Uniqueness and Numerical Method for Determining a Spatial Source Term in a Time-Fractional Diffusion Wave Equation

Abstract

This paper is devoted to recovering a spatial source term in a time-fractional diffusion-wave equation by using an additional final time observation. Based on the regularity of solution for the direct problem, the existence and uniqueness of the spatial source term are obtained. For this ill-posed problem, the generalized quasi-boundary value regularization method is proposed, and the convergence rate of the regularized solution under an a priori choice rule of the regularization parameter is achieved. Specially, for the modified quasi-boundary value regularized problem, we propose a numerical method in terms of combining the backward Euler convolution quadrature scheme for approximating the time-fractional derivative and the piecewise linear finite element scheme for dealing with the space variables. Based on a complicated numerical analysis, we give an error estimate between the numerical source solution in a fully discrete regularized problem and the exact source function under certain reasonable smoothness assumptions to the given data including the initial functions and a time-dependent source term. Some numerical experiments are provided for one- and two-dimensional cases to verify the theoretical result and illustrate the efficient of the proposed numerical algorithm.

Appendix

In this appendix, we present the deduction of fully discrete solution in (6.3) and the proof of Lemma 6.2. The following two Lemmas will be utilized.

Lemma 8.1

[10] Let $\Sigma _\theta =\{z\in {\mathbb {C}}:|z|\ne 0, |\arg {z}|\le \theta \}$ for $\pi /2<\theta <\pi /\alpha$, there exists positive constants $c=c(\theta , \alpha )$ s.t.

$$\begin{aligned} \Vert (z^\alpha -L)^{-1}\Vert \le c|z|^{-\alpha },\quad \Vert (z^\alpha -L_h)^{-1}\Vert \le c|z|^{-\alpha },\quad z\in \Sigma _{\theta }. \end{aligned}$$

Lemma 8.2

[15] For any $\theta \in (\pi /2, \pi )$, there exists $\theta ^{\prime }\in (\pi /2, \pi )$ and positive constants $c, c_1, c_2$ independent of $\tau$ such that for all $z\in \Gamma _{\theta ,\sigma }^{\tau }$

$$\begin{aligned}&c_1|z|\le |\delta _{\tau }(e^{-z\tau })|\le c_2|z|,\quad \delta _{\tau }(e^{-z\tau })\in \Sigma _{\theta ^{\prime }},\\&|\delta _{\tau }(e^{-z\tau })-z|\le c\tau |z|^2,\quad |\delta _{\tau }(e^{-z\tau })^{\alpha }-z^{\alpha }|\le c\tau z^{\alpha +1}. \end{aligned}$$

Deduction of fully discrete solution (6.4) Denote $w_n={\tilde{U}}_n-\phi _h-t_n\psi _h$, then $w_0=0$. Multiplying $\xi ^n$ on both sides of the equation in the fully discrete problem (6.3) and summing over n from 1 to $\infty$, there is

$$\begin{aligned} \sum _{n=0}^{\infty }\overline{\partial }_\tau ^{\alpha }w_n\xi ^n-\sum _{n=0}^{\infty }L_hw_n\xi ^n&=\sum _{n=1}^{\infty }L_h(\phi _h+t_n\psi _h)\xi ^n\nonumber \\&\quad +{\tilde{f}}_{h,\tau }\sum _{n=1}^{\infty }(p(0)+t_np^{\prime }(0)+\overline{\partial }_\tau ^{(-2)} p^{\prime \prime }(t_n))\xi ^n. \end{aligned}$$

(8.1)

In terms of the convolution quadrature (6.1), we have

$$\begin{aligned} \sum _{n=0}^{\infty }\overline{\partial }_\tau ^{\alpha }w_n\xi ^n=\sum _{n=0}^{\infty }\sum _{j=0}^{n}\omega _j^{(\alpha )}w_{n-j}\xi ^n=\sum _{j=0}^{\infty }\omega _j^{(\alpha )}\xi ^{j}\sum _{n=0}^{\infty }w_n\xi ^n=\delta _{\tau }(\xi )^\alpha \sum _{n=0}^{\infty }w_n\xi ^n, \end{aligned}$$

and similarly by the definition of $\overline{\partial }_\tau ^{(-2)}$, we have

$$\begin{aligned} \sum _{n=1}^{\infty }\overline{\partial }_\tau ^{(-2)} p^{\prime \prime }(t_n)\xi ^n=\sum _{n=0}^{\infty }\overline{\partial }_\tau ^{(-2)} p^{\prime \prime }(t_n)\xi ^n-{\tilde{\omega }}_0 p^{\prime \prime }(0)=\delta _{\tau }(\xi )^{-2}\sum _{n=0}^{\infty }p^{\prime \prime }(t_n)\xi ^n-\tilde{\omega }_0 p^{\prime \prime }(0). \end{aligned}$$

Notice that $\sum _{n=1}^{\infty }\xi ^n=\frac{\xi }{1-\xi }$ and $\sum _{n=1}^{\infty }t_n \xi ^n=\frac{\tau \xi }{(1-\xi )^2}$, by (8.1), it implies

$$\begin{aligned} (\delta _{\tau }(\xi )^\alpha -L_h)\sum _{n=0}^{\infty }w_n\xi ^n&=\frac{\xi }{1-\xi }L_h\phi _h+\frac{\tau \xi }{(1-\xi )^2}L_h\psi _h +[p(0)\frac{\xi }{1-\xi }+p^{\prime }(0)\frac{\tau \xi }{(1-\xi )^2} \nonumber \\&\quad +\delta _{\tau }(\xi )^{-2}\sum _{n=0}^{\infty }p^{\prime \prime }(t_n)\xi ^n-\tilde{\omega }_0 p^{\prime \prime }(0)]{\tilde{f}}_{h,\tau }. \end{aligned}$$

(8.2)

In view of the relationship of $w_n$ and ${{\tilde{U}}}_n$, then ${\tilde{U}}_n$ satisfies

$$\begin{aligned}&(\delta _{\tau }(\xi )^\alpha -L_h)\sum _{n=0}^{\infty }{\tilde{U}}_n\xi ^n\nonumber \\&\quad =\delta _{\tau }(\xi )^\alpha \frac{\xi }{1-\xi }\phi _h+\delta _{\tau }(\xi )^\alpha \frac{\tau \xi }{(1-\xi )^2}\psi _h+(\delta _{\tau }(\xi )^\alpha -L_h)\phi _h\nonumber \\&\qquad +\left( \frac{\xi }{1-\xi }p(0)+ \frac{\tau \xi }{(1-\xi )^2}p^{\prime }(0)+\delta _{\tau }(\xi )^{-2}\sum _{n=0}^{\infty }p^{\prime \prime }(t_n)\xi ^n-\tilde{\omega }_0 p^{\prime \prime }(0)\right) {\tilde{f}}_{h,\tau } :={\tilde{W}}(\xi ). \end{aligned}$$

(8.3)

By the Cauchy integral formula, we have

$$\begin{aligned} {\tilde{U}}_n=\frac{1}{2\pi i}\int _{|\xi |=\rho }(\delta _{\tau }(\xi )^\alpha -L_h)^{-1}{\tilde{W}}(\xi )\xi ^{-n-1}d\xi , \end{aligned}$$

(8.4)

where $1>\rho >0$ is a small constant.

According to the Cauchy integral theorem, we know $\frac{1}{2\pi i}\int _{|\xi |=\rho } k(\xi )d\xi =0$ for analytic function $k(\xi )$ over $|\xi |\le \rho$, then we have

$$\begin{aligned}&\frac{1}{2\pi i}\int _{|\xi |=\rho }(\delta _{\tau }(\xi )^\alpha -L_h)^{-1}\delta _{\tau }(\xi )^{-2}\sum _{j=0}^{\infty }p^{\prime \prime }(t_j)\xi ^j\xi ^{-n-1}d\xi \\&\quad = \frac{1}{2\pi i}\sum _{j=0}^{n}\int _{|\xi |=\rho }(\delta _{\tau }(\xi )^\alpha -L_h)^{-1}\delta _{\tau }(\xi )^{-2}p^{\prime \prime }(t_j)\xi ^{j-n-1}d\xi , \end{aligned}$$

since for small enough $\rho$ and $j\ge n+1$, $(\delta _{\tau }(\xi )^\alpha -L_h)^{-1}\delta _{\tau }(\xi )^{-2}\xi ^{j-n-1}$ is analytic over the disk $|\xi |\le \rho$, refer to [32]. It is well-known that the following holds

$$\begin{aligned} \frac{1}{2\pi i}\int _{|\xi |=\rho }\xi ^{-n-1}d\xi \phi _h=0,\quad n\ge 1. \end{aligned}$$

Then we can denote

$$\begin{aligned} W(\xi )={\tilde{W}}(\xi )-(\delta _{\tau }(\xi )^\alpha -L_h)\phi _h. \end{aligned}$$

(8.5)

By the transform $\xi =e^{-z\tau }$ and the Cauchy integral theorem, we have

$$\begin{aligned} {\tilde{U}}_n=\frac{1}{2\pi i}\int _{\Gamma _{\theta ,\delta }^{\tau }}e^{zt_n}(\delta _{\tau }(e^{-z\tau })^\alpha -L_h)^{-1}W(e^{-z\tau })\tau dz. \end{aligned}$$

(8.6)

Therefore, the solution of the fully discrete problem can be obtained easily as (6.4).

Proof of Lemma 6.2

Proof

In terms of (6.5), one knows

$$\begin{aligned} F_j^{h,\tau }(T)&=\frac{1}{2\pi i}\int _{\Gamma ^{\tau }_{\theta ,\sigma }} e^{zT}\delta _{\tau }(e^{-z\tau })^{-1}(\delta _{\tau }(e^{-z\tau })^\alpha +\lambda _n)^{-1}e^{-z\tau }dzp(0)\nonumber \\&\quad +\frac{1}{2\pi i}\int _{\Gamma ^{\tau }_{\theta ,\sigma }} e^{zT}\delta _{\tau }(e^{-z\tau })^{-2}(\delta _{\tau }(e^{-z\tau })^\alpha +\lambda _n)^{-1}e^{-z\tau }dzp^{\prime }(0)\nonumber \\&\quad +\frac{\tau }{2\pi i}\sum _{k=1}^{N}\int _{\Gamma ^{\tau }_{\theta ,\sigma }} e^{zt_{N-k}}\delta _{\tau }(e^{-z\tau })^{-2}(\delta _{\tau }(e^{-z\tau })^\alpha +\lambda _n)^{-1}dzp^{\prime \prime }(t_k)\nonumber \\&\quad +\frac{\tau }{2\pi i}\int _{\Gamma ^{\tau }_{\theta ,\sigma }} e^{zT}\delta _{\tau }(e^{-z\tau })^{-2}(\delta _{\tau }(e^{-z\tau })^\alpha +\lambda _n)^{-1}dzp^{\prime \prime }(0)\nonumber \\&\quad -\frac{\tau \tilde{\omega }_0}{2\pi i}\int _{\Gamma ^{\tau }_{\theta ,\sigma }} e^{zT}(\delta _{\tau }(e^{-z\tau })^\alpha +\lambda _n)^{-1}dzp^{\prime \prime }(0)\nonumber \\&=:I_1^{h,\tau }+I_2^{h,\tau }+I_3^{h,\tau }+I_4^{h,\tau }+I_5^{h,\tau }. \end{aligned}$$

(8.7)

Due to (6.9), it follows

$$\begin{aligned} F_j^h(T)&=\frac{1}{2\pi i}\int _{\Gamma _{\theta ,\sigma }} e^{zT}z^{-1}(z^\alpha +\lambda _j^h)^{-1}dzp(0)+\frac{1}{2\pi i}\int _{\Gamma _{\theta ,\sigma }} e^{zT}z^{-2}(z^\alpha +\lambda _j^h)^{-1}dzp^{\prime }(0)\nonumber \\&\quad +\frac{1}{2\pi i}\int _0^{T} \int _{\Gamma _{\theta ,\sigma }} e^{z t_{N-s}}z^{-2}(z^\alpha +\lambda _j^h)^{-1}p^{\prime \prime }(s)ds\nonumber \\&=:I_1^h+I_2^h+I_3^h. \end{aligned}$$

(8.8)

In the following, we try to prove

$$\begin{aligned} \lambda _j^h|F_j^{h,\tau }(t_n)-F_j^h(t_n)|&\le \lambda _j^h|I_1^{h,\tau }-I_1^h|+\lambda _j^h|I_2^{h,\tau }-I_2^h|+\lambda _j^h|I_3^{h,\tau }-I_3^h|\\&\quad +\lambda _j^h|I_4^{h,\tau }|+\lambda _j^h|I_5^{h,\tau }|\le C\tau . \end{aligned}$$

Firstly, it is easy to deduce

$$\begin{aligned}&\lambda _j^h(I_1^h-I_1^{h,\tau })\\&\quad =\frac{1}{2\pi i}\int _{\Gamma _{\theta ,\sigma }\backslash \Gamma ^{\tau }_{\theta ,\sigma }} e^{zT}\lambda _j^hz^{-1}(z^\alpha +\lambda _j^h)^{-1}dzp(0)\\&\qquad +\frac{1}{2\pi i}\int _{\Gamma ^{\tau }_{\theta ,\sigma }} e^{zT}\lambda _j^h[z^{-1}(z^\alpha +\lambda _j^h)^{-1}- \delta _{\tau }(e^{-z\tau })^{-1}(\delta _{\tau }(e^{-z\tau })^\alpha +\lambda _j^h)^{-1}]dzp(0)\\&\qquad +\frac{1}{2\pi i}\int _{\Gamma ^{\tau }_{\theta ,\sigma }} e^{zT}\lambda _j^h\delta _{\tau }(e^{-z\tau })^{-1}(\delta _{\tau }(e^{-z\tau })^\alpha +\lambda _j^h)^{-1}(1-e^{-z\tau })dzp(0)\\&\quad =:e_{11}+e_{12}+e_{13}. \end{aligned}$$

According to Lemma 8.1, it can be proved that

$$\begin{aligned} | \lambda _j^h(z^\alpha +\lambda _j^h)^{-1}|=|(\lambda _j^h+z^\alpha -z^\alpha )(z^\alpha +\lambda _j^h)^{-1}|=|1-z^\alpha (z^\alpha +\lambda _j^h)^{-1}|\le 2. \end{aligned}$$

Hence, by simple calculations, we can obtain

$$\begin{aligned} |e_{11}|&\le \frac{|p(0)|}{\pi }\int _{\Gamma _{\theta ,\sigma }\backslash \Gamma ^{\tau }_{\theta ,\sigma }}|e^{zT}z^{-1}||dz| \le C\int _{\frac{\pi }{\tau \sin {\theta }}}^{\infty }e^{\rho \cos {\theta }T}\rho ^{-1} d\rho \le C\tau \int _{0}^{\infty }e^{\rho \cos {\theta }T}d\rho \\&\le C\tau , \end{aligned}$$

where $C>0$ depends on $\Vert p\Vert _{C[0,T]}$.

By Lemma 8.2, it is easy to prove $|z^{-1}-\delta _{\tau }(e^{-z\tau })^{-1}|\le C\tau$ and

$$\begin{aligned} |z^{\alpha -1}(z^\alpha +\lambda _j^h)^{-1}-\delta _{\tau }(e^{-z\tau })^{\alpha -1}(\delta _{\tau }(e^{-z\tau })^\alpha +\lambda _j^h)^{-1}|\le C\tau . \end{aligned}$$

Then by taking $\sigma =T^{-1}$, noting that $\cos \theta <0$, one can derive at

$$\begin{aligned} |e_{12}|&\le \frac{|p(0)|}{2\pi }\int _{\Gamma _{\theta ,\sigma }^{\tau }}|e^{zT}(z^{-1}-z^{\alpha -1}(z^\alpha +\lambda _j^h)^{-1} -\delta _{\tau }(e^{-z\tau })^{-1}\\&\quad +\delta _{\tau }(e^{-z\tau })^{\alpha -1}(\delta _{\tau }(e^{-z\tau })^\alpha +\lambda _j^h)^{-1}||dz|\\&\le C\tau \int _{\Gamma ^{\tau }_{\theta ,\sigma }} |e^{zT}||dz| \le C\tau \left( \int _{\sigma }^{\frac{\pi }{ \tau \sin {\theta }}}e^{\rho {\textit{cos}}{\theta }T}d\rho + \int _{-\theta }^{\theta }e^{\sigma {\textit{cos}}{\varphi }T}\sigma d\varphi \right) \\&\le C\tau \left( T^{-1}\int _{1}^{\infty }e^{\cos {\theta }y}dy+ T^{-1}\int _{-\theta }^{\theta }e^{ \cos {\varphi }}d\varphi \right) \le C\tau . \end{aligned}$$

Further, we have

$$\begin{aligned} |e_{13}|&\le \frac{|p(0)|}{2\pi }\int _{\Gamma ^{\tau }_{\theta ,\sigma }} |e^{zT}\lambda _j^h(\delta _{\tau }(e^{-z\tau })^\alpha +\lambda _j^h)^{-1}\tau ||dz| \le C\tau \int _{\Gamma ^{\tau }_{\theta ,\sigma }} |e^{zT}||dz|\\&\le C\tau . \end{aligned}$$

Therefore, we have $\lambda _j^h|I_1^h-I_1^{h,\tau }|\le C\tau$ where $C>0$ depending on $\Vert p\Vert _{C[0,T]}$.

Secondly, we estimate $\lambda _j^h(I_2^h-I_2^{h,\tau })$, it can be divided into

$$\begin{aligned}&\lambda _j^h(I_2^h-I_2^{h,\tau }) \\&\quad =\frac{1}{2\pi i}\int _{\Gamma _{\theta ,\sigma }\backslash \Gamma ^{\tau }_{\theta ,\sigma }} e^{zT}\lambda _j^hz^{-2}(z^\alpha +\lambda _j^h)^{-1}dz p^{\prime }(0)\\&\qquad +\frac{1}{2\pi i}\int _{\Gamma ^{\tau }_{\theta ,\sigma }} e^{zT}\lambda _j^h[z^{-2}(z^\alpha +\lambda _j^h)^{-1}dz- \delta _{\tau }(e^{-z\tau })^{-2}(\delta _{\tau }(e^{-z\tau })^\alpha +\lambda _j^h)^{-1}]dz p^{\prime }(0)\\&\qquad +\frac{1}{2\pi i}\int _{\Gamma ^{\tau }_{\theta ,\sigma }} e^{zT}\lambda _j^h\delta _{\tau }(e^{-z\tau })^{-2}(\delta _{\tau }(e^{-z\tau })^\alpha +\lambda _j^h)^{-1}(1-e^{-z\tau })dzp^{\prime }(0)\\&\quad =:e_{21}+e_{22}+e_{23}. \end{aligned}$$

Each term is estimated respectively as follows

$$\begin{aligned} |e_{21}|&\le \frac{|p^{\prime }(0)|}{2\pi }\int _{\Gamma _{\theta ,\sigma }\backslash \Gamma ^{\tau }_{\theta ,\sigma }} |e^{zT}z^{-2}||dz| \le C\int _{\frac{\pi }{\tau \sin {\theta }}}^{\infty }e^{\rho \cos {\theta }T}\rho ^{-2} d\rho \le C\tau ^2\int _0^{\infty }e^{\rho \cos {\theta }T}d\rho \\&\le C\tau ^2. \end{aligned}$$

And by Lemma 8.2, it is easy to prove $|z^{-2}-\delta _{\tau }(e^{-z\tau })^{-2}|\le C\tau |z|^{-1}$ and

$$\begin{aligned} |z^{\alpha -2}(z^\alpha +\lambda _j^h)^{-1}-\delta _{\tau }(e^{-z\tau })^{\alpha -2}(\delta _{\tau }(e^{-z\tau })^\alpha +\lambda _j^h)^{-1}|\le C\tau |z|^{-1}, \end{aligned}$$

from which we have

$$\begin{aligned} |e_{22}|&=\frac{|p^{\prime }(0)|}{2\pi }\int _{\Gamma ^{\tau }_{\theta ,\sigma }} |e^{zT}(z^{-2}-z^{\alpha -2}(z^\alpha +\lambda _j^h)^{-1} -\delta _{\tau }(e^{-z\tau })^{-2}\\&\quad +\delta _{\tau }(e^{-z\tau })^{\alpha -2} (\delta _{\tau }(e^{-z\tau })^\alpha +\lambda _j^h)^{-1})||dz|\\&\le C\tau \int _{\Gamma ^{\tau }_{\theta ,\sigma }} |e^{zT}z^{-1}||dz| \le C\tau \left( \int _{\sigma }^{\frac{\pi }{\tau \sin {\theta }}}e^{\rho {\textit{cos}}{\theta }T}\rho ^{-1}d\rho + \int _{-\theta }^{\theta }e^{\sigma {\textit{cos}}{\varphi }T} d\varphi \right) \\ {}&\le C\tau \left( \int _{1}^{\infty }e^{{\textit{cos}}{\theta }y}y^{-1}dy+\int _{-\theta }^{\theta }e^{ {\textit{cos}}{\varphi }}d\varphi \right) \le C\tau , \end{aligned}$$

and note that $1-e^{-\tau z}=\delta _{\tau }(e^{-z\tau })\tau$, we have

$$\begin{aligned} |e_{23}|&\le |\frac{|p^{\prime }(0)|}{2\pi }\int _{\Gamma ^{\tau }_{\theta ,\sigma }} |e^{zT}\lambda _j^h\delta _{\tau }(e^{-z\tau })^{-1}(\delta _{\tau }(e^{-z\tau })^\alpha +\lambda _j^h)^{-1}\tau ||dz| \le C\tau \int _{\Gamma ^{\tau }_{\theta ,\sigma }} |e^{zT}z^{-1}||dz|\\&\le C\tau . \end{aligned}$$

Therefore, we have $\lambda _j^h|I_2^h-I_2^{h,\tau }|\le C\tau$ where $C>0$ depending on $\Vert p'\Vert _{C[0,T]}$.

Thirdly, we estimate $\lambda _j^h(I_3^h-I_3^{h,\tau })$ as follow.

$$\begin{aligned}&\lambda _j^h(I_3^h-I_3^{h,\tau })\\&\quad =\frac{1}{2\pi i}\int _0^T\int _{\Gamma _{\theta ,\sigma }} e^{z(T-s)}\lambda _j^hz^{-2}(z^\alpha +\lambda _j^h)^{-1}dz p^{\prime \prime }(s)ds\\&\qquad -\frac{1}{2\pi i}\sum _{k=1}^{N}\int _{\Gamma _{\theta ,\sigma }^{\tau }} \tau e^{z t_{N-k}}\lambda _j^h\delta _{\tau }(e^{-z\tau })^{-2}(\delta _{\tau }(e^{-z\tau })^\alpha +\lambda _j^h)^{-1}dz p^{\prime \prime }(t_k)\\&\quad =\frac{1}{2\pi i}\int _0^T\int _{\Gamma _{\theta ,\sigma }\backslash \Gamma _{\theta ,\sigma }^{\tau }} e^{z(T-s)}\lambda _j^hz^{-2}(z^\alpha +\lambda _j^h)^{-1}dz p^{\prime \prime }(s)ds\\&\qquad +\frac{1}{2\pi i}\int _0^T\int _{\Gamma _{\theta ,\sigma }^{\tau }} e^{z(T-s)}\lambda _j^h[z^{-2}(z^\alpha +\lambda _j^h)^{-1}{-}\delta _{\tau }(e^{-z\tau })^{-2}(\delta _{\tau }(e^{-z\tau })^\alpha {+}\lambda _j^h)^{-1}]dz p^{\prime \prime }(s)ds\\&\qquad +\frac{1}{2\pi i}\int _0^T\int _{\Gamma _{\theta ,\sigma }^{\tau }} e^{z(T-s)}\lambda _j^h\delta _{\tau }(e^{-z\tau })^{-2}(\delta _{\tau }(e^{-z\tau })^\alpha +\lambda _j^h)^{-1}dz p^{\prime \prime }(s)ds \\&\qquad -\frac{1}{2\pi i}\sum _{k=1}^{N}\int _{\Gamma _{\theta ,\sigma }^{\tau }} \tau e^{zt_{N-k}}\lambda _j^h\delta _{\tau }(e^{-z\tau })^{-2}(\delta _{\tau }(e^{-z\tau })^\alpha +\lambda _j^h)^{-1}dz p^{\prime \prime }(t_k)\\&\quad =:e_{31}+e_{32}+e_{33}, \end{aligned}$$

where $e_{33}$ contains the last two terms. Then we analyse each term as follows.

$$\begin{aligned} |e_{31}|&\le \frac{1}{2\pi }\int _0^T\int _{\frac{\pi }{\tau \sin {\theta }}}^{\infty }e^{\rho \cos {\theta }(T-s)}\rho ^{-2}d\rho |p^{\prime \prime }(s)|ds \le C\int _0^T\int _{\frac{\pi }{\tau \sin {\theta }}}^{\infty }\rho ^{-2}d\rho |p^{\prime \prime }(s)|ds\\&\le C\tau \Vert p^{\prime \prime }(t)\Vert _{L^1(0,T)}, \end{aligned}$$

and for estimating $e_{32}$, we take $\sigma =(T-s)^{-1}$, then we have

$$\begin{aligned} |e_{32}|&\le \frac{1}{2\pi }\int _0^T\int _{\Gamma _{\theta ,\sigma }^{\tau }} |e^{z(T-s)}[z^{-2}-z^{\alpha -2}(z^\alpha +\lambda _j^h)^{-1}\\&\quad -\delta _{\tau }(e^{-z\tau })^{-2} +\delta _{\tau }(e^{-z\tau })^{\alpha -2}(\delta _{\tau }(e^{-z\tau })^\alpha +\lambda _j^h)^{-1}]||dz||p^{\prime \prime }(s)|ds\\&\le C\tau \int _0^T\int _{\Gamma _{\theta ,\sigma }^{\tau }} |e^{z(T-s)}z^{-1}||dz||p^{\prime \prime }(s)|ds \le C\tau \int _0^T\bigg (\int _\sigma ^{\frac{\pi }{\tau \sin {\theta }}}e^{\rho \cos {\theta }(T-s)}\rho ^{-1}d\rho \\&\quad +\int _{-\theta }^\theta e^{\sigma \cos {\varphi }(T-s)}d\varphi \bigg )|p^{\prime \prime }(s)|ds\\&\le C\tau \int _0^T\left( \int _1^{\infty }e^{\cos {\theta }y}y^{-1}dy+\int _{-\theta }^\theta e^{\cos {\varphi }}d\varphi \right) |p^{\prime \prime }(s)|ds \le C\tau \Vert p^{\prime \prime }(t)\Vert _{L^1(0,T)}. \end{aligned}$$

As for $e_{33}$, we have

$$\begin{aligned} e_{33}&=\frac{1}{2\pi i}\sum _{k=1}^{N}\int _{t_{k-1}}^{t_k}\int _{\Gamma _{\theta ,\sigma }^{\tau }} e^{z(T-s)}\lambda _j^h\delta _{\tau }(e^{-z\tau })^{-2}(\delta _{\tau }(e^{-z\tau })^\alpha +\lambda _j^h)^{-1}dz p^{\prime \prime }(s)ds\\&\quad -\frac{1}{2\pi i}\sum _{k=1}^{N}\int _{\Gamma _{\theta ,\sigma }^{\tau }} \tau e^{zt_{N-k}}\lambda _j^h\delta _{\tau }(e^{-z\tau })^{-2}(\delta _{\tau }(e^{-z\tau })^\alpha +\lambda _j^h)^{-1}dz p^{\prime \prime }(t_k), \end{aligned}$$

noting that $e^{-zs}p^{\prime \prime }(s)=e^{-zt_k}p^{\prime \prime }(t_k)-\int _{s}^{t_k}e^{-z\xi }(-zp^{\prime \prime }(\xi )+p^{\prime \prime \prime }(\xi ))d\xi$ for $s\in [t_{k-1},t_k]$, and substituting it into $e_{33}$, we have

$$\begin{aligned} |e_{33}|&\le \frac{1}{2\pi }\sum _{k=1}^{N}|\int _{t_{k-1}}^{t_k}\int _{\Gamma _{\theta ,\sigma }^{\tau }} \int _{s}^{t_k} e^{z(T-\xi )} \lambda _j^h\delta _{\tau }(e^{-z\tau })^{-2}(\delta _{\tau }(e^{-z\tau })^\alpha +\lambda _j^h)^{-1}\\&\quad \times (-zp^{\prime \prime }(\xi )+p^{\prime \prime \prime }(\xi )) d\xi dzds|\\&\le C\sum _{k=1}^{N}\int _{t_{k-1}}^{t_k}\int _{s}^{t_k} \int _{\Gamma _{\theta ,\sigma }^{\tau }} |e^{z(T-\xi )}|(|z|^{-1}|p^{\prime \prime }(\xi )|+|z|^{-2}|p^{\prime \prime \prime }(\xi )|)|dz| d\xi ds\\&\le C\sum _{k=1}^{N}\int _{t_{k-1}}^{t_k}\int _{s}^{t_k} \bigg [\left( \int _{\sigma }^{\frac{\pi }{\tau \sin {\theta }}} e^{\rho \cos {\theta }(T-\xi )}\rho ^{-1}d\rho +\int _{-\theta }^\theta e^{\sigma \cos {\varphi }(T-\xi )}d\varphi \right) |p^{\prime \prime }(\xi )|\\&\quad +\left( \int _{\sigma }^{\frac{\pi }{\tau \sin {\theta }}} e^{\rho \cos {\theta }(T-\xi )}\rho ^{-2}d\rho +\int _{-\theta }^\theta e^{\sigma \cos {\varphi }(T-\xi )}\sigma ^{-1}d\varphi \right) |p^{\prime \prime \prime }(\xi )|\bigg ]d\xi ds, \end{aligned}$$

then by taking $\sigma =(T-\xi )^{-1}$, $y=\rho (T-\xi )$, it follows

$$\begin{aligned} |e_{33}|&\le C\sum _{k=1}^{N}\int _{t_{k-1}}^{t_k}\int _{s}^{t_k} \bigg [\left( \int _1^{\infty } e^{\cos {\theta }y}y^{-1}dy+\int _{-\theta }^\theta e^{\cos {\varphi }}d\varphi \right) |p^{\prime \prime }(\xi )|\\&\quad +(T-\xi )\left( \int _1^{\infty } e^{\cos {\theta }y}y^{-2}dy+\int _{-\theta }^\theta e^{\cos {\varphi }}d\varphi \right) |p^{\prime \prime \prime }(\xi )|)\bigg ]d\xi ds\\&\le C\sum _{k=1}^{N}\int _{t_{k-1}}^{t_k}\int _{s}^{t_k}(|p^{\prime \prime }(\xi )|+(T-\xi )|p^{\prime \prime \prime }(\xi )|)d\xi ds\\&=C\sum _{k=1}^{N}\int _{t_{k-1}}^{t_k}\int _{t_{k-1}}^{\xi }(|p^{\prime \prime }(\xi )|+(T-\xi )|p^{\prime \prime \prime }(\xi )|)ds d\xi \\&\le C\sum _{k=1}^{N}\int _{t_{k-1}}^{t_k}(\xi -t_{k-1})(|p^{\prime \prime }(\xi )|+(T-\xi )|p^{\prime \prime \prime }(\xi )|)d\xi \\&\le C\tau (\Vert p^{\prime \prime }(t)\Vert _{L^1(0,T)}+\Vert p^{\prime \prime \prime }(t)\Vert _{L^1(0,T)}). \end{aligned}$$

Therefore, we have $\lambda _j^h|I_3^h-I_3^{h,\tau }|\le C\tau$ where $C>0$ depending on $\Vert p^{\prime \prime }\Vert _{L^1(0,T)}+\Vert p^{\prime \prime \prime }\Vert _{L^1(0,T)}$.

Finally, we deduce the estimates of the last two terms.

$$\begin{aligned} |\lambda _j^hI_4^{h,\tau }|&\le \frac{|p^{\prime \prime }(0)|\tau }{2\pi }\int _{\Gamma ^{\tau }_{\theta ,\sigma }} |e^{zT}\lambda _j^h\delta _{\tau }(e^{-z\tau })^{-2}(\delta _{\tau }(e^{-z\tau })^\alpha +\lambda _j)^{-1}||dz|\\&\le \frac{|p^{\prime \prime }(0)|\tau }{2\pi }\int _{\Gamma ^{\tau }_{\theta ,\sigma }} |e^{zT}z^{-2}||dz|\\ {}&\le C\tau \left( \int _\sigma ^\infty e^{\rho \cos {\theta }T}\rho ^{-2}d\rho +\int _{-\theta }^\theta e^{\sigma \cos {\varphi }T}\sigma ^{-1}d\varphi \right) \\ {}&\le C\tau \left( T\int _1^\infty e^{\cos {\theta }y}y^{-2}dy +T\int _{-\theta }^\theta e^{\cos {\varphi }}d\varphi \right) \le C\tau , \end{aligned}$$

where $C>0$ depends on $\Vert p^{\prime \prime }\Vert _{C[0,T]}$.

It is known the convolution weight ${\tilde{\omega }}_0=\tau ^2$, then we have

$$\begin{aligned} |\lambda _j^hI_5^{h,\tau }|&\le \frac{\tau \tilde{\omega }_0|p^{\prime \prime }(0)|}{2\pi }\int _{\Gamma ^{\tau }_{\theta ,\sigma }} |e^{zT}\lambda _j^h(\delta _{\tau }(e^{-z\tau })^\alpha +\lambda _j)^{-1}||dz| \le C\tau ^3\int _{\Gamma ^{\tau }_{\theta ,\sigma }}\! |e^{zT}||dz| \le C\tau ^3. \end{aligned}$$

Combining the estimates above for each term, one can obtain the following estimate

$$\begin{aligned} |F_j^h(T)-F_j^{h,\tau }(T)|\le \frac{C\tau }{\lambda _j^h}. \end{aligned}$$

By $F_j^h(T)\ge \frac{C}{\lambda _j^h}$, for a small enough $\tau$, there exists a positive constant C such that

$$\begin{aligned} F_j^{h,\tau }(T)=F_j^h(T)-F_j^h(T)+F_j^{h,\tau }(T)\ge F_j^h(T)-|F_j^h(T)-F_j^{h,\tau }(T)|\ge \frac{C}{\lambda _j^h}>0. \end{aligned}$$

$\square$