^2}{4\tau }$ is necessary and sufficient for the stability of scheme (25) [24,31].

In the case (2) $c_0=1$, $d_0=0$, $a_1=-b_1=1$, $a_0=0$, $b_0=\alpha \ne 0$. The difference scheme (24) has the form (for $a_1=-1$ slightly different)

$$\left\{\begin{array}{c}{y}_t=\Lambda y^{\left(\sigma \right)},\\ {(\Lambda y)}_i=y_{\overline{x}x,i},i=\overline{1,N-1},\\ {(\Lambda y)}_N=2h^{-1}(\pm y_{x,0}-y_{\overline{x},N}-\alpha y_N),\\ y_i^0=g\left(x_i\right),i=\overline{0,N};y_0=0.\end{array}\right.$$

(26)

Difference scheme (26) as well as the original differential problem are unstable for $\alpha <0$. For $\alpha >0$, the energy norm is constructed under which the scheme is stable and the equivalence constants of this norm and the $L_2^h$-norms independent on $h>0$ are determined; the necessary and sufficient conditions for the stability of the scheme in $L_2^h$-norm are received [31].

Example 2.

Consider difference schemes with boundary conditions of $III-IV$ types: $c_0\pm d_0=0$, $a_1\mp b_1\ne 0$. The differential problem (22) is rewritten in the form

$$\left\{\begin{array}{c}{u}_t(x,t)=u_{xx}(x,t),0<x<1,0<t\le T,\\ u(x,0)=g\left(x\right),0\le x\le 1,t=0,\\ u_x(0,t)\pm a{u}_x(1,t)+cu(0,t)=0,\\ u(0,t)\pm u(1,t)=0.\end{array}\right.$$

(27)

Let $a=1$. The approximating difference scheme for problem (27) has the form

$$\left\{\begin{array}{c}{y}_t=\Lambda y^{\left(\sigma \right)},\\ {(\Lambda y)}_i=y_{\overline{x}x,i},i=\overline{1,N-1},\\ {(\Lambda y)}_N=2h^{-1}(y_{x,0}\pm y_{\overline{x},N}+c{y}_0),\\ y_i^0=g\left(x_i\right),i=\overline{0,N};y_0\pm y_N=0.\end{array}\right.$$

(28)

Schemes (28) are (anti-) periodic difference schemes ($c=0$) and (anti-) periodic schemes with perturbation ($c\ne 0$). The main difference operator of scheme (28) is self-adjointed, and the scheme itself is stable ([28], p. 54). Because the boundary conditions in the problem are combined, the sweep method cannot be applied directly to find its solution. To solve the problem, a cyclic sweep method is developed which is correct and stable in this case ([28], p. 77). For $c\ne 0$ another version of the sweep method is developed ([28], p. 81).

8. Conclusions

The examples from Section 7 confirm the effectiveness of the approach which we propose to prove the stability and convergence of nonlocal difference schemes approximating initial-boundary value problems for the heat equation with not strongly regular boundary conditions. The main advantage of the algorithm suggested is that it can be directly used to obtain solution to the difference problem. There is no need to know in advance whether or not a Riesz basis of a spatial differential operator exists, or to determine its spectral properties, or to establish the stability of the difference scheme. According to the algorithm, all schemes of the form (9) with not strongly regular boundary conditions are stable and converge in $L_2^h$-norm. The necessary and sufficient condition for the stability of nonlocal difference schemes is the same as for local ones: $\sigma \ge 0.5-\frac{h^2}{4\tau }$. The use of the algorithm being applied for differential problems [20] and difference schemes, in a certain sense, completely solves the issue of constructing solutions to nonlocal heat conduction problems with not strongly regular boundary conditions. It is proven that the solutions exist, are unique, and continuously depend on the initial data of the problem. The solution of the differential problem (1)–(2), (7), is defined as the sum of two series [20]. The numerical solution of the problems (1)–(2), (7) is given as the sum of the solutions of two appropriate local difference schemes or directly using the scheme (9).

We emphasize once again the limitation of the present study: the proposed algorithm for reducing the solving nonlocal difference scheme to the successive solving two local difference schemes with Sturm-type boundary conditions is not feasible for other classes of boundary conditions, but only for the class of not strongly regular ones.

Thus, the present study shows that the problems arising in the simulation of nonlocal physical processes, including heat conduction processes, in closed contours with ideal/nonideal contact at the ends can be solved numerically when not strongly regular boundary conditions are required.

Note that this algorithm is also applicable to solving nonlocal problems for a second order ordinary differential equation with not strongly regular boundary conditions of the form

$$\begin{array}{c}Lv\left(x\right)\equiv -v^{″}\left(x\right)+q\left(x\right)v\left(x\right)=f\left(x\right),0<x<1,\\ \left\{\begin{array}{c}{a}_1{v}^{\prime }\left(0\right)+b_1{v}^{\prime }\left(1\right)+a_{0}v\left(0\right)+b_{0}v\left(1\right)=0,\\ c_{0}v\left(0\right)+d_{0}v\left(1\right)=0.\end{array}\right.\end{array}$$

Here, $q\left(x\right)$ and $f\left(x\right)$ are given real functions, $q\left(x\right)>0$ is a symmetric function: $q\left(x\right)=q(1-x)$, and the boundary conditions satisfy one of the types $I-IV$. The difference scheme with weights for this problem is stable for the same values of the parameter $\sigma$.

In recent years, there have been several papers on fractional-order dynamical systems [34,35,36,37,38,39]. As one of the future directions of the research, the authors are going to extend the proposed algorithm for solving nonlocal time-fractional boundary value problems with not strongly regular boundary conditions in the spatial variable.