1 Introduction

The differential equations with unknown parameters are widely used in mathematical modelling of real-life systems in different fields of science and have been studied extensively by many researchers (see [17, 19, 20, 22, 23, 25, 26, 29] and the references therein). These type of problems are called inverse, since the differential equation contains unknown source term (parameter) and therefore its solution cannot be determined uniquely from imposed initial and/or boundary conditions. Typically, one needs to introduce some additional condition(s) into the problem to gain a well-posedness of a mathematical model.

In general, the differential equations with unknown parameters are not solvable analytically and therefore one needs to tackle them by using numerical approaches. Construction of accurate and efficient numerical methods for differential equations with unknown parameters has been a subject of study by many authors (see, for example, [1, 2, 4, 8, 11, 12, 21, 27, 28, 31]).

There is a great deal of work in the analysis of direct problems for hyperbolic–parabolic equations (see [18, 24, 30]), as well as in construction of difference schemes for such problems (see [5,6,7, 10]). However, the inverse problems for hyperbolic–parabolic equations with unknown parameters have not been well-investigated in general.

Various local and nonlocal boundary value problems for hyperbolic–parabolic equations with unknown parameter can be reduced to the boundary value problem for the differential equation with parameter p

$$\begin{aligned} \left\{ \begin{array}{l} u''(t) + Au(t) = p + f(t), ~ 0<t<1, \\ u'(t) + Au(t) = p + g(t), ~ -1<t<0, \\ u(0+)=u(0-), ~ u'(0+) = u'(0-), \\ u(-1)=\varphi , ~ u(\lambda )=\psi , ~ -1<\lambda \le 1 \end{array} \right. \end{aligned}$$
(1)

in a Hilbert space H with self-adjoint positive definite operator A satisfying \(A\ge \delta I\), where \(\delta >0\). The last condition in (1) is considered in order to compensate the uncertainty in the problem due to unknown term p.

The unique solvability of problem (1) in the space C(H) of the continuous H-valued functions u(t) defined on \([-1,1]\), equipped with the norm

$$\begin{aligned} \Vert u \Vert _{C(H)} = \max \limits _{-1\le t\le 1} \Vert u(t) \Vert _{H} \end{aligned}$$

was established in [3] and the following theorem on continuous dependence of the solution on the given data was proven.

Theorem 1

([3]) Assume that \(\varphi \), \(\psi \in D(A)\). Let f(t) and g(t) be continuously differentiable functions on [0, 1] and \([-1,0]\), respectively. Then, for the solution \(\{u(t),p\}\) of problem (1) in \(C(H)\times H\) the following stability inequalities

$$\begin{aligned} \Vert u \Vert _{C(H)} + \Vert A^{-1}p \Vert _{H} \le M \Big [ \Vert \varphi \Vert _{H} + \Vert \psi \Vert _{H} + \max _{0\le t\le 1} \Vert f(t)\Vert _{H} + \max _{-1\le t\le 0} \Vert g(t)\Vert _{H} \Big ], \\ \begin{array}{l} \displaystyle \max _{-1\le t\le 0} \Vert u'(t)\Vert _{H} + \max _{0\le t\le 1} \Vert u''(t)\Vert _{H} + \Vert Au \Vert _{C(H)} + \Vert p\Vert _{H} \le M \Big [ \Vert A\varphi \Vert _{H} + \Vert \varphi \Vert _{H} \\ \displaystyle \quad + \Vert A\psi \Vert _{H} + \Vert \psi \Vert _{H} + \Vert f(0)\Vert _{H} + \max _{0\le t\le 1} \Vert f'(t)\Vert _{H} + \Vert g(0)\Vert _{H} + \max _{-1\le t \le 0} \Vert g'(t)\Vert _{H} \Big ] \end{array} \end{aligned}$$

hold, where M does not depend on \(\varphi \), \(\psi \), f(t) and g(t) but may depend on \(\delta \) and \(\lambda \).

The first and second order of accuracy difference schemes for the approximate solution of abstract problem (1) were initially constructed in [16] and [13], respectively. Numerical algorithm for implementation of both schemes when applied to one-dimensional hyperbolic–parabolic equation with Dirichlet and Neumann boundary conditions were discussed in [14] and [15], respectively. We would like to emphasize that previous research was mainly focused on numerical studies for some applications of abstract problem (1), i.e. problems with specified operator A. To the best of our knowledge, the rigorous analysis of stable difference schemes for problem (1) in the abstract form is still lacking. The main goal of this study is to investigate the first and second order of accuracy stable difference schemes for the approximate solution of problem (1). We prove the unique solvability of these difference schemes and obtain the stability estimates for their solutions. The analysis relies on the operator approach and the proofs of the stability estimates are based on the spectral representation of the self-adjoint positive definite operator in a Hilbert space.

2 First order of accuracy stable difference scheme

Let \(\tau =1/N\) be a sufficiently small positive number such that \(\lambda \ge -1+\tau \). We define the grid points \(t_{k}=k\tau ,~-N\le k\le N\). For the approximate solution of problem (1) we consider the first order of accuracy stable difference scheme

$$\begin{aligned} \left\{ \begin{array}{l} \frac{u_{k+1}-2u_{k}+u_{k-1}}{\tau ^{2}} + Au_{k+1} = p + f_{k}, \quad 1\le k \le N-1, \\ \frac{u_{k}-u_{k-1}}{\tau } + Au_{k} = p + g_{k}, \quad -N+1 \le k\le 0, \\ \frac{u_{1}-u_{0}}{\tau } = p - Au_{0} + g_{0}, \quad u_{-N}=\varphi , \quad u_{\ell } = \psi , \end{array} \right. \end{aligned}$$
(2)

where \(\ell =\lfloor \lambda /\tau \rfloor \), \(f_{k}=f(t_{k})\), \(1\le k \le N-1\) and \(g_{k}=g(t_{k})\), \(-N+1\le k\le 0\).

Let us first give some lemmas which we will need in the remaining part of this section. Here and everywhere else, we denote

$$\begin{aligned} R=\left( I+i\tau A^{1/2}\right) ^{-1}, \quad {\tilde{R}}=\left( I-i\tau A^{1/2}\right) ^{-1}, \quad Q=\left( I+\tau A\right) ^{-1}. \end{aligned}$$

Lemma 1

([9]) The following estimates hold

$$\begin{aligned}&\Vert R \Vert _{H\rightarrow H} \le 1, \quad \Vert {\tilde{R}} \Vert _{H\rightarrow H} \le 1, \quad \Vert {\tilde{R}}^{-1}R \Vert _{H\rightarrow H} \le 1, \quad \big \Vert R^{-1}{\tilde{R}}\big \Vert _{H\rightarrow H} \le 1, \nonumber \\&\quad \Vert \tau A^{1/2} R \Vert _{H\rightarrow H} \le 1, \quad \Vert \tau A^{1/2} {\tilde{R}} \Vert _{H\rightarrow H} \le 1. \end{aligned}$$
(3)

Lemma 2

([9]) The following estimates hold

$$\begin{aligned} \Vert Q^{m} \Vert _{H\rightarrow H} \le 1, \quad \Vert A^{1/2} Q^{m} \Vert _{H\rightarrow H} \le \frac{1}{2\sqrt{m\tau }}, \quad m\ge 1. \end{aligned}$$
(4)

Lemma 3

([9]) If \(-1+\tau \le \lambda < \tau \), then \(-N+1\le \ell \le 0\) and the following estimate holds

$$\begin{aligned}{} & {} \left\| \left( I-Q^{N+\ell }\right) ^{-1} \right\| _{H\rightarrow H} \le M_{1}(\delta ,\lambda ). \end{aligned}$$
(5)

Lemma 4

The following estimate

$$\begin{aligned} \left\| \left[ \frac{R^{m-1}+{\tilde{R}}^{m-1}}{2} + \frac{A^{1/2}}{2i}\left( {\tilde{R}}^{-1}R^{m-1}-R^{-1}{\tilde{R}}^{m-1}\right) \right] Q^{N}\right\| _{H\rightarrow H} < 1 \end{aligned}$$
(6)

holds for any \(m\ge 1\).

Proof

Since

$$\begin{aligned}&\frac{R^{m-1}+{\tilde{R}}^{m-1}}{2} + \frac{A^{1/2}}{2i}\left( {\tilde{R}}^{-1}R^{m-1}-R^{-1}{\tilde{R}}^{m-1}\right) \\&\quad = \left( I-\tau A - iA^{1/2}\right) \frac{R^{m-1}}{2} + \left( I-\tau A + iA^{1/2}\right) \frac{{\tilde{R}}^{m-1}}{2}, \end{aligned}$$

by using (3) and the following estimates

$$\begin{aligned} \left\| \left( I-\tau A \pm iA^{1/2}\right) Q^{N} \right\| _{H\rightarrow H} < 1, \end{aligned}$$
(7)

we obtain (6). The proof of estimates (7) is based on the spectral representation of the self-adjoint positive definite operator A in a Hilbert space H [9]. \(\square \)

Lemma 5

If \(\tau \le \lambda \le 1\), then \(1\le \ell \le N\) and the following estimate holds

$$\begin{aligned}{} & {} \left\| \left( I-\left[ \frac{R^{\ell -1}+{\tilde{R}}^{\ell -1}}{2} + \frac{A^{1/2}}{2i}\left( {\tilde{R}}^{-1}R^{\ell -1}-R^{-1}{\tilde{R}}^{\ell -1}\right) \right] Q^{N}\right) ^{-1}\right\| _{H\rightarrow H}\nonumber \\{} & {} \quad \le M_{2}(\delta ,\lambda ). \end{aligned}$$
(8)

Proof

The proof of estimate (8) is based on the estimate (6). \(\square \)

Now, we give the main theorem for the solution of the first order of accuracy difference scheme (2).

Theorem 2

The difference scheme (2) has a unique solution and the following stability estimate holds

$$\begin{aligned} \max _{-N\le k\le N} \Vert u_{k}\Vert _{H} + \big \Vert A^{-1}p\big \Vert _{H}&\le M^{*}(\delta ,\lambda ) \Big [ \Vert \varphi \Vert _{H} + \Vert \psi \Vert _{H} \nonumber \\&\quad + \max _{1\le k \le N-1}\Vert f_{k}\Vert _{H} + \max _{-N+1 \le k\le 0} \Vert g_{k}\Vert _{H} \Big ], \end{aligned}$$
(9)

where \(M^{*}(\delta ,\lambda )\) is independent of \(\varphi \), \(\psi \), \(\tau \), \(f_{k}\) and \(g_{k}\).

Proof

Let us denote

$$\begin{aligned} u_{k}=v_{k}+A^{-1}p, \quad -N\le k\le N. \end{aligned}$$
(10)

Then, the difference scheme (2) results in the following auxiliary difference scheme

$$\begin{aligned} \left\{ \begin{array}{l} \frac{v_{k+1}-2v_{k}+v_{k-1}}{\tau ^{2}} + Av_{k+1} = f_{k}, \quad 1\le k \le N-1, \\ \frac{v_{k}-v_{k-1}}{\tau } + Av_{k} = g_{k}, \quad -N+1 \le k\le 0, \\ \frac{v_{1}-v_{0}}{\tau } = - Av_{0} + g_{0}, \quad v_{\ell } = v_{-N} + \psi - \varphi . \end{array} \right. \end{aligned}$$
(11)

First, we obtain the formulas for the solution of scheme (11). It is known that for the given \(v_{0}\) the difference scheme

$$\begin{aligned} \left\{ \begin{array}{l} \frac{v_{k+1}-2v_{k}+v_{k-1}}{\tau ^{2}} + Av_{k+1} = f_{k}, \quad 1\le k \le N-1, \\ \frac{v_{1}-v_{0}}{\tau } = - Av_{0} + g_{0} \end{array} \right. \end{aligned}$$

has a solution

$$\begin{aligned} v_{k} =&\frac{R^{k-1}+{\tilde{R}}^{k-1}}{2}v_{0}+\left( R-{\tilde{R}}\right) ^{-1}\tau \left( R^{k}-{\tilde{R}}^{k}\right) \left( - Av_{0} + g_{0}\right) \nonumber \\&- \sum \limits _{j=1}^{k}\frac{A^{-1/2}}{2i}\left( R^{k-j}-{\tilde{R}}^{k-j}\right) f_{j}\tau , \quad 1\le k\le N. \end{aligned}$$
(12)

Additionally, for the given \(v_{-N}\) the difference scheme

$$\begin{aligned} \frac{v_{k}-v_{k-1}}{\tau } + Av_{k} = g_{k}, ~ -N+1\le k\le 0 \end{aligned}$$

has a solution

$$\begin{aligned} v_{k}=Q^{N+k}v_{-N} + \sum \limits _{j=-N+1}^{k}Q^{k-j+1}g_{j}\tau , \quad -N+1\le k\le 0. \end{aligned}$$
(13)

In particular, with \(k=0\) in (13) we have

$$\begin{aligned} v_{0}=Q^{N}v_{-N} + \sum \limits _{j=-N+1}^{0}Q^{-j+1}g_{j} \tau . \end{aligned}$$

Then, by putting it in (12), we obtain

$$\begin{aligned} v_{k} =&\left[ \frac{R^{k-1}+{\tilde{R}}^{k-1}}{2} - \tau A\left( R-{\tilde{R}}\right) ^{-1}\left( R^{k}-{\tilde{R}}^{k}\right) \right] \\&\times \left[ Q^{N}v_{-N} + \sum \limits _{j=-N+1}^{0}Q^{-j+1}g_{j} \tau \right] + \left( R-{\tilde{R}}\right) ^{-1}\left( R^{k}-{\tilde{R}}^{k}\right) \tau g_{0} \\&- \sum \limits _{j=1}^{k}\frac{A^{-1/2}}{2i}\left( R^{k-j}-{\tilde{R}}^{k-j}\right) f_{j}\tau , \quad k=1,2,\ldots ,N. \end{aligned}$$

Using \(R-{\tilde{R}}=-2i\tau A^{1/2}R{\tilde{R}}\), we have

$$\begin{aligned} v_{k} =&\left[ \frac{R^{k-1}+{\tilde{R}}^{k-1}}{2} + \frac{A^{1/2}}{2i}\left( {\tilde{R}}^{-1}R^{k-1}-R^{-1}{\tilde{R}}^{k-1}\right) \right] \nonumber \\&\times \left[ Q^{N}v_{-N} + \sum \limits _{j=-N+1}^{0}Q^{-j+1}g_{j} \tau \right] - \frac{A^{-1/2}}{2i}\left( {\tilde{R}}^{-1}R^{k-1}-R^{-1}{\tilde{R}}^{k-1}\right) g_{0} \nonumber \\&- \sum \limits _{j=1}^{k}\frac{A^{-1/2}}{2i}\left( R^{k-j}-{\tilde{R}}^{k-j}\right) f_{j}\tau , \quad k=1,2,\ldots ,N. \end{aligned}$$
(14)

If \(-1+\tau \le \lambda < \tau \), then \(-N+1\le \ell \le 0\) and therefore from (11) and (13) it follows

$$\begin{aligned} v_{\ell } = v_{-N} + \psi - \varphi = Q^{N+\ell }v_{-N} + \sum \limits _{j=-N+1}^{\ell }Q^{\ell -j+1}g_{j}\tau , \end{aligned}$$

so that

$$\begin{aligned} v_{-N} = \left( I-Q^{N+\ell }\right) ^{-1} \left( \sum \limits _{j=-N+1}^{\ell }Q^{\ell -j+1}g_{j}\tau + \varphi - \psi \right) . \end{aligned}$$
(15)

If \(\tau \le \lambda \le 1\), then \(1\le \ell \le N\) and, therefore, from (11) and (14) it follows

$$\begin{aligned} v_{\ell }&= v_{-N} + \psi - \varphi = \left( \frac{R^{\ell -1}+{\tilde{R}}^{\ell -1}}{2} + \frac{A^{1/2}}{2i}\left( {\tilde{R}}^{-1}R^{\ell -1}-R^{-1}{\tilde{R}}^{\ell -1}\right) \right) \\&\quad \times \left( Q^{N}v_{-N} + \sum \limits _{j=-N+1}^{0}Q^{-j+1}g_{j} \tau \right) - \frac{A^{-1/2}}{2i}\left( {\tilde{R}}^{-1}R^{\ell -1}-R^{-1}{\tilde{R}}^{\ell -1}\right) g_{0} \\&\quad - \sum \limits _{j=1}^{\ell }\frac{A^{-1/2}}{2i}\left( R^{\ell -j}-{\tilde{R}}^{\ell -j}\right) f_{j}\tau , \end{aligned}$$

so that

$$\begin{aligned} v_{-N}&= \left( I-\left\{ \frac{R^{\ell -1}+{\tilde{R}}^{\ell -1}}{2} + \frac{A^{1/2}}{2i}\left( {\tilde{R}}^{-1}R^{\ell -1}-R^{-1}{\tilde{R}}^{\ell -1}\right) \right\} Q^{N}\right) ^{-1} \nonumber \\&\quad \times \left[ \left( \frac{R^{\ell -1}+{\tilde{R}}^{\ell -1}}{2} + \frac{A^{1/2}}{2i}\left( {\tilde{R}}^{-1}R^{\ell -1}-R^{-1}{\tilde{R}}^{\ell -1}\right) \right) \sum \limits _{j=-N+1}^{0}Q^{-j+1}g_{j} \tau \right. \nonumber \\&\quad - \frac{A^{-1/2}}{2i}\left( {\tilde{R}}^{-1}R^{\ell -1}-R^{-1}{\tilde{R}}^{\ell -1}\right) g_{0} \nonumber \\&\left. \quad - \sum \limits _{j=1}^{\ell }\frac{A^{-1/2}}{2i}\left( R^{\ell -j}-{\tilde{R}}^{\ell -j}\right) f_{j}\tau + \varphi - \psi \right] . \end{aligned}$$
(16)

Hence, for the solution of the auxiliary difference scheme (11) we have formulas (13) and (14), with \(v_{-N}\) being found by formula (15) when \(-1+\tau \le \lambda < \tau \) and formula (16) when \(\tau \le \lambda \le 1\). Now, taking into account that \(u_{-N}=\varphi \), we have \(A^{-1}p=\varphi - v_{-N}\). Then, by using (10), we obtain the solution of difference scheme (2).

Now, let us obtain the estimate (9). By using (15) and the estimates (4) and (5), we obtain

$$\begin{aligned} \left\| v_{-N}\right\| _{H} \le M_{1}(\delta ,\lambda )\Big [ \left\| \varphi \right\| _{H} + \left\| \psi \right\| _{H} + \max _{-N+1 \le k\le 0} \left\| g_{k}\right\| _{H} \Big ]. \end{aligned}$$
(17)

Similarly, by using (16) and the estimates (3), (4) and (8), we obtain

$$\begin{aligned} \left\| v_{-N}\right\| _{H} \le M_{2}(\delta ,\lambda ) \Big [ \left\| \varphi \right\| _{H} + \left\| \psi \right\| _{H} + \max _{1\le k \le N-1} \left\| f_{k}\right\| _{H} + \max _{-N+1 \le k\le 0} \left\| g_{k}\right\| _{H} \Big ].\nonumber \\ \end{aligned}$$
(18)

Then, using (13) and the estimates (4), (17) and (18), we get

$$\begin{aligned} \left\| v_{k}\right\| _{H} \le&\left\| v_{-N}\right\| _{H} + \max _{-N+1 \le k\le 0} \left\| g_{k}\right\| _{H} \nonumber \\ \le&M_{3}(\delta ,\lambda ) \Big [ \left\| \varphi \right\| _{H} + \left\| \psi \right\| _{H} + \max _{1\le k \le N-1} \left\| f_{k}\right\| _{H} + \max _{-N+1 \le k\le 0} \left\| g_{k}\right\| _{H} \Big ] \end{aligned}$$
(19)

for \(k=-N+1,\ldots ,0\). Using (14) and the estimates (3), (4), (6), (17) and (18), we obtain

$$\begin{aligned} \left\| v_{k}\right\| _{H} \le&\left\| v_{-N}\right\| _{H} + M_{4}(\delta ) \Big (\max _{1\le k \le N-1} \left\| f_{k}\right\| _{H} + \max _{-N+1 \le k\le 0} \left\| g_{k}\right\| _{H} \Big ) \nonumber \\ \le&M_{5}(\delta ,\lambda ) \Big [ \left\| \varphi \right\| _{H} + \left\| \psi \right\| _{H} + \max _{1\le k \le N-1} \left\| f_{k}\right\| _{H} + \max _{-N+1 \le k\le 0} \left\| g_{k}\right\| _{H} \Big ] \end{aligned}$$
(20)

for \(k=1,\ldots ,N\). Since \(A^{-1}p=\varphi - v_{-N}\), by using (17) and (18), we have

$$\begin{aligned} \big \Vert A^{-1}p\big \Vert _{H}&\le \left\| \varphi \right\| _{H} + \left\| v_{-N}\right\| _{H} \nonumber \\ \le&M_{6}(\delta ,\lambda ) \Big [ \left\| \varphi \right\| _{H} + \left\| \psi \right\| _{H} + \max _{1\le k \le N-1} \left\| f_{k}\right\| _{H} + \max _{-N+1 \le k\le 0} \left\| g_{k}\right\| _{H} \Big ]. \end{aligned}$$
(21)

So, using (10), (19), (20) and (21), we prove the estimates

$$\begin{aligned} \Vert u_{k}\Vert _{H} \le&\big \Vert A^{-1}p\big \Vert _{H} + \left\| v_{k}\right\| _{H} \nonumber \\ \le&M_{7}(\delta ,\lambda ) \Big [ \left\| \varphi \right\| _{H} + \left\| \psi \right\| _{H} + \max _{1\le k \le N-1} \left\| f_{k}\right\| _{H} + \max _{-N+1 \le k\le 0} \left\| g_{k}\right\| _{H} \Big ] \end{aligned}$$
(22)

for \(k=-N,\ldots ,N\). Estimate (9) follows from (21) and (22). \(\square \)

3 Second order of accuracy stable difference scheme

In this section, for the approximate solution to a problem (1) we consider the second order of accuracy stable difference scheme

$$\begin{aligned} \left\{ \begin{array}{l} \frac{u_{k+1}-2u_{k}+u_{k-1}}{\tau ^{2}} + Au_{k}+ \frac{\tau ^{2}}{4} A^{2}u_{k+1} = (I+\frac{\tau ^2}{4}A)p + f_{k}, ~ 1\le k \le N-1, \\ \frac{u_{k}-u_{k-1}}{\tau } + A (I+\frac{\tau }{2}A) u_{k} = (I+\frac{\tau }{2}A) (p+g_{k}), ~ -N+1\le k\le 0, \\ (I+\tau ^{2}A)\frac{u_{1}-u_{0}}{\tau } = p-Au_{0}+g_{0} + \frac{\tau }{2} (p-Au_{0}+f_{0}), \\ u_{-N}=\varphi ,\quad u_{\ell } + (\lambda -\tau \ell ) \frac{u_{\ell }-u_{\ell -1}}{\tau } =\psi , \end{array} \right. \end{aligned}$$
(23)

where \(f_{k}=f(t_{k})\), \(0\le k \le N-1\), \(g_{k}=g(t_{k}-\frac{\tau }{2})\), \(-N+1\le k\le 0\) and \(\ell =\lfloor \lambda /\tau \rfloor \). We assume here for simplicity that \(\tau \) is a sufficiently small positive number such that \(\lambda \ge -1+2\tau \).

Let us first give some lemmas which we will need in the remaining part of this section. Here and everywhere else, we denote

$$\begin{aligned} R_{1}= & {} \left( I-\frac{\tau ^{2}A}{2}+i\tau A^{1/2}\right) ^{-1}, \quad R_{2}=\left( I-\frac{\tau ^{2}A}{2}-i\tau A^{1/2}\right) ^{-1}, \\ {\tilde{Q}}= & {} \left( I+\tau A + \frac{\tau ^{2}A^{2}}{2}\right) ^{-1}. \end{aligned}$$

Lemma 6

([9]) The following estimates hold

$$\begin{aligned}&\Vert R_{1}\Vert _{H\rightarrow H} \le 1, \quad \Vert R_{2}\Vert _{H\rightarrow H} \le 1, \quad \left\| R_{2}^{-1}R_{1}\right\| _{H\rightarrow H} \le 1, \nonumber \\&\left\| R_{1}^{-1}R_{2} \right\| _{H\rightarrow H} \le 1, \quad \left\| \tau A^{1/2} R_{1} \right\| _{H\rightarrow H} \le 1, \quad \left\| \tau A^{1/2} R_{2} \right\| _{H\rightarrow H} \le 1, \nonumber \\&\left\| \left( I\pm i\tau A^{1/2}\right) ^{-1} \right\| _{H\rightarrow H} \le 1, \quad \left\| \left( I\pm \frac{i\tau A^{1/2}}{2}\right) ^{-1} \right\| _{H\rightarrow H} \le 1, \nonumber \\&\left\| \tau A^{1/2}\left( I\pm i\tau A^{1/2}\right) ^{-1} \right\| _{H\rightarrow H} \le 1. \end{aligned}$$
(24)

Lemma 7

( [9]) The following estimates hold

$$\begin{aligned} \left\| \left( I+\frac{\tau A}{2}\right) {\tilde{Q}} \right\| _{H\rightarrow H} \le 1, \quad \big \Vert {\tilde{Q}}^{m} \big \Vert _{H\rightarrow H} \le 1, \quad m\ge 1. \end{aligned}$$
(25)

Lemma 8

If \(-1+2\tau \le \lambda < \tau \), then \(-N+2\le \ell \le 0\) and the following estimate holds

$$\begin{aligned} \left\| \Bigg (I-\Big [I-(\lambda -\tau \ell )\Big (I+\frac{\tau A}{2}\Big )A\Big ]{\tilde{Q}}^{N+\ell } \Bigg )^{-1}\right\| _{H\rightarrow H}\le M_{1}(\delta ,\lambda ). \end{aligned}$$
(26)

Proof

The proof of the estimate (26) is based on the following estimate

$$\begin{aligned} \left\| \Big [ I-(\lambda -\tau \ell )\Big (I+\frac{\tau A}{2}\Big )A \Big ]{\tilde{Q}} \right\| _{H\rightarrow H} < 1. \end{aligned}$$

The proof of the last estimate is based on spectral representation of the self-adjoint positive definite operator A in a Hilbert space H [9]. \(\square \)

Lemma 9

If \(\tau \le \lambda < 2\tau \), then \(\ell =1\) and the following estimate holds

$$\begin{aligned} \left\| \Bigg (I-\Big [I-\lambda \big (1+\frac{\tau }{2}\big )\left( I+\tau ^{2}A\right) ^{-1}A\Big ]{\tilde{Q}}^{N} \Bigg )^{-1}\right\| _{H\rightarrow H}\le M_{2}(\delta ,\lambda ). \end{aligned}$$
(27)

Proof

We have

$$\begin{aligned} \bigg (I-\lambda \big (1+\frac{\tau }{2}\big )\left( I+\tau ^{2}A\right) ^{-1}A \bigg ){\tilde{Q}} =&\left( I+\tau ^{2}A\right) ^{-1} \bigg (I+\tau ^{2}A - \frac{\lambda }{\tau }\big (1+\frac{\tau }{2}\big ) \tau A \bigg ) {\tilde{Q}} \\ =&\left( I+\tau ^{2}A\right) ^{-1} \big (I - \alpha \tau A \big ) {\tilde{Q}}, \end{aligned}$$

where \(\alpha =\frac{\lambda }{\tau }\big (1+\frac{\tau }{2}\big ) - \tau \). Since \(\tau \le \lambda < 2\tau \), it follows that \(1-\frac{\tau }{2}\le \alpha < 2\). Then, using the estimates (24) and the following estimate

$$\begin{aligned} \left\| \left( I-\alpha \tau A\right) {\tilde{Q}} \right\| _{H\rightarrow H} < 1, \end{aligned}$$
(28)

we obtain

$$\begin{aligned} \left\| \bigg (I-\lambda \big (1+\frac{\tau }{2}\big )\left( I+\tau ^{2}A\right) ^{-1}A \bigg ){\tilde{Q}} \right\| _{H\rightarrow H} < 1, \end{aligned}$$

which allows us to prove the estimate (27). The proof of estimates (28) is based on spectral representation of the self-adjoint positive definite operator A in a Hilbert space H [9]. \(\square \)

Lemma 10

If \(2\tau \le \lambda \le 1\), then \(2\le \ell \le N\) and the following estimate holds

$$\begin{aligned}&\Bigg \Vert \Bigg (I - \Bigg [\bigg (I+(\lambda -\tau \ell )\Big (\frac{\tau A}{2}-iA^{1/2}\Big )\bigg ) R_{1}^{\ell } \nonumber \\&\qquad - \frac{1}{4}\bigg \{\bigg (I+(\lambda -\tau \ell )\Big (\frac{\tau A}{2}-iA^{1/2}\Big )\bigg ) R_{2}^{-1}R_{1}^{\ell -1} \nonumber \\&\qquad - \bigg (I+(\lambda -\tau \ell )\Big (\frac{\tau A}{2}+iA^{1/2}\Big )\bigg ) R_{1}^{-1}R_{2}^{\ell -1}\bigg \} \nonumber \\&\qquad \left. \times \bigg (2R_{1}+i\tau A^{1/2}R_{1} + \left( I+\tau ^{2}A\right) ^{-1}\left( 2iA^{1/2} + i\tau A^{1/2}\right) \bigg ) \Bigg ] {\tilde{Q}}^{N}\Bigg )^{-1} \right\| _{H\rightarrow H} \nonumber \\&\quad \le M_{3}(\delta ,\lambda ). \end{aligned}$$
(29)

Proof

Denote

$$\begin{aligned} P_{\ell }&= \Bigg [\bigg (I+(\lambda -\tau \ell )\Big (\frac{\tau A}{2}-iA^{1/2}\Big )\bigg ) R_{1}^{\ell } \\&\quad - \frac{1}{4}\bigg \{\bigg (I+(\lambda -\tau \ell )\Big (\frac{\tau A}{2}-iA^{1/2}\Big )\bigg ) R_{2}^{-1}R_{1}^{\ell -1} \\&\quad - \bigg (I+(\lambda -\tau \ell )\Big (\frac{\tau A}{2}+iA^{1/2}\Big )\bigg ) R_{1}^{-1}R_{2}^{\ell -1}\bigg \} \\&\quad \times \bigg (2R_{1}+i\tau A^{1/2}R_{1} + \left( I+\tau ^{2}A\right) ^{-1}\left( 2iA^{1/2} + i\tau A^{1/2}\right) \bigg ) \Bigg ] {\tilde{Q}}^{N}. \end{aligned}$$

Applying the identity

$$\begin{aligned} (I - P_{\ell })^{-1}&= \bigg (I-\Big (c(\tau \ell )-As(\tau \ell )\Big )e^{-A}\bigg )^{-1} \\&\quad + \Bigg [ \bigg \{R_{1}^{\ell } - \frac{1}{4} \bigg ( R_{2}^{-1}R_{1}^{\ell -1} - R_{1}^{-1}R_{2}^{\ell -1}\bigg ) \\&\quad \times \bigg (2R_{1}+i\tau A^{1/2}R_{1} + \left( I+\tau ^{2}A\right) ^{-1}\left( 2iA^{1/2} + i\tau A^{1/2}\right) \bigg ) \bigg \}{\tilde{Q}}^{N} \\&\quad - \Big (c(\tau \ell )-As(\tau \ell )\Big )e^{-A} \Bigg ] (I-P_{\ell })^{-1} \bigg (I-\Big (c(\tau \ell )-As(\tau \ell )\Big )e^{-A}\bigg )^{-1} \\&\quad + (\lambda -\tau \ell ) \Bigg [\Big (\frac{\tau A}{2}-iA^{1/2}\Big ) R_{1}^{\ell } \\&\quad - \frac{1}{4}\left\{ \Big (\frac{\tau A}{2}-iA^{1/2}\Big ) R_{2}^{-1}R_{1}^{\ell -1} - \Big (\frac{\tau A}{2}+iA^{1/2}\Big ) R_{1}^{-1}R_{2}^{\ell -1}\right\} \\&\quad \times \bigg (2R_{1}+i\tau A^{1/2}R_{1} + \left( I+\tau ^{2}A\right) ^{-1}\left( 2iA^{1/2} + i\tau A^{1/2}\right) \bigg ) \Bigg ] {\tilde{Q}}^{N} \\&\quad \times \left( I-P_{\ell }\right) ^{-1} \bigg (I-\Big (c(\tau \ell )-As(\tau \ell )\Big )e^{-A}\bigg )^{-1} \end{aligned}$$

and the following estimates

$$\begin{aligned}{} & {} \left\| \bigg (I-\Big (c(\tau \ell )-As(\tau \ell )\Big )e^{-A}\bigg )^{-1} \right\| _{H\rightarrow H} \le M(\delta ,\lambda ), \\{} & {} \left\| \left[ R_{1}^{\ell } - \frac{1}{2} \Big ( R_{2}^{-1}R_{1}^{\ell } - R_{2}^{\ell -1}\Big ) \right] {\tilde{Q}}^{N} - c(\tau \ell )e^{-A} \right\| _{H\rightarrow H} \le M_{1}\tau , \end{aligned}$$
$$\begin{aligned}&\bigg \Vert \frac{1}{4} \Big ( R_{2}^{-1}R_{1}^{\ell -1} - R_{1}^{-1}R_{2}^{\ell -1}\Big ) \bigg [i\tau A^{1/2} R_{1} + \left( I+\tau ^{2}A\right) ^{-1}\left( 2iA^{1/2} + i\tau A^{1/2}\right) \bigg ] {\tilde{Q}}^{N} \\&\quad + As(\tau \ell )e^{-A} \bigg \Vert _{H\rightarrow H} \le M_{1}\tau , \end{aligned}$$
$$\begin{aligned}&\bigg \Vert \bigg [ \Big (\frac{\tau A}{2}-iA^{1/2}\Big ) R_{1}^{\ell } - \frac{1}{4}\left\{ \Big (\frac{\tau A}{2}-iA^{1/2}\Big ) R_{2}^{-1}R_{1}^{\ell -1} - \Big (\frac{\tau A}{2}+iA^{1/2}\Big ) R_{1}^{-1}R_{2}^{\ell -1}\right\} \\&\quad \times \bigg (2R_{1}+i\tau A^{1/2}R_{1} + \left( I+\tau ^{2}A\right) ^{-1}\left( 2iA^{1/2} + i\tau A^{1/2}\right) \bigg ) \bigg ] {\tilde{Q}}^{N} \bigg \Vert _{H\rightarrow H} \le M_{2}, \end{aligned}$$

we get

$$\begin{aligned}&\left\| (I-P_{\ell })^{-1} \right\| _{H\rightarrow H} \le \left\| \bigg (I-\Big (c(\tau \ell )-As(\tau \ell )\Big )e^{-A}\bigg )^{-1} \right\| _{H\rightarrow H} \\&\qquad + \Big (2M_{1}\tau +M_{2}(\lambda -\tau \ell )\Big ) \left\| \left( I-P_{\ell }\right) ^{-1} \right\| _{H\rightarrow H} \left\| \bigg (I-\Big (c(\tau \ell )-As(\tau \ell )\Big )e^{-A}\bigg )^{-1} \right\| _{H\rightarrow H} \\&\quad \le M(\delta ,\lambda ) + \tau \big (2M_{1}+M_{2}\big ) M(\delta ,\lambda ) \left\| \left( I-P_{\ell }\right) ^{-1} \right\| _{H\rightarrow H}. \end{aligned}$$

From this estimate it follows that

$$\begin{aligned} \left\| \left( I-P_{\ell }\right) ^{-1} \right\| _{H\rightarrow H} \le \frac{M(\delta ,\lambda )}{1-\tau \big (2M_{1}+M_{2}\big ) M(\delta ,\lambda )} \end{aligned}$$

and therefore, for small \(\tau \) we obtain (29). \(\square \)

Now, we give the main theorem for the solution of the difference scheme (23).

Theorem 3

The difference scheme (23) has a unique solution and the following stability estimate holds

$$\begin{aligned} \max _{-N\le k\le N} \left\| u_{k}\right\| _{H} + \Vert A^{-1}p\Vert _{H} \nonumber&\le M^{*}(\delta ,\lambda ) \Big [ \left\| \varphi \right\| _{H}+\left\| \psi \right\| _{H} \\&\quad +\max _{0\le k\le N-1}\left\| f_{k}\right\| _{H} + \max _{-N+1 \le k\le 0}\left\| g_{k}\right\| _{H} \Big ], \end{aligned}$$
(30)

where \(M^{*}(\delta ,\lambda )\) is independent of \(\varphi \), \(\psi \), \(\tau \), \(f_{k}\) and \(g_{k}\).

Proof

With substitution (10), the difference scheme (23) results in the following auxiliary difference scheme

$$\begin{aligned} \left\{ \begin{array}{l} \frac{v_{k+1}-2v_{k}+v_{k-1}}{\tau ^{2}} + Av_{k}+ \frac{\tau ^{2}}{4} A^{2}v_{k+1} = f_{k}, \quad 1\le k \le N-1, \\ \frac{v_{k}-v_{k-1}}{\tau } + A(I+\frac{\tau }{2}A) v_{k} = (I+\frac{\tau }{2}A) g_{k}, \quad -N+1 \le k\le 0, \\ (I+\tau ^{2}A)\frac{v_{1}-v_{0}}{\tau } = -Av_{0}+g_{0} + \frac{\tau }{2} (-Av_{0}+f_{0}), \\ v_{\ell } + (\lambda -\tau \ell ) \frac{v_{\ell }-v_{\ell -1}}{\tau } =v_{-N}+\psi -\varphi . \end{array} \right. \end{aligned}$$
(31)

First, we obtain the formulas for the solution of the scheme (31). It is known that for the given \(v_{0}\) the difference scheme

$$\begin{aligned} \left\{ \begin{array}{l} \frac{v_{k+1}-2v_{k}+v_{k-1}}{\tau ^{2}} + Av_{k}+ \frac{\tau ^{2}}{4} A^{2}v_{k+1} = f_{k}, \quad 1\le k \le N-1, \\ (I+\tau ^{2}A)\frac{v_{1}-v_{0}}{\tau } = -Av_{0}+g_{0} + \frac{\tau }{2}(-Av_{0}+f_{0}) \end{array} \right. \end{aligned}$$

has a solution

$$\begin{aligned} v_{k}&= \left( R_{1}-R_{2}\right) ^{-1} \bigg [ R_{1}R_{2}\left( R_{2}^{k-1}-R_{1}^{k-1}\right) \\&\quad + \left( R_{1}^{k}-R_{2}^{k}\right) \left( I+\tau ^{2}A\right) ^{-1}\left( I-\tau A + \frac{\tau ^{2}A}{2}\right) \bigg ]v_{0} \\&\quad + \left( R_{1}-R_{2}\right) ^{-1}\left( R_{1}^{k}-R_{2}^{k}\right) \left( I+\tau ^{2}A\right) ^{-1}\left( \tau g_{0} + \frac{\tau ^{2}}{2}f_{0}\right) \\&\quad + \sum \limits _{j=1}^{k} \left( R_{1}-R_{2}\right) ^{-1} R_{1}R_{2}\left( R_{1}^{k-j}-R_{2}^{k-j}\right) \tau ^{2}f_{j}, \quad k=1,\ldots ,N. \end{aligned}$$

Since \(R_{1}-R_{2}=-2i\tau A^{1/2}R_{1}R_{2}\), we have

$$\begin{aligned} v_{k}&= \left( R_{1}-R_{2}\right) ^{-1} \bigg [R_{1}R_{2}\left( R_{2}^{k-1}-R_{1}^{k-1}\right) \nonumber \\&\quad + \left( R_{1}^{k}-R_{2}^{k}\right) \left( I+\tau ^{2}A\right) ^{-1}\left( I-\tau A + \frac{\tau ^{2}A}{2}\right) \bigg ]v_{0} \nonumber \\&\quad + \frac{iA^{-1/2}}{2}\left( R_{2}^{-1}R_{1}^{k-1}-R_{1}^{-1}R_{2}^{k-1}\right) \left( I+\tau ^{2}A\right) ^{-1}\left( g_{0} + \frac{\tau }{2}f_{0}\right) \nonumber \\&\quad + \sum \limits _{j=1}^{k} \frac{iA^{-1/2}}{2}\left( R_{1}^{k-j}-R_{2}^{k-j}\right) f_{j}\tau , \quad k=1,\ldots ,N. \end{aligned}$$
(32)

Let us rearrange the first term in the right-hand side of (32)

$$\begin{aligned}&(R_{1} - R_{2})^{-1} \bigg [R_{1}R_{2}\left( R_{2}^{k-1}-R_{1}^{k-1}\right) \\&\qquad + \left( R_{1}^{k}-R_{2}^{k}\right) \left( I+\tau ^{2}A\right) ^{-1}\left( I-\tau A + \frac{\tau ^{2}A}{2}\right) \bigg ]v_{0} \\&\quad = (R_{1}-R_{2})^{-1} \bigg [R_{2}^{k}R_{1}-R_{1}^{k}R_{2} + R_{1}^{k} - R_{2}^{k} \\&\qquad - \left( R_{1}^{k}-R_{2}^{k}\right) \left( I+\tau ^{2}A\right) ^{-1}\left( \tau A + \frac{\tau ^{2}A}{2}\right) \bigg ]v_{0} \\&\quad = \left( R_{1}-R_{2}\right) ^{-1} \bigg [R_{1}^{k}\left( R_{1}-R_{2}\right) + \left( R_{1}^{k}-R_{2}^{k}\right) \left( I-R_{1}\right) \\&\qquad - \left( R_{1}^{k}-R_{2}^{k}\right) \left( I+\tau ^{2}A\right) ^{-1}\left( \tau A + \frac{\tau ^{2}A}{2}\right) \bigg ]v_{0} \\&\quad = \left[ R_{1}^{k} + \left( R_{1}-R_{2}\right) ^{-1}\left( R_{1}^{k}-R_{2}^{k}\right) \left\{ I-R_{1} - \left( I+\tau ^{2}A\right) ^{-1}\left( \tau A + \frac{\tau ^{2}A}{2}\right) \right\} \right] v_{0} \\&\quad = \Bigg [R_{1}^{k} - \left( R_{2}^{-1}R_{1}^{k-1}-R_{1}^{-1}R_{2}^{k-1}\right) \\&\qquad \times \left\{ \frac{iA^{-1/2}}{2\tau }\left( I-R_{1}^{-1}\right) R_{1} + \left( I+\tau ^{2}A\right) ^{-1}\left( \frac{iA^{1/2}}{2} + \frac{i\tau A^{1/2}}{4}\right) \right\} \Bigg ]v_{0} \\&\quad = \Bigg [R_{1}^{k} - \left( R_{2}^{-1}R_{1}^{k-1}-R_{1}^{-1}R_{2}^{k-1}\right) \\&\qquad \times \left\{ \frac{R_{1}}{2}+\frac{i\tau A^{1/2}}{4}R_{1} + \left( I+\tau ^{2}A\right) ^{-1}\left( \frac{iA^{1/2}}{2} + \frac{i\tau A^{1/2}}{4}\right) \right\} \Bigg ]v_{0}. \end{aligned}$$

Then, (32) becomes

$$\begin{aligned} v_{k}&= \Bigg [R_{1}^{k} - \left( R_{2}^{-1}R_{1}^{k-1}-R_{1}^{-1}R_{2}^{k-1}\right) \nonumber \\&\quad \times \left\{ \frac{R_{1}}{2}+\frac{i\tau A^{1/2}}{4}R_{1} + \left( I+\tau ^{2}A\right) ^{-1}\left( \frac{iA^{1/2}}{2} + \frac{i\tau A^{1/2}}{4}\right) \right\} \Bigg ]v_{0} \nonumber \\&\quad + \frac{iA^{-1/2}}{2}\left( R_{2}^{-1}R_{1}^{k-1}-R_{1}^{-1}R_{2}^{k-1}\right) \left( I+\tau ^{2}A\right) ^{-1}\left( g_{0} + \frac{\tau }{2}f_{0}\right) \nonumber \\&\quad + \sum \limits _{j=1}^{k} \frac{iA^{-1/2}}{2}\left( R_{1}^{k-j}-R_{2}^{k-j}\right) f_{j} \tau , \quad k=1,\ldots ,N. \end{aligned}$$
(33)

It is known that for the given \(v_{-N}\) the difference scheme

$$\begin{aligned} \frac{v_{k}-v_{k-1}}{\tau } + A\left( I+\frac{\tau }{2}A\right) v_{k} = \left( I+\frac{\tau }{2}A\right) g_{k}, ~ -N+1\le k\le 0 \end{aligned}$$

has a solution

$$\begin{aligned} v_{k}={\tilde{Q}}^{N+k}v_{-N} + \sum \limits _{j=-N+1}^{k}\left( I+\frac{\tau A}{2}\right) {\tilde{Q}}^{k-j+1}g_{j}\tau , \quad k=-N+1,\ldots ,0. \end{aligned}$$
(34)

In particular, with \(k=0\) we have

$$\begin{aligned} v_{0}={\tilde{Q}}^{N}v_{-N} + \sum \limits _{j=-N+1}^{0}\left( I+\frac{\tau A}{2}\right) {\tilde{Q}}^{-j+1}g_{j}\tau . \end{aligned}$$
(35)

Then, by putting (35) in (33), we obtain

$$\begin{aligned} v_{k}&= \Bigg [R_{1}^{k} - \frac{1}{4}\left( R_{2}^{-1}R_{1}^{k-1}-R_{1}^{-1}R_{2}^{k-1}\right) \nonumber \\&\quad \times \bigg (2R_{1}+i\tau A^{1/2}R_{1} + \left( I+\tau ^{2}A\right) ^{-1}\left( 2iA^{1/2} + i\tau A^{1/2}\right) \bigg ) \Bigg ] \nonumber \\&\quad \times \Bigg ({\tilde{Q}}^{N}v_{-N} + \sum \limits _{j=-N+1}^{0}\left( I+\frac{\tau A}{2}\right) {\tilde{Q}}^{-j+1}g_{j}\tau \Bigg ) \nonumber \\&\quad + \frac{iA^{-1/2}}{2}\left( R_{2}^{-1}R_{1}^{k-1}-R_{1}^{-1}R_{2}^{k-1}\right) \left( I+\tau ^{2}A\right) ^{-1}\left( g_{0} + \frac{\tau }{2}f_{0}\right) \nonumber \\&\quad + \sum \limits _{j=1}^{k} \frac{iA^{-1/2}}{2}\left( R_{1}^{k-j}-R_{2}^{k-j}\right) f_{j} \tau , \quad k=1,\ldots ,N. \end{aligned}$$
(36)

If \(-1+2\tau \le \lambda < \tau \), then \(-N+2\le \ell \le 0\) and therefore from (31) and (34) it follows

$$\begin{aligned} v_{\ell } +&\left( \lambda -\tau \ell \right) \frac{v_{\ell }-v_{\ell -1}}{\tau } = v_{-N} + \psi - \varphi \\&\quad = \left( I-\left( \lambda -\tau \ell \right) \Big (I+\frac{\tau A}{2}\Big )A\right) {\tilde{Q}}^{N+\ell }v_{-N} + \big (\tau +\lambda -\tau \ell \big )\Big (I+\frac{\tau A}{2}\Big ){\tilde{Q}}g_{\ell } \\&\qquad + \sum \limits _{j=-N+1}^{\ell -1} \Big (I+\frac{\tau A}{2}\Big ) \left( I-\left( \lambda -\tau \ell \right) \Big (I+\frac{\tau A}{2}\Big )A\right) {\tilde{Q}}^{\ell -j+1}g_{j}\tau , \end{aligned}$$

so that

$$\begin{aligned} v_{-N}&= \Bigg (I-\Big [I-(\lambda -\tau \ell )\Big (I+\frac{\tau A}{2}\Big )A\Big ]{\tilde{Q}}^{N+\ell } \Bigg )^{-1} \nonumber \\&\quad \times \Bigg [\varphi - \psi + \big (\tau +\lambda -\tau \ell \big )\Big (I+\frac{\tau A}{2}\Big ){\tilde{Q}}g_{\ell } \nonumber \\&\quad + \sum \limits _{j=-N+1}^{\ell -1} \Big (I+\frac{\tau A}{2}\Big ) \left( I-\left( \lambda -\tau \ell \right) \Big (I+\frac{\tau A}{2}\Big )A\right) {\tilde{Q}}^{\ell -j+1}g_{j}\tau \Bigg ]. \end{aligned}$$
(37)

If \(\tau \le \lambda < 2\tau \), then \(\ell =1\) and therefore from (31) and (35) it follows

$$\begin{aligned}&v_{1} + (\lambda -\tau ) \frac{v_{1}-v_{0}}{\tau } = v_{-N} + \psi - \varphi \\ {}&\quad = \Bigg (I-\lambda \Big (1+\frac{\tau }{2}\Big )A\left( I+\tau ^{2}A\right) ^{-1}\Bigg ){\tilde{Q}}^{N}v_{-N} + \lambda \left( I+\tau ^{2}A\right) ^{-1} \Big (g_{0}+\frac{\tau }{2}f_{0}\Big ) \\&\qquad + \sum \limits _{j=-N+1}^{0} \Big (I+\frac{\tau A}{2}\Big ) \Bigg (I-\lambda \Big (1+\frac{\tau }{2}\Big )A\left( I+\tau ^{2}A\right) ^{-1}\Bigg ) {\tilde{Q}}^{-j+1}g_{j}\tau , \end{aligned}$$

so that

$$\begin{aligned} v_{-N}&= \Bigg (I-\Big [I-\lambda \Big (1+\frac{\tau }{2}\Big )A\left( I+\tau ^{2}A\right) ^{-1}\Big ]{\tilde{Q}}^{N} \Bigg )^{-1} \nonumber \\&\quad \times \Bigg [\varphi - \psi + \lambda \left( I+\tau ^{2}A\right) ^{-1} \Big (g_{0}+\frac{\tau }{2}f_{0}\Big ) \nonumber \\&\quad + \sum \limits _{j=-N+1}^{0} \Big (I+\frac{\tau A}{2}\Big ) \bigg (I-\lambda \Big (1+\frac{\tau }{2}\Big )A\left( I+\tau ^{2}A\right) ^{-1}\bigg ) {\tilde{Q}}^{-j+1}g_{j}\tau \Bigg ]. \end{aligned}$$
(38)

If \(2\tau \le \lambda \le 1\), then \(2\le \ell \le N\) and therefore from (31) and (36) it follows

$$\begin{aligned}&v_{\ell } + \left( \lambda -\tau \ell \right) \frac{v_{\ell }-v_{\ell -1}}{\tau } = v_{-N} + \psi - \varphi = \Bigg [ \bigg (I+(\lambda -\tau \ell )\Big (\frac{\tau A}{2}-iA^{1/2}\Big )\bigg )R_{1}^{\ell } \\&\quad - \frac{1}{4}\bigg \{\left( I+(\lambda -\tau \ell )\Big (\frac{\tau A}{2}-iA^{1/2}\Big )\right) R_{2}^{-1}R_{1}^{\ell -1} \\&\quad - \bigg (I+(\lambda -\tau \ell )\Big (\frac{\tau A}{2}+iA^{1/2}\Big )\bigg ) R_{1}^{-1}R_{2}^{\ell -1}\bigg \} \\&\quad \times \bigg \{2R_{1}+i\tau A^{1/2}R_{1} + \left( I+\tau ^{2}A\right) ^{-1}\left( 2iA^{1/2} + i\tau A^{1/2}\right) \bigg \} \Bigg ] \\&\quad \times \Bigg ({\tilde{Q}}^{N}v_{-N} + \sum \limits _{j=-N+1}^{0}\left( I+\frac{\tau A}{2}\right) {\tilde{Q}}^{-j+1}g_{j}\tau \Bigg ) \\&\quad + \frac{iA^{-1/2}}{2}\Bigg [\bigg (I+(\lambda -\tau \ell )\Big (\frac{\tau A}{2}-iA^{1/2}\Big )\bigg ) R_{2}^{-1}R_{1}^{\ell -1} \\&\quad - \bigg (I+(\lambda -\tau \ell )\Big (\frac{\tau A}{2}+iA^{1/2}\Big )\bigg ) R_{1}^{-1}R_{2}^{\ell -1}\Bigg ] \left( I+\tau ^{2}A\right) ^{-1}\left( g_{0} + \frac{\tau }{2}f_{0}\right) \\&\quad + \sum \limits _{j=1}^{\ell -1} \frac{iA^{-1/2}}{2}\Bigg [\bigg (I+(\lambda -\tau \ell )\Big (\frac{\tau A}{2}-iA^{1/2}\Big )\bigg ) R_{1}^{\ell -j} \\&\quad - \bigg (I+(\lambda -\tau \ell )\Big (\frac{\tau A}{2}+iA^{1/2}\Big )\bigg ) R_{2}^{\ell -j}\Bigg ] f_{j}\tau , \end{aligned}$$

so that

$$\begin{aligned} v_{-N}&= \Bigg (I-\Bigg [\bigg (I+(\lambda -\tau \ell )\Big (\frac{\tau A}{2}-iA^{1/2}\Big )\bigg ) R_{1}^{\ell } \nonumber \\&\quad - \frac{1}{4} \bigg \{\bigg (I+(\lambda -\tau \ell )\Big (\frac{\tau A}{2}-iA^{1/2}\Big )\bigg ) R_{2}^{-1}R_{1}^{\ell -1} \nonumber \\&\quad - \bigg (I+(\lambda -\tau \ell )\Big (\frac{\tau A}{2}+iA^{1/2}\Big )\bigg ) R_{1}^{-1}R_{2}^{\ell -1} \bigg \} \nonumber \\&\quad \times \bigg (2R_{1}+i\tau A^{1/2}R_{1} + \left( I+\tau ^{2}A\right) ^{-1}\left( 2iA^{1/2} + i\tau A^{1/2}\right) \bigg ) \Bigg ] {\tilde{Q}}^{N}\Bigg )^{-1} \nonumber \\&\quad \times \Bigg (\varphi - \psi + \Bigg [ \bigg (I+(\lambda -\tau \ell )\Big (\frac{\tau A}{2}-iA^{1/2}\Big )\bigg ) R_{1}^{\ell } \nonumber \\&\quad - \frac{1}{4} \bigg \{\bigg (I+(\lambda -\tau \ell )\Big (\frac{\tau A}{2}-iA^{1/2}\Big )\bigg ) R_{2}^{-1}R_{1}^{\ell -1} \nonumber \\&\quad - \bigg (I+(\lambda -\tau \ell )\Big (\frac{\tau A}{2}+iA^{1/2}\Big )\bigg ) R_{1}^{-1}R_{2}^{\ell -1} \bigg \} \nonumber \\&\quad \times \bigg (2R_{1}+i\tau A^{1/2}R_{1} + \left( I+\tau ^{2}A\right) ^{-1}\left( 2iA^{1/2} + i\tau A^{1/2}\right) \bigg ) \Bigg ] \nonumber \\&\quad \times \sum \limits _{j=-N+1}^{0}\left( I+\frac{\tau A}{2}\right) {\tilde{Q}}^{-j+1}g_{j}\tau \nonumber \\&\quad + \frac{iA^{-1/2}}{2}\Bigg [\bigg (I+(\lambda -\tau \ell )\Big (\frac{\tau A}{2}-iA^{1/2}\Big )\bigg ) R_{2}^{-1}R_{1}^{\ell -1} \nonumber \\&\quad - \bigg (I+(\lambda -\tau \ell )\Big (\frac{\tau A}{2}+iA^{1/2}\Big )\bigg ) R_{1}^{-1}R_{2}^{\ell -1}\Bigg ] \left( I+\tau ^{2}A\right) ^{-1}\left( g_{0} + \frac{\tau }{2}f_{0}\right) \nonumber \\&\quad + \sum \limits _{j=1}^{\ell -1} \frac{iA^{-1/2}}{2}\Bigg [\bigg (I+(\lambda -\tau \ell )\Big (\frac{\tau A}{2}-iA^{1/2}\Big )\bigg ) R_{1}^{\ell -j} \nonumber \\&\quad - \bigg (I+(\lambda -\tau \ell )\Big (\frac{\tau A}{2}+iA^{1/2}\Big )\bigg ) R_{2}^{\ell -j}\Bigg ] f_{j}\tau \Bigg ). \end{aligned}$$
(39)

Hence, for the solution of auxiliary difference scheme (31) we have formulas (34) and (36), with \(v_{-N}\) being found by formula (37) when \(-1+2\tau \le \lambda < \tau \), formula (38) when \(\tau \le \lambda <2\tau \) and formula (39) when \(2\tau \le \lambda \le 1\). Then, by using (10), we obtain the solution of difference scheme (23).

Now, let us prove the estimate (30). By using (37) and the estimates (25) and (26), we obtain

$$\begin{aligned} \left\| v_{-N}\right\| _{H} \le M_{1}(\delta ,\lambda ) \Big [ \left\| \varphi \right\| _{H} + \left\| \psi \right\| _{H} + \max _{-N+1 \le k\le 0} \left\| g_{k}\right\| _{H} \Big ]. \end{aligned}$$
(40)

By using (38) and the estimates (25) and (27), we obtain

$$\begin{aligned} \left\| v_{-N}\right\| _{H} \le M_{2}(\delta ,\lambda ) \big [ \left\| \varphi \right\| _{H} + \left\| \psi \right\| _{H} + \left\| f_{0}\right\| _{H} + \max _{-N+1 \le k\le 0} \left\| g_{k}\right\| _{H} \big ]. \end{aligned}$$
(41)

By using (39) and the estimates (24), (25) and (29), we obtain

$$\begin{aligned} \left\| v_{-N}\right\| _{H} \le M_{3}(\delta ,\lambda ) \Big [ \left\| \varphi \right\| _{H} + \left\| \psi \right\| _{H} + \max _{0\le k \le N-1} \left\| f_{k}\right\| _{H} + \max _{-N+1 \le k\le 0} \left\| g_{k}\right\| _{H} \Big ].\nonumber \\ \end{aligned}$$
(42)

Then, using (34) and the estimates (25), (40), (41) and (42), we obtain

$$\begin{aligned} \left\| v_{k}\right\| _{H}&\le \left\| v_{-N}\right\| _{H} + \max _{-N+1 \le k\le 0} \left\| g_{k}\right\| _{H} \nonumber \\&\le M_{4}(\delta ,\lambda ) \Big [ \left\| \varphi \right\| _{H} + \left\| \psi \right\| _{H} + \max _{0\le k \le N-1} \left\| f_{k}\right\| _{H} + \max _{-N+1 \le k\le 0} \left\| g_{k}\right\| _{H} \Big ] \end{aligned}$$
(43)

for \(k=-N+1,\ldots ,0\). Using (36) and the estimates (24), (25), (40), (41) and (42), we obtain

$$\begin{aligned} \left\| v_{k}\right\| _{H}&\le M_{5}(\delta ,\lambda ) \Big ( \left\| v_{-N}\right\| _{H} + \max _{0\le k \le N-1} \left\| f_{k}\right\| _{H} + \max _{-N+1 \le k\le 0} \left\| g_{k}\right\| _{H} \Big ) \nonumber \\&\le M_{6}(\delta ,\lambda ) \Big [ \left\| \varphi \right\| _{H} + \left\| \psi \right\| _{H} + \max _{0\le k \le N-1} \left\| f_{k}\right\| _{H} + \max _{-N+1 \le k\le 0} \left\| g_{k}\right\| _{H} \Big ] \end{aligned}$$
(44)

for \(k=1,\ldots ,N\). Since \(A^{-1}p=\varphi - v_{-N}\), by using (40), (41) and (42), we obtain

$$\begin{aligned} \big \Vert A^{-1}p\big \Vert _{H}&\le \left\| \varphi \right\| _{H} + \left\| v_{-N}\right\| _{H} \nonumber \\&\le M_{7}(\delta ,\lambda ) \Big [ \left\| \varphi \right\| _{H} + \left\| \psi \right\| _{H} + \max _{0\le k \le N-1} \left\| f_{k}\right\| _{H} + \max _{-N+1 \le k\le 0} \left\| g_{k}\right\| _{H} \Big ]. \end{aligned}$$
(45)

So, using (10), (43), (44) and (45), we prove the estimates

$$\begin{aligned} \left\| u_{k}\right\| _{H}&\le \big \Vert A^{-1}p\big \Vert _{H} + \left\| v_{k}\right\| _{H} \nonumber \\&\le M_{8}(\delta ,\lambda ) \Big [ \left\| \varphi \right\| _{H} + \left\| \psi \right\| _{H} + \max _{0\le k \le N-1} \left\| f_{k}\right\| _{H} + \max _{-N+1 \le k\le 0} \left\| g_{k}\right\| _{H} \Big ] \end{aligned}$$
(46)

for \(k=-N,\ldots ,N\). Estimate (30) follows from (45) and (46). \(\square \)

4 Conclusion

In the present paper, we conducted the analysis for the first and second order of accuracy difference schemes (2) and (23) for the approximate solution of problem (1). We have shown the unique solvability of these difference schemes and obtained the stability estimates for their solutions. It has been achieved by reducing the difference schemes to corresponding auxiliary difference schemes, having no parameter involved, and using the operator method in the proofs.

Stable difference schemes (2) and (23) are constructed for the abstract problem (1) and therefore the results are readily applicable for a broad range of problems that can be reduced to form (1).

We note that the second order of accuracy scheme (23) involves \(A^{2}\) and that can undesirable in applications when one needs to proceed with the spatial discretization of a multi-dimensional problem. It is of practical importance to construct the second order of accuracy stable difference scheme that involves A but no higher powers of A. Finally, the future work also concerns construction and analysis of stable difference schemes for the boundary value problem for hyperbolic–parabolic equations with unknown parameter dependent on t.