Numerical investigation of generalized tempered-type integrodifferential equations with respect to another function

Abstract

This paper studies two efficient numerical methods for the generalized tempered integrodifferential equation with respect to another function. The proposed methods approximate the unknown solution through two phases. First, the backward Euler (BE) method and first-order interpolation quadrature rule are adopted to approximate the temporal derivative and generalized tempered integral term to construct a semi-discrete BE scheme. Second, the backward differentiation formula (BDF) and second-order interpolation quadrature rule are adopted to establish a semi-discrete second-order BDF (BDF2) scheme. Additionally, the stability and convergence of two semi-discrete methods are deduced in detail. To further demonstrate the effectiveness of proposed techniques, fully discrete BE and BDF2 finite difference schemes are formulated. Subsequently, the theoretical results of two fully discrete difference schemes are presented. Finally, the numerical results demonstrate the accuracy and competitiveness of the theoretical analysis.

1 Introduction

Fractional calculus (FC) generalises integer differential calculus, containing integrals and derivatives of any real (or complex) order, which has been widely applied in various fields such as, signal analysis, quantum mechanics, continuum mechanics, and elasticity [9, 14, 15, 17,18,19, 29, 31]. Indeed, fractional differential equations (FDEs) have attracted a great interest in analytical and numerical approaches over the last few decades [1,2,3]. Among different definitions of fractional derivatives and integrals, the Caputo fractional derivative is one of the most commonly used definitions [6, 21]. Almeida [4] introduced a new fractional derivative with respect to another function, in the sense of the Caputo derivative and derived some important properties of this new operator. Zaky et al. [30] proposed a mapping transformation that converts $\psi$-Caputo fractional differential equations with respect to another function $\psi$ to their Caputo counterparts, where the function $\psi$ is strictly monotone. Rashid et al. [27] introduced the generalized proportional fractional integral (GPFI) with respect to another function $\Psi$ and proved some several inequalities of the GPFI with respect to another function $\Psi$. Restrepo et al. [28] established the explicit solutions of differential equations of complex fractional orders with respect to functions including continuous variable coefficients. Kosztołowicz and Dutkiewicz [16] proposed a subdiffusion model including Caputo fractional time derivative in the sense of another function to explain subdiffusion in a medium having a structure evolving over time. Babakhani and Frederico [5] introduced the definition of Caputo pseudo-fractional derivatives of order with respect to another function based on a semiring. Mali et al. [20] developed the theory of tempered fractional integrals and derivatives of a function with respect to another function. Fahad and Fernandez [11] considered the operators of Riemann-Liouville fractional differentiation of a function with respect to another function. Later, Fahad and Fernandez [10] developed the theory of Mikusinski’s operational calculus to Caputo fractional derivatives of a function with respect to another function.

This paper considers the generalized integrodifferential equations with the tempered singular kernel as follows

$$\begin{aligned} \begin{aligned} \frac{\partial v}{\partial t}&+ Av + \left( ^{\psi }\mathcal {J}^{\alpha ,\lambda } [v]\right) (t) = g(t), \quad 0< t \le T, \\&v(0)=v_0, \end{aligned} \end{aligned}$$

(1)

where A is a self-adjoint positive-definite linear operator in the Hilbert space, the generalized tempered fractional integral can be defined by

$$\begin{aligned} \begin{aligned} \left( ^{\psi }\mathcal {J}^{\alpha ,\lambda } [\rho ]\right) (t) =\int _0^t \beta _{\alpha ,\lambda }\left( \psi (t)-\psi (s)\right) \psi '(s) \rho (s) ds, \quad t >0, \end{aligned} \end{aligned}$$

from which, the g(t) and $v_0$ are prescribed, see [8, 22]; the tempered singular kernel [23, 25] satisfies

$$\begin{aligned} \begin{aligned} \beta _{\alpha ,\lambda }(t)=\frac{e^{-\lambda t} t^{\alpha -1}}{\varGamma (\alpha )}, \quad 0<\alpha <1, \quad \lambda \ge 0, \end{aligned} \end{aligned}$$

where the notation $\varGamma (\cdot )$ represents the Gamma function. In addition, we suppose that $\psi (t)$ is a positive monotone-increasing bounded convex function and satisfies $\psi '''(t)\ge 0$ on [0, T].

Problems of type (1) can be considered as the model arising in heat transfer theory with memory, population dynamics and viscoelastic materials; see, e.g., [12, 13] and references therein.

The main goal of this work is to introduce two efficient numerical methods for the generalized tempered integrodifferential equation with respect to another function $\psi$, from which, BE scheme and first-order interpolating quadrature are implemented to construct a semi-discrete BE scheme, moreover, second order accurate backward differentiation formula (BDF2) and second-order interpolating quadrature were employed to formulate a semi-discrete BDF2 scheme. After that, the stability and convergence of two semi-discrete approaches are proved based on the discrete energy norm. In addition, we establish two fully discrete schemes by the finite difference approximations and above two semi-discrete schemes, and establish their theoretical results.

Following these ideas, this work has been organized as follows. Section 2 provides the time-discrete BE scheme for the problem (1). Section 3 establishes the BDF2 scheme and gives the corresponding theoretical result for the problem (1). Section 4 constructs two fully discrete finite difference schemes based on spatial finite difference approximations and studies their convergence and stability results. Section 5 presents two numerical examples to highlight in computational terms of the accuracy and effectiveness of the proposed method. Finally, Section 6 presents some concluding remarks.

Throughout this paper, the notation C denotes a positive constant that is independent of the spacial and temporal step sizes, and may be not necessarily the same on each occurrence.

2 The BE scheme

Here, we establish the time-discrete BE approach of the problem (1). For this aim, let us define

$$\begin{aligned} \begin{aligned} u(t) = v(t)e^{\lambda \psi (t)}, \quad f(t) = g(t)e^{\lambda \psi (t)}, \quad \lambda \ge 0, \quad t \ge 0. \end{aligned} \end{aligned}$$

(2)

Thus, the problem (1) can be transformed into

$$\begin{aligned} \begin{aligned} \frac{\partial u}{\partial t}&+ Au + \left( ^{\psi }\mathcal {J}^{\alpha ,0} [u]\right) (t) -\lambda \psi '(t) u = f(t), \quad t >0, \\&u(0)=e^{\lambda \psi (0)}v_0=u_0. \end{aligned} \end{aligned}$$

(3)

In what follows, we shall solve (3) in order to obtain numerical solutions of problem (1). After that, we denote some helpful notations as follows

$$\begin{aligned} \begin{aligned} t_n = nk, \quad k=T/N, \quad 0=t_0<t_1<\cdots <t_N=T, \end{aligned} \end{aligned}$$

(4)

where k is the time uniform step size and N is a positive integer. Further, we denote

$$\begin{aligned} \begin{aligned} \delta _t w^n = \frac{w^n-w^{n-1}}{k}, \quad 1\le n \le N, \quad \delta _t^{(2)} w^n = \frac{3}{2}\delta _t w^n-\frac{1}{2}\delta _t w^{n-1}, \quad 2\le n \le N. \end{aligned} \end{aligned}$$

2.1 Construction of the BE scheme

Here, we adopt the BE method to discretize (3). At first, to approximate the integral term $\left( ^{\psi }\mathcal {J}^{\alpha ,0} [u]\right) (t_n)$, defining the following first-order interpolation quadrature rule

$$\begin{aligned} \begin{aligned} q_n^{(\alpha )}(u)&= \sum \limits _{j=1}^{n}\int _{t_{j-1}}^{t_j}\beta _{\alpha ,0}\left( \psi (t_n)-\psi (s) \right) \psi '(s)u(t_j) ds = \sum \limits _{j=1}^{n} w_{n,j} u(t_j), \end{aligned} \end{aligned}$$

(5)

where

$$\begin{aligned} \begin{aligned} w_{n,j}=\frac{(\psi _n-\psi _{j-1})^{\alpha }-(\psi _n-\psi _{j})^{\alpha }}{\varGamma (\alpha +1)}, \quad \psi _n=\psi (t_n), \quad 1\le j \le n. \end{aligned} \end{aligned}$$

(6)

Then, we obtain the quadrature error

$$\begin{aligned} \begin{aligned} \epsilon _n(u)&= q_n^{(\alpha )}(u)- \left( ^{\psi }\mathcal {J}^{\alpha ,0} [u]\right) (t_n)\\&= \sum \limits _{j=1}^{n}\int _{t_{j-1}}^{t_j}\beta _{\alpha ,0}\left( \psi (t_n)-\psi (s) \right) \psi '(s)[u(t_j)-u(s)] ds, \end{aligned} \end{aligned}$$

(7)

therefore, we have

$$\begin{aligned} \begin{aligned} |\epsilon _n(u)|&\le \sum \limits _{j=1}^{n}\int _{t_{j-1}}^{t_j}\beta _{\alpha ,0}\left( \psi (t_n)-\psi (s) \right) \psi '(s)\int _{t_{j-1}}^{t_j}|u'(\vartheta )|d\vartheta ds\\ {}&\le \sum \limits _{j=1}^{n}w_{n,j} \int _{t_{j-1}}^{t_j}|u'(\vartheta )|d\vartheta . \end{aligned} \end{aligned}$$

(8)

Then, we exchange summation metrics to yield

$$\begin{aligned} \begin{aligned} \mathcal {E}_n(u) := k\sum \limits _{n=1}^{N}\Vert \epsilon _n(u)\Vert&\le k\sum \limits _{j=1}^{N}\left( \sum \limits _{n=j}^{N}w_{n,j}\right) \int _{t_{j-1}}^{t_j}\Vert u_t\Vert \textrm{d}t , \end{aligned} \end{aligned}$$

from which, we utilize the fact that

$$\begin{aligned} \begin{aligned} (\psi _{s}-\psi _{j-1}) - (\psi _{s+1}-\psi _{j}) = (\psi _{s}-\psi _{s+1}) - (\psi _{j-1}-\psi _{j}) \le 0, \\ s=j,j+1,\ldots ,N-1, \quad \Longrightarrow (\psi _{s}-\psi _{j-1})^\alpha \le (\psi _{s+1}-\psi _{j})^\alpha . \end{aligned} \end{aligned}$$

Consequently,

$$\begin{aligned} \begin{aligned} \sum \limits _{n=j}^{N}w_{n,j}&= \frac{1}{\varGamma (\alpha +1)} \left\{ (\psi _{N}-\psi _{j-1})^\alpha + \sum \limits _{s=j}^{N-1} \left[ (\psi _{s}-\psi _{j-1})^\alpha - (\psi _{s+1}-\psi _{j})^\alpha \right] \right\} \\&\le \frac{(\psi _{N}-\psi _{j-1})^\alpha }{\varGamma (\alpha +1)}, \end{aligned} \end{aligned}$$

(9)

which can obtain

$$\begin{aligned} \begin{aligned} \mathcal {E}_n(u) \le k\sum \limits _{n=1}^{N}\frac{(\psi _{N}-\psi _{j-1})^\alpha }{\varGamma (\alpha +1)}\int _{t_{j-1}}^{t_j}\Vert u_t\Vert \textrm{d}t \le \frac{(\psi _{N}-\psi _{0})^\alpha }{\varGamma (\alpha +1)} k\int _{0}^{T}\Vert u_t\Vert \textrm{d}t . \end{aligned} \end{aligned}$$

(10)

Next, considering (3) at $t=t_n$, then we apply the BE method and quadrature rule (5) to get

$$\begin{aligned} \begin{aligned} \delta _t u^n&+ Au^n + \sum \limits _{j=1}^{n} w_{n,j} u^j -\lambda \psi '(t_n) u^n = f^n + R_1^n + R_2^n, \quad 1\le n \le N, \end{aligned} \end{aligned}$$

(11)

where $u^n=u(t_n)$, $f^n=f(t_n)$, $R_1^n=\epsilon _n(u)$, and

$$\begin{aligned} \begin{aligned} R_2^n=\delta _t u^n-\frac{\partial u}{\partial t}(t_n) = \frac{1}{k} \int _{t_{n-1}}^{t_n} (t-t_{n-1}) u_{tt}(t)\textrm{d}t , \quad n\ge 1. \end{aligned} \end{aligned}$$

By ignoring the small term and replacing $u^n$ with its numerical approximation $U^n$, we can obtain the following time semi-discrete BE scheme

$$\begin{aligned} \begin{aligned} \delta _t U^n&+ AU^n + \sum \limits _{j=1}^{n} w_{n,j} U^j -\lambda \psi '(t_n) U^n = f^n, \quad 1\le n \le N, \end{aligned} \end{aligned}$$

(12)

with the initial value $U^0=u_0$.

2.2 Analysis of the BE scheme

This section provides the stability and estimate error of the BE scheme. In what follows, we give the following stability results.

Theorem 1

The BE scheme (12) is unconditionally stable and the numerical solution $U^n$ satisfies

$$\begin{aligned} \begin{aligned} \max \limits _{1\le n \le N} \Vert U^n\Vert \le {C(T,\psi )} \left( \Vert u_0\Vert + k\sum \limits _{n=1}^{N} \Vert f^n\Vert \right) , \end{aligned} \end{aligned}$$

where k denotes the time-uniform step size, and $C(T,\psi )$ depends only on T and $\psi$.

Proof

At first, taking the inner product of (12) with $kU^n$, and using the positiveness of the operator A, we obtain

$$\begin{aligned} \begin{aligned} {k}(\delta _t U^n,U^n)&+{k}\sum \limits _{j=1}^{n} w_{n,j} (U^j,U^n) -\lambda \psi '(t_n){k} (U^n,U^n) = {k}(f^n,U^n), \end{aligned} \end{aligned}$$

(13)

from which, we define

$$\begin{aligned} \begin{aligned} c_0=\max \limits _{0\le t \le T}\{\psi '(t)\}, \quad \left\| U^K\right\| = \max \limits _{0\le n \le N} \Vert U^n\Vert , \end{aligned} \end{aligned}$$

(14)

and sum for (13) from 1 to N, then $(w_{n,j}\ge 0)$

$$\begin{aligned} \begin{aligned} \left\| U^K\right\| ^2&\le \left\| U^0\right\| ^2 + 2 k\sum \limits _{n=1}^{K} \max \limits _{1\le j \le n}\left\| U^j\right\| \Vert U^n\Vert \sum \limits _{j=1}^{n} w_{n,j} \\&\quad + 2\lambda c_0 k\sum \limits _{n=1}^{K}\Vert U^n\Vert ^2 + 2k\sum \limits _{n=1}^{K}\Vert U^n\Vert \left\| f^n\right\| \\&\le \Vert U^0\Vert \left\| U^K\right\| + 2k\sum \limits _{n=1}^{N} \left\| U^K\right\| \Vert U^n\Vert \sum \limits _{j=1}^{n} w_{n,j} \\&\quad + 2\lambda c_0 k\sum \limits _{n=1}^{N}\Vert U^n\Vert \left\| U^K\right\| + 2k\sum \limits _{n=1}^{N}\left\| U^K\right\| \Vert f^n\Vert , \end{aligned} \end{aligned}$$

(15)

which can yield that

$$\begin{aligned} \begin{aligned} \left\| U^N\right\| \le \left\| U^K\right\|&\le \left\| U^0\right\| + 2k\sum \limits _{n=1}^{N} \left[ \frac{(\psi _n-\psi _0)^{\alpha }}{\varGamma (\alpha +1)} + \lambda c_0 \right] \Vert U^n\Vert + 2k\sum \limits _{n=1}^{N} \Vert f^n\Vert \\&\le \left( \left\| U^0\right\| + 2k\sum \limits _{n=1}^{N} \Vert f^n\Vert \right) + c_1 k\sum \limits _{n=1}^{N} \Vert U^n\Vert , \end{aligned} \end{aligned}$$

(16)

where $c_1= \frac{2(\psi _N-\psi _0)^{\alpha }}{\varGamma (\alpha +1)} + 2\lambda c_0$ and $U^0=u_0$. This completes the proof by the Grönwall inequality. $\square$

Now, we establish the convergence result for (12). For this aim, we denote

$$\begin{aligned} \begin{aligned} \rho ^n = u(t_n) - U^n, \quad {\tilde{e}}^n = v(t_n) - V^n, \end{aligned} \end{aligned}$$

(17)

where $V^n = e^{-\lambda \psi _n}U^n$, which means that ${\tilde{e}}^n=e^{-\lambda \psi _n}\rho ^n$. Then, subtracting (12) from (11), we yield the following error equations

$$\begin{aligned} \begin{aligned} \delta _t \rho ^n&+ A\rho ^n + \sum \limits _{j=1}^{n} w_{n,j} \rho ^j -\lambda \psi '(t_n) \rho ^n = R_1^n + R_2^n, \quad 1\le n \le N, \\&\rho ^0 = 0. \end{aligned} \end{aligned}$$

(18)

Similar to the analysis of Theorem 1, we get the following results.

Theorem 2

Let $U^n$ and $u(t_n)$ be the solutions of (12) and (3), respectively. Then, it holds that

$$\begin{aligned} \begin{aligned} \max \limits _{1\le n \le N}\Vert U^n-u(t_n)\Vert \le {C(T,\psi )} \left( k\int _{0}^{T}\Vert u_{t}\Vert \textrm{d}t + k\int _{0}^{T}\Vert u_{tt}\Vert \textrm{d}t \right) , \end{aligned} \end{aligned}$$

in which $C(T,\psi )$ independent of the time step-size k.

Proof

By Theorem 1 and (18), we first have

$$\begin{aligned} \begin{aligned} \max \limits _{1\le n \le N}\Vert \rho ^n\Vert&\le {C(T,\psi )} \left( \Vert \rho ^0\Vert + k\sum \limits _{n=1}^{N} \left\| R_1^n + R_2^n\right\| \right) \\&\le {C(T,\psi )} \left( k\sum \limits _{n=1}^{N} \left\| R_1^n\right\| + k\sum \limits _{n=1}^{N} \left\| R_2^n\right\| \right) , \end{aligned} \end{aligned}$$

(19)

from which, by applying (10) and

$$\begin{aligned} \begin{aligned} k\sum \limits _{n=1}^{N} \left\| R_2^n\right\| \le k\sum \limits _{n=1}^{N}\int _{t_{n-1}}^{t_n} \Vert u_{tt}\Vert \textrm{d}t = k\int _{0}^{T}\Vert u_{tt}\Vert \textrm{d}t , \end{aligned} \end{aligned}$$

which completes the proof.

3 Second-order BDF scheme

This section establishes the BDF2 scheme and gives the corresponding theoretical result for problem (3).

3.1 Construction of the BDF2 scheme

Herein, we implement the BDF2 approach to discretize (3). Firstly, in order to approximate the integral term $\left( ^{\psi }\mathcal {J}^{\alpha ,0} [u]\right) (t_n)$, we define the second-order interpolation quadrature rule as follows

$$\begin{aligned} \begin{aligned} {\tilde{q}}_n^{(\alpha )}(u)&= \sum \limits _{j=1}^{n}\int _{t_{j-1}}^{t_j}\beta _{\alpha ,0}\left( \psi (t_n)-\psi (s) \right) \psi '(s){N_1(s)}\textrm{d}s = \sum \limits _{j=0}^{n} \tilde{\kappa }_{n,j} u(t_j), \end{aligned} \end{aligned}$$

(20)

where ${N_1(s)=u(t_{j-1})+\frac{s-t_{j-1}}{k}\left[ u(t_j)-u(t_{j-1})\right] }$ and

$$\begin{aligned} \begin{aligned}&\tilde{\kappa }_{n,0}= \frac{1}{k\varGamma (\alpha +1)}\left[ k(\psi _n-\psi _0)^{\alpha } - \int _{0}^{t_1}( \psi _n-\psi (s) )^{\alpha }\textrm{d}s \right] , \\ {}&\tilde{\kappa }_{n,j}= \frac{1}{k\varGamma (\alpha +1)}\left( \int _{t_{j-1}}^{t_{j}}-\int _{t_{j}}^{t_{j+1}} \right) ( \psi _n-\psi (s) )^{\alpha }\textrm{d}s, \quad 1\le j \le n-1, \\ {}&\tilde{\kappa }_{n,n}= \frac{1}{k\varGamma (\alpha +1)} \int _{t_{n-1}}^{t_{n}}( \psi _n-\psi (s) )^{\alpha }\textrm{d}s. \end{aligned} \end{aligned}$$

(21)

Further, we deduce the following quadrature error by utilizing Newton’s form of the remainder for linear interpolation,

$$\begin{aligned} \begin{aligned} \tilde{\epsilon }_n(u)&= {\tilde{q}}_n^{(\alpha )}(u)- \left( ^{\psi }\mathcal {J}^{\alpha ,0} [u]\right) (t_n)\\&= \sum \limits _{j=1}^{n}\int _{t_{j-1}}^{t_j}\beta _{\alpha ,0}\left( \psi _n-\psi (s) \right) \psi '(s) (s-t_{j-1})(t_j-s) u[t_{j-1},t_j,s] \textrm{d}s \\&= \sum \limits _{j=2}^{n}\int _{t_{j-1}}^{t_j}\beta _{\alpha ,0}\left( \psi _n-\psi (s) \right) \psi '(s) (s-t_{j-1})(t_j-s) u[t_{j-1},t_j,s] \textrm{d}s \\&\quad + \int _{0}^{k}\beta _{\alpha ,0}\left( \psi _n-\psi (s) \right) \psi '(s) (k-s) su[0,k,s] \textrm{d}s, \end{aligned} \end{aligned}$$

(22)

from which we use the fact that $N_1(s)-u(s)=(s-t_{j-1})(t_j-s) u[t_{j-1},t_j,s]$. Then, we will estimate the quadrature error (22). First, we have

$$\begin{aligned}&\begin{aligned} \left| su[0,k,s]\right| = \left| u[s,k] - u[0,k] \right| \le \frac{2}{k-s}\int _{0}^{k} |u_t(t)|\textrm{d}t , \quad 0<s<k, \end{aligned} \end{aligned}$$

(23)

$$\begin{aligned}&\begin{aligned} \big |u[t_{j-1},t_j,s]\big | \le \frac{1}{k} \left( \int _{s}^{t_{j}}+\int _{t_{j-1}}^{s} \right) |u_{tt}(t)|\textrm{d}t , \quad t_{j-1}< s < t_j, \quad 2\le j \le n. \end{aligned} \end{aligned}$$

(24)

Thus by the assumptions $\psi ^{(n)}(t) \ge 0$, $n=0,1,2,3$, we can yield

$$\begin{aligned} \begin{aligned} \big |\tilde{\epsilon }_n(u) \big | \le 2w_{n,1}\int _{0}^{k}|u_t(t)|\textrm{d}t + k\sum \limits _{j=2}^{n}w_{n,j} \int _{t_{j-1}}^{t_j} |u_{tt}(t)|\textrm{d}t, \end{aligned} \end{aligned}$$

(25)

where $w_{n,j}$ defined in (6). This naturally gets

$$\begin{aligned} \begin{aligned} \widetilde{\mathcal {E}}_n(u) := k\sum \limits _{n=1}^{N}\Vert \tilde{\epsilon }_n(u)\Vert&\le k\sum \limits _{n=1}^{N}w_{n,1}\int _{0}^{k}\Vert u_t(t)\Vert \textrm{d}t \\ {}&\quad + k^2\sum \limits _{n=1}^{N}\left( \sum \limits _{j=2}^{n}w_{n,j}\right) \int _{t_{j-1}}^{t_j} \Vert u_{tt}(t)\Vert \textrm{d}t , \end{aligned} \end{aligned}$$

(26)

from which, we have $\sum \limits _{n=1}^{N}w_{n,1}=\frac{\psi _N-\psi _0}{\varGamma (\alpha +1)}$, and swap summation metrics that

$$\begin{aligned} \begin{aligned} \sum \limits _{n=1}^{N}&\left( \sum \limits _{j=2}^{n}w_{n,j}\right) \int _{t_{j-1}}^{t_j} \Vert u_{tt}(t)\Vert \textrm{d}t = \sum \limits _{j=2}^{N} \left( \sum \limits _{n=j}^{N}w_{n,j}\right) \int _{t_{j-1}}^{t_j} \Vert u_{tt}(t)\Vert \textrm{d}t \\ {}&\le \sum \limits _{j=2}^{N} \left( \frac{\psi _N-\psi _{j-1}}{\varGamma (\alpha +1)} \right) \int _{t_{j-1}}^{t_j} \Vert u_{tt}(t)\Vert \textrm{d}t \\ {}&{ \quad \le \frac{\psi _N-\psi _{1}}{\varGamma (\alpha +1)} \int _{k}^{t_N} \Vert u_{tt}(t)\Vert \textrm{d}t} . \end{aligned} \end{aligned}$$

(27)

Consequently,

$$\begin{aligned} \begin{aligned} \widetilde{\mathcal {E}}_n(u) \le \frac{2(\psi _N-\psi _0)}{\varGamma (\alpha +1)} k \int _{0}^{k}\Vert u_t(t)\Vert \textrm{d}t + \frac{\psi _N-\psi _{1}}{\varGamma (\alpha +1)} k^2 \int _{k}^{t_N} \Vert u_{tt}(t)\Vert \textrm{d}t . \end{aligned} \end{aligned}$$

(28)

Note that the integrals in (21) will not be easy to compute when the function $\psi (t)$ is complicated. Thus we shall use the medium rectangle formula to approximate them. First, we define

$$\begin{aligned} \begin{aligned} g_n(s)=(\psi _n-\psi (s))^{\alpha }, \quad t_{j-1/2}=\frac{1}{2}(t_j+t_{j-1}), \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \Lambda _{n,j}= \frac{1}{\varGamma (\alpha +1)} \left[ \frac{1}{k} \int _{t_{j-1}}^{t_{j}}g_n(s)ds - \left( \psi _n-\psi (t_{j-1/2})\right) ^{\alpha } \right] . \end{aligned} \end{aligned}$$

(29)

By differentiating $g_n(s)$ with respect to s, we arrive at

$$\begin{aligned} \begin{aligned} g_n'(s)&= -\alpha \psi '(s)(\psi _n-\psi (s))^{\alpha -1} \le 0, \quad 0<\alpha <1, \\ g_n''(s)&= - \alpha \left\{ (1-\alpha )\left[ \psi '(s)\right] ^2 (\psi _n-\psi (s))^{\alpha -2} + \psi ''(s)(\psi _n-\psi (s))^{\alpha -1} \right\} \le 0, \\ g_n'''(s)&= - \alpha (1-\alpha )(2-\alpha ) \left[ \psi '(s) (\psi _n-\psi (s))^{\alpha -3}\right] \\ {}&\quad - 3\alpha (1-\alpha )\left[ \psi '(s)\psi ''(s) (\psi _n-\psi (s))^{\alpha -2} \right] \\ {}&\quad - \alpha \left[ \psi '''(s) (\psi _n-\psi (s))^{\alpha -1} \right] \le 0. \end{aligned} \end{aligned}$$

(30)

Then we denote the following modified quadrature rule to approximate (20), i.e.,

$$\begin{aligned} \begin{aligned} {\hat{q}}_n^{(\alpha )}(u)&= \sum \limits _{j=0}^{n} \hat{\kappa }_{n,j} u(t_j), \end{aligned} \end{aligned}$$

(31)

in which,

$$\begin{aligned} \begin{aligned}&\hat{\kappa }_{n,0}= \frac{1}{\varGamma (\alpha +1)}\left[ (\psi _n-\psi _0)^{\alpha } - ( \psi _n-\psi (t_{1/2}) )^{\alpha } \right] , \\&\hat{\kappa }_{n,j}= \frac{1}{\varGamma (\alpha +1)}\left[ ( \psi _n-\psi (t_{j-1/2}) )^{\alpha } - ( \psi _n-\psi (t_{j+1/2}) )^{\alpha } \right] , \quad 1\le j \le n-1, \\&\hat{\kappa }_{n,n}= \frac{1}{\varGamma (\alpha +1)} ( \psi _n-\psi (t_{n-1/2}) )^{\alpha }. \end{aligned} \end{aligned}$$

(32)

Next, we present the error estimate of (20) and (31). By the medium rectangle formula, we have

$$\begin{aligned} \begin{aligned} \Lambda _{n,j}=\frac{g_n''(\xi _j)k^2}{24\varGamma (\alpha +1)}\le 0, \quad t_{j-1}< \xi _j<t_j, \quad 1\le j \le n, \end{aligned} \end{aligned}$$

(33)

where we employ $g_n^{(m)}(s)\le 0$ with $m=1,2,3$ (see (30)), based on appropriate assumptions of the function $\psi$. Hence, we get

$$\begin{aligned} \begin{aligned} \left\| \Theta _n(u)\right\| = \left\| {\hat{q}}_n^{(\alpha )}(u) - {\tilde{q}}_n^{(\alpha )}(u)\right\| \le \max \limits _{0\le j \le n}\left\| u(t_j)\right\| \sum \limits _{j=0}^{n} \left| \hat{\kappa }_{n,j}-\tilde{\kappa }_{n,j} \right| , \end{aligned} \end{aligned}$$

(34)

from which we deduce

$$\begin{aligned} \begin{aligned} {\widetilde{R}}_n := \sum \limits _{j=0}^{n} \left| \hat{\kappa }_{n,j}-\tilde{\kappa }_{n,j} \right|&= -\Lambda _{n,1} + \sum \limits _{j=1}^{n-1}(\Lambda _{n,j}-\Lambda _{n,j+1}) - \Lambda _{n,n}\\ {}&= \frac{-g_n''(\xi _n)k^2}{12\varGamma (\alpha +1)}, \quad t_{n-1}< \xi _n<t_n, \end{aligned} \end{aligned}$$

where $g_n''(s)$ denoted in (30). First supposing that $\psi ^{(m)}(s)$ $(s=0,1,2,3)$ is bounded, we have

$$\begin{aligned} \begin{aligned}&k {\widetilde{R}}_1 \le C_{\alpha ,\psi }k^3 (\psi _1-\psi (\xi _1))^{\alpha -2}\left[ \psi '(\xi _1)\right] ^2 \\ {}&\quad \le C_{\alpha ,\psi }k^{1+\alpha } \left[ \frac{\psi '(\xi _1)}{\psi '(\hat{\xi }_1)}\right] ^2, \quad t_{0}< \xi _1<\hat{\xi }_1<t_1, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} k\sum \limits _{n=2}^{N} {\widetilde{R}}_n&\le C_{\alpha ,\psi }k^3\sum \limits _{n=2}^{N} (\psi _n-\psi (\xi _n))^{\alpha -2}\left[ \psi '(\xi _n)\right] ^2 \\ {}&\le C_{\alpha ,\psi }k^3\sum \limits _{n=2}^{N} t_n^{\alpha -2} \left[ \frac{\psi '(\xi _n)}{\psi '(\hat{\xi }_n)}\right] ^2 \le C_{\alpha ,\psi }k^2 \int _{k}^{t_N}s^{\alpha -2}\textrm{d}s \\ {}&\le C_{\alpha ,\psi }k^{1+\alpha }, \quad t_{n-1}< \xi _n<\hat{\xi }_n<t_n, \quad 2\le n\le N{.} \end{aligned} \end{aligned}$$

Thus, we yield that

$$\begin{aligned} \begin{aligned} k\sum \limits _{n=1}^{N} \left\| \Theta _n(u)\right\| \le \max \limits _{0\le n \le N}\left\| u(t_n)\right\| \left( k\sum \limits _{n=1}^{N} {\widetilde{R}}_n \right) \le C_{\alpha ,\psi }k^{1+\alpha } \max \limits _{0\le n \le N}\left\| u(t_n)\right\| . \end{aligned} \end{aligned}$$

(35)

Then, considering (3) at $t=t_n$, applying the BDF2 method and using quadrature rule (31), we have

$$\begin{aligned}&\begin{aligned} \delta _t u^1&+ Au^1 + \sum \limits _{j=0}^{1} \hat{\kappa }_{1,j} u^1 -\lambda \psi '(t_1) u^1 = f^1 + R_3^1 + R_4^1, \end{aligned} \end{aligned}$$

(36)

$$\begin{aligned}&\begin{aligned} \delta _t^{(2)} u^n&+ Au^n + \sum \limits _{j=0}^{n} \hat{\kappa }_{n,j} u^j -\lambda \psi '(t_n) u^n = f^n + R_3^n + R_4^n, \quad 2\le n \le N, \end{aligned} \end{aligned}$$

(37)

where $R_3^n=\tilde{\epsilon }_n(u)+\Theta _n(u)$ with $n\ge 1$, $R_4^1=R_2^1$ and

$$\begin{aligned} \begin{aligned} k\sum \limits _{n=2}^{N} \Vert R_4^n\Vert \le 4\left( k\int _{0}^{2k}\Vert u_{tt}\Vert \textrm{d}t + k^2\int _{k}^{t_N}\Vert u_{ttt}\Vert \textrm{d}t \right) . \end{aligned} \end{aligned}$$

(38)

Then, omitting the small term and replacing $u^n$ by $U^n$, we obtain the following time semi-discrete BDF2 scheme

$$\begin{aligned}&\delta _t U^1 + AU^1 + \sum \limits _{j=0}^{1} \hat{\kappa }_{1,j} U^1 -\lambda \psi '(t_1) U^1 = f^1, \end{aligned}$$

(39)

$$\begin{aligned}&\delta _t^{(2)} U^n + AU^n + \sum \limits _{j=0}^{n} \hat{\kappa }_{n,j} U^j -\lambda \psi '(t_n) U^n = f^n, \quad 2\le n \le N, \end{aligned}$$

(40)

with the initial data $U^0=u_0$.

3.2 Analysis of the BDF2 scheme

Herein, the stability and estimate error of scheme (39)-(40) will be derived. First, we shall establish the following stability.

Theorem 3

The BDF2 scheme (39)-(40) is unconditionally stable and the following inequality holds

$$\begin{aligned} \begin{aligned} \left\| U^N\right\| \le {C(T,\psi )} \left( \Vert u_0\Vert + k\sum \limits _{n=1}^{N} \Vert f^n\Vert \right) , \quad N\ge 1, \end{aligned} \end{aligned}$$

in which $C(T,\psi )$ independent of the time step-size k.

Proof

Applying the inner product of (39)-(40) with $kU^1$ and $kU^n$, respectively, and employing the positiveness of the operator A arrives at

$$\begin{aligned} \begin{aligned} k(\delta _t U^1,U^1)&+ k\sum \limits _{j=0}^{1} \hat{\kappa }_{1,j} (U^j,U^1) - k\lambda \psi '(t_1) \Vert U^1\Vert ^2 {\le \,} k(f^1,U^1), \end{aligned} \end{aligned}$$

(41)

$$\begin{aligned} \begin{aligned} k(\delta _t^{(2)} U^n,U^n)&+ k\sum \limits _{j=0}^{n} \hat{\kappa }_{n,j} (U^j,U^n) - k\lambda \psi '(t_n) \Vert U^n\Vert ^2 {\le \,} k(f^n,U^n), \quad n\ge 2. \end{aligned} \end{aligned}$$

(42)

Summing for (42) from 2 to N, and adding (41) leads to

$$\begin{aligned} \begin{aligned}&k(\delta _t U^1,U^1) + k\sum \limits _{n=2}^{N}(\delta _t^{(2)} U^n,U^n) + k\sum \limits _{n=1}^{N}\sum \limits _{j=0}^{n} \hat{\kappa }_{n,j} (U^j,U^n) \\ {}&\quad -k\lambda \sum \limits _{n=1}^{N}\psi '(t_n)\Vert U^n\Vert ^2 {\le \,} k\sum \limits _{n=1}^{N}(f^n,U^n). \end{aligned} \end{aligned}$$

(43)

Then from [26, (4.8)], it is easy to arrive at

$$\begin{aligned} \begin{aligned} k(\delta _t U^1,U^1)&+ k\sum \limits _{n=2}^{N}(\delta _t^{(2)} U^n,U^n) \\&\ge \frac{3}{4}\Vert U^{N}\Vert ^2 - \frac{1}{4}\left( \Vert U^{N-1}\Vert ^2 +\Vert U^{1}\Vert ^2 +\Vert U^{0}\Vert ^2\right) . \end{aligned} \end{aligned}$$

(44)

Besides, notice that $\hat{\kappa }_{n,j}\ge 0$, which can get $\Xi _n:=\sum \limits _{j=0}^{n} \hat{\kappa }_{n,j}=\frac{(\psi _n-\psi _0)^{\alpha }}{\varGamma (\alpha +1)}$. Thus, with above analyses, then (42) becomes

$$\begin{aligned} \begin{aligned} \frac{3}{4}\Vert U^{N}\Vert ^2&\le \frac{1}{4}\left( \Vert U^{N-1}\Vert ^2 +\Vert U^{1}\Vert ^2 +\Vert U^{0}\Vert ^2\right) + k\sum \limits _{n=1}^{N} \Xi _N \Vert U^n\Vert \max \limits _{0\le j \le n} \Vert U^j\Vert \\ {}&+ \lambda \psi '(t_N) k\sum \limits _{n=1}^{N} \Vert U^n\Vert ^2 + k\sum \limits _{n=1}^{N} \Vert f^n\Vert \Vert U^n\Vert . \end{aligned} \end{aligned}$$

(45)

Using (14), then we have

$$\begin{aligned} \begin{aligned} \frac{3}{4}\Vert U^{K}\Vert ^2&\le \frac{1}{4}\left( \Vert U^{K-1}\Vert ^2 +\Vert U^{1}\Vert ^2 +\Vert U^{0}\Vert ^2\right) + k\sum \limits _{n=1}^{K} \Xi _K \Vert U^n\Vert \Vert U^{K}\Vert \\ {}&\quad + \lambda \psi '(t_K) k\sum \limits _{n=1}^{K} \Vert U^n\Vert ^2 + k\sum \limits _{n=1}^{K} \Vert f^n\Vert \Vert U^n\Vert \\ {}&\le \frac{1}{4}\Vert U^{K}\Vert ^2 + \frac{1}{4}\left( \Vert U^{1}\Vert +\Vert U^{0}\Vert \right) \Vert U^{K}\Vert + \Xi _N k\sum \limits _{n=1}^{N} \Vert U^n\Vert \Vert U^{K}\Vert \\ {}&\quad + \lambda c_0 k\sum \limits _{n=1}^{N} \Vert U^n\Vert \Vert U^K\Vert + k\sum \limits _{n=1}^{N} \Vert f^n\Vert \Vert U^K\Vert , \end{aligned} \end{aligned}$$

which can yield

$$\begin{aligned} \begin{aligned} \left\| U^{N}\right\| \le \left\| U^{K}\right\|&\le \frac{\Vert U^{1}\Vert +\Vert U^{0}\Vert }{2} + 2k\sum \limits _{n=1}^{N} \Vert f^n\Vert + 2\left( \Xi _N + \lambda c_0\right) k\sum \limits _{n=1}^{N} \Vert U^n\Vert . \end{aligned} \end{aligned}$$

(46)

Employing familiar Grönwall inequality and $U^0=u_0$, we obtain

$$\begin{aligned} \begin{aligned} \left\| U^{N}\right\| \le {C(T,\psi )} \left( \Vert U^{1}\Vert +\Vert u_0\Vert + k\sum \limits _{n=1}^{N} \Vert f^n\Vert \right) . \end{aligned} \end{aligned}$$

(47)

In order to finish the proof, we need to estimate $\Vert U^{1}\Vert$. Based on above analyses, we use (41) to get

$$\begin{aligned} \begin{aligned} \frac{\Vert U^{1}\Vert ^2 - \Vert U^{0}\Vert ^2 }{2}&\le k \Xi _1 \Vert U^{1} \Vert \left( \Vert U^{1} \Vert + \Vert U^{0} \Vert \right) + k\lambda \psi '(t_1)\Vert U^{1} \Vert ^2 + k \left\| f^1\right\| \Vert U^{1} \Vert \\ {}&\le k \Xi _1 \Vert U^{1} \Vert \left( \Vert U^{1} \Vert + \Vert U^{0} \Vert \right) \\ {}&\quad + \left( k\lambda \psi '(t_1)\Vert U^{1} \Vert + k \left\| f^1\right\| \right) \left( \Vert U^{1} \Vert + \Vert U^{0} \Vert \right) , \end{aligned} \end{aligned}$$

thence we yield

$$\begin{aligned} \begin{aligned} \frac{\Vert U^{1}\Vert - \Vert U^{0}\Vert }{2}&\le k \left( \lambda \psi '(t_1) + \Xi _1 \right) \Vert U^{1} \Vert + k \left\| f^1\right\| . \end{aligned} \end{aligned}$$

Then, if $k\le \frac{1}{4\left( \lambda \psi '(t_1) + k \Xi _1 \right) }$, we can obtain

$$\begin{aligned} \begin{aligned} \Vert U^{1}\Vert \le \Vert U^{0}\Vert + 2 k \left\| f^1\right\| = \Vert u_0\Vert + 2 k \left\| f^1\right\| , \end{aligned} \end{aligned}$$

which combines (47), the proof of this theorem is completed.

Then, subtracting (12) from (11), we obtain the error equations as

$$\begin{aligned} \begin{aligned}&\delta _t \rho ^1 + A\rho ^1 + \sum \limits _{j=0}^{1} \hat{\kappa }_{1,j} \rho ^1 -\lambda \psi '(t_1) \rho ^1 = R^1_3 + R^1_4, \\ {}&\delta _t^{(2)} \rho ^n + A\rho ^n + \sum \limits _{j=0}^{n} \hat{\kappa }_{n,j} \rho ^j -\lambda \psi '(t_n) \rho ^n = R^n_3 + R^n_4, \quad 2\le n \le N, \\ {}&\rho ^0 = 0. \end{aligned} \end{aligned}$$

(48)

Then, analogous to the analysis of Theorem 3, the following theorem holds.

Theorem 4

Let $U^n$ and $u(t_n)$ be the solutions of (39)-(40) and (3), respectively. Then, it follows that

$$\begin{aligned} \begin{aligned}&\left\| U^N-u(t_N)\right\| \le {C(T,\psi )}\\ {}&\quad \left( k\int _{0}^{2k}\Vert u_{tt}\Vert \textrm{d}t + k^2\int _{k}^{T}\Vert u_{ttt}\Vert \textrm{d}t + k^{1+\alpha }\max \limits _{0\le n \le N}\left\| u(t_n)\right\| \right) . \end{aligned} \end{aligned}$$

Proof

Based on Theorem 3, we have

$$\begin{aligned} \begin{aligned} \left\| \rho ^N\right\|&\le {C(T,\psi )} \left( k\sum \limits _{n=1}^{N} \left\| R_3^n + R_4^n\right\| \right) \le {C(T,\psi )} \left( k\sum \limits _{n=1}^{N} \left\| R_3^n\right\| + k\sum \limits _{n=1}^{N} \left\| R_4^n\right\| \right) , \end{aligned} \end{aligned}$$

(49)

from which, using (28), (34)-(35) and (38), we complete the proof.

4 Applications

In this section, we set $A=-\Delta =-\frac{\partial ^2}{\partial x^2}$ over the domain $\Omega =(0,L)$ with homogeneous Dirichlet boundary conditions, which implies that $Au=-u_{xx}$. We shall establish two fully discrete finite difference schemes by using spatial finite difference approximations.

After that, let $x_i=ih$ and $h=\frac{L}{M}$, where $M\in \mathbb {Z}^+$ and $0\le i \le M$. For any grid function $\omega =\{\omega _i^n|0\le i \le M,0\le n \le N\}$, we denote some notations as follows

$$\begin{aligned} \begin{aligned} \delta _x \omega _{i-\frac{1}{2}}^n:=\frac{1}{h}\left( \omega _i^n-\omega _{i-1}^n\right) , \quad \delta _x^2 \omega _{i}^n:=\frac{1}{h}\left( \delta _x \omega _{i+\frac{1}{2}}^n-\delta _x \omega _{i-\frac{1}{2}}^n\right) . \end{aligned} \end{aligned}$$

Besides, set $\Omega _h:=\{\omega |\omega =(\omega _0,\omega _1,\ldots ,\omega _M )\}$ and $\mathring{\Omega }_h:=\{\omega |\omega \in \Omega _h, \omega _0=\omega _M=0 \}$. For any $\omega ,\upsilon \in \mathring{\Omega }_h$, denote the following discrete inner product and $L_2$ norm

$$\begin{aligned} \begin{aligned} \langle \omega ,\upsilon \rangle :=h\sum \limits _{i=1}^{M-1}\omega _i \upsilon _i, \quad \Vert \omega \Vert _2 :=\sqrt{\langle \omega ,\omega \rangle }. \end{aligned} \end{aligned}$$

Then, the following lemma holds based on the Taylor expansion with integral remainder.

Lemma 1

[32] Assume that $u(x,\cdot )\in C^{4}\big ([0,L]\big )$ and $u_i=u(x_i,\cdot )$. Then for $1\le i \le M-1$, we have

$$\begin{aligned} \begin{aligned} \frac{\partial ^2u}{\partial x^2}(x_i,\cdot )=\delta _x^2u_i -\frac{1}{6}h^2\int _0^1\left[ \frac{\partial ^4u}{\partial x^4}(x_i+\xi h,\cdot )+\frac{\partial ^4u}{\partial x^4}(x_i-\xi h,\cdot )\right] (1-\xi )^3d\xi . \end{aligned} \end{aligned}$$

4.1 Fully discrete BE difference scheme

Here, based on scheme (12), we will provide the fully discrete BE difference approach of the problem (3). First considering (3) at the point $(x_i,t_n)$, we employ (11) and Lemma 1 to obtain

$$\begin{aligned} \begin{aligned} \delta _t u^n_i - \delta _x^2 u^n_i&+ \sum \limits _{j=1}^{n} w_{n,j} u^j_i -\lambda \psi '(t_n) u^n_i = f^n_i + (\mathcal {R}_{1})^n_i,\\ {}&\quad 1\le i \le M-1, \qquad 1\le n \le N, \end{aligned} \end{aligned}$$

(50)

where $u_i^n=u(x_i,t_n)$, $(\mathcal {R}_{1})^n_i= (R_1^n)_i + (R_2^n)_i + \mathcal {O}(h^2)$, and

$$\begin{aligned} \begin{aligned}&u^n_0 = u^n_M = 0, \qquad 0\le n \le N, \\&u^0_i = u_0(x_i), \qquad 1\le i \le M-1. \end{aligned} \end{aligned}$$

(51)

Dropping the truncation error and replacing $u_i^n$ with its numerical solution $U_i^n$, we get the fully discrete BE finite difference scheme

$$\begin{aligned} \begin{aligned} \delta _t U^n_i - \delta _x^2 U^n_i&+ \sum \limits _{j=1}^{n} w_{n,j} U^j_i -\lambda \psi '(t_n) U^n_i = f^n_i,\\ {}&\quad 1\le i \le M-1, \qquad 1\le n \le N \end{aligned} \end{aligned}$$

(52)

with initial and boundary value conditions

$$\begin{aligned} \begin{aligned}&U^n_0 = U^n_M = 0, \qquad 0\le n \le N, \\ {}&U^0_i = u_0(x_i), \qquad 1\le i \le M-1. \end{aligned} \end{aligned}$$

(53)

Then, similar to the proof of Theorems 1 and 2, we obtain the following stability and convergence results.

Theorem 5

The fully discrete BE finite difference scheme (52)-(53) is unconditionally stable. We have

$$\begin{aligned} \begin{aligned} \max \limits _{1\le n \le N} \Vert U^n\Vert _2 \le {C(T,\psi )} \left( \Vert U^0\Vert _2 + k\sum \limits _{n=1}^{N} \Vert f^n\Vert _2 \right) . \end{aligned} \end{aligned}$$

Theorem 6

Let $U^n_i$ and $u_i^n$ be the solutions of the BE scheme (52)-(53) and (3), respectively. Then, it holds that

$$\begin{aligned} \begin{aligned} \max \limits _{1\le n \le N}\Vert U^n-u^n\Vert _2 \le {C(T,\psi )} \left( h^2 + k\int _{0}^{T}\Vert u_{tt}\Vert _2 \textrm{d}t \right) . \end{aligned} \end{aligned}$$

4.2 Fully discrete BDF2 difference scheme

Below based on scheme (39)-(40), we will establish the fully discrete BDF2 difference scheme for problem (3). Firstly, we consider (3) at the point $(x_i,t_n)$, and use (36)-(37) and Lemma 1 to get

$$\begin{aligned}&\begin{aligned} \delta _t u^1_i - \delta _x^2 u^1_i&+ \sum \limits _{j=0}^{1} \hat{\kappa }_{1,j} u^j_i -\lambda \psi '(t_1) u^1_i = f^1_i + (\mathcal {R}_{2})^1_i,\quad 1\le i \le M-1, \end{aligned} \end{aligned}$$

(54)

$$\begin{aligned}&\begin{aligned} \delta _t^{(2)} u^n_i - \delta _x^2 u^n_i&+ \sum \limits _{j=0}^{n} \hat{\kappa }_{n,j} u^j_i -\lambda \psi '(t_n) u^n_i = f^n_i + (\mathcal {R}_{2})^n_i,\\ {}&\quad 1\le i \le M-1, \qquad 2 \le n \le N, \end{aligned} \end{aligned}$$

(55)

from which $(\mathcal {R}_{2})^n_i= (R_3^n)_i + (R_4^n)_i + \mathcal {O}(h^2)$ with $1 \le n \le N$.

Then, omitting the truncation error, replacing $u_i^n$ with its numerical solution $U_i^n$ and combining initial-boundary value conditions, we yield the following fully discrete BDF2 difference scheme

$$\begin{aligned}&\begin{aligned} \delta _t U^1_i - \delta _x^2 U^1_i&+ \sum \limits _{j=0}^{1} \hat{\kappa }_{1,j} U^j_i -\lambda \psi '(t_1) U^1_i = f^1_i,\quad 1\le i \le M-1, \end{aligned} \end{aligned}$$

(56)

$$\begin{aligned}&\begin{aligned} \delta _t^{(2)} U^n_i - \delta _x^2 U^n_i&+ \sum \limits _{j=0}^{n} \hat{\kappa }_{n,j} U^j_i -\lambda \psi '(t_n) U^n_i = f^n_i,\\ {}&\quad 1\le i \le M-1, \qquad 2 \le n \le N, \end{aligned} \end{aligned}$$

(57)

$$\begin{aligned}&\begin{aligned}&U^n_0 = U^n_M = 0, \qquad 0\le n \le N, \\&U^0_i = u_0(x_i), \qquad 1\le i \le M-1. \end{aligned} \end{aligned}$$

(58)

Then, analogous to the analyses of Theorems 3 and 4, we get the following theoretical results.

Theorem 7

The fully discrete BDF2 finite difference scheme (56)-(58) is unconditionally stable. It holds that

$$\begin{aligned} \begin{aligned} \left\| U^N\right\| _2 \le {C(T,\psi )} \left( \Vert U^0\Vert _2 + k\sum \limits _{n=1}^{N} \Vert f^n\Vert _2 \right) . \end{aligned} \end{aligned}$$

Theorem 8

Let $U^n_i$ and $u_i^n$ be the solutions of the BDF2 scheme (56)-(58) and (3), respectively. Then, we obtain

$$\begin{aligned} \begin{aligned}&\left\| U^N-u^N\right\| _2 \\ {}&\quad \le {C(T,\psi )} \left( h^2 + k\int _{0}^{2k}\Vert u_{tt}\Vert _2 \textrm{d}t + k^2\int _{k}^{T}\Vert u_{ttt}\Vert _2 \textrm{d}t + k^{1+\alpha }\max \limits _{0\le n \le N}\left\| u^n\right\| _2 \right) . \end{aligned} \end{aligned}$$

Remark 1

Noting (17), we arrive at

$$\begin{aligned} \begin{aligned} {\tilde{e}}^n = v(t_n) - V^n = e^{-\lambda \psi _n}\left( u(t_n) - U^n \right) , \end{aligned} \end{aligned}$$

thus we can utilize the theoretical estimates of $U^n$ to yield the stability and convergence of $V^n$ (that is, numerical solutions of the problem (1)).

5 Numerical simulations and discussion

This section considers the one-dimensional case of (1) for illustrating proposed methods and choose the parameters $T=L=1$. Then, noting that $V^n=e^{-\lambda \psi _n}U^n$ for $0\le n \le N$, we denote the errors and the temporal convergence orders

$$\begin{aligned} E_{\textrm{BE}}(k,h) = \max \limits _{1\le j \le M-1} \left| V_j^N(k,h)-V_j^{2N}(k/2,h)\right| , \quad \textrm{rate}_{\textrm{BE}}^k = \log _{2} \left( \frac{E_{\textrm{BE}}(2k,h)}{E_{\textrm{BE}}(k,h)}\right) , \end{aligned}$$

and we define the error and the spatial convergence order

$$\begin{aligned} G_{\textrm{BE}}(k,h) = \max \limits _{1\le j \le M-1} \left| V_j^N(k,h)-V_{2j}^{N}(k,h/2)\right| , \quad \textrm{rate}_\textrm{BE}^h = \log _{2} \left( \frac{G_{\textrm{BE}}(k,2h)}{G_\textrm{BE}(k,h)}\right) {,} \end{aligned}$$

such that the notations $E_{\textrm{BDF2}}$, $G_{ \mathrm BDF2}$, $\textrm{rate}_{\textrm{BDF2}}^h$ and $\textrm{rate}_{\textrm{BDF2}}^k$ can be indicated similarly.

Example 1

The exact solution of (1) is unknown. Then, we give the initial condition $u_0(x)=x^{1/2}(1-x)^{1/2}\in L_2(\Omega )$ and the source term $f(x,t)=t^{\alpha }x^{1/2}(1-x)^{1/2}$.

Table 1 presents the errors and time convergence rates of the BE scheme (52)-(53) by fixing $\lambda =2$, $M=16$ and $\psi (t)=t^3+t^2+1$, from which, we can see that the proposed scheme can reach the first order for time. Furthermore, Table 2 reports the spatial second-order accuracy of the proposed scheme. These are in accordance with the theory (see Theorem 6). Tables 3 and 4 list the errors and temporal-spatial convergence rates of the BDF2 scheme (56)-(58) by fixing some parameters, respectively. Then, we can observe that the proposed method can yield the order $1+\alpha$ for time in Table 3 and the second order for space in Table 4, which verifies the theoretical results (see Theorem 8).

Table 1 The errors and time convergence rates of the BE scheme with $\lambda =2$, $M=16$ and $\psi (t)=t^3+t^2+1$ in Example 1

Table 2 The errors and space convergence rates of the BE scheme with $\lambda =2$, $N=128$ and $\psi (t)=t^3+t^2+1$ in Example 1

Table 3 The errors and time convergence rates of the BDF2 scheme with $\lambda =1$, $M=16$ and $\psi (t)=t^3+t^2+1$ in Example 1

Table 4 The errors and space convergence rates of the BDF2 scheme with $\lambda =1$, $N=128$ and $\psi (t)=t^3+t^2+1$ in Example 1

Example 2

In this example, we consider (1) with $A=\lambda =0$ and the forcing term $f(x,t)=0$, and let the exact solution be unknown. Then, we set the initial data $u_0=1$.

Herein, we only consider (1) with the time variable. In view of Tables 5 and 6, we can see that the BE scheme (12) and BDF2 scheme (39)-(40) obtain the first-order and order $1+\alpha$ in the time direction, respectively. In addition, we discuss three cases for the values of function $\psi (t)$, from which, the first two cases meet the assumptions of $\psi (t)$, and the third does not meet its assumptions. However, in Table 7, temporal order $1+\alpha$ of the BDF2 scheme (39)-(40) can be achieved in all three cases with $\alpha =0.5$, which means that theoretical analyses need to be further improved and relaxed.

Table 5 The errors and time convergence rates of the BE scheme with $\psi (t)=e^t$ in Example 2

Table 6 The errors and time convergence rates of the BDF2 scheme with $\psi (t)=e^{t/2}$ in Example 2

Table 7 The errors and time convergence rates of the BDF2 scheme with $\alpha =0.50$ in Example 2

6 Summary

This paper proposed and analyzed two numerical schemes for solving generalized tempered-type integrodifferential equations with respect to another function $\psi$, from which, the BE method and first-order interpolating quadrature were employed to construct a semi-discrete BE scheme, moreover, the BDF2 method and second-order interpolating quadrature were employed to formulate a semi-discrete BDF2 scheme. Then, the stability and convergence of two semi-discrete schemes were proved by the energy argument. Further, we also established two fully discrete schemes by the finite difference approximations and above two semi-discrete schemes, and gave their theoretical results. Finally, numerical examples validated the effectiveness of the proposed methods. In the future work, we will apply the nonuniform meshes to overcome the singular behavior of the solution (arising from integral term), so as to achieve the accurate temporal second order [7, 24].