Fractional Tikhonov Regularization Method for an Inverse Boundary Value Problem of the Fractional Elliptic Equation

Zhang Xiao,, Zhang Hongwu,*

School of Mathematics and Information Science, North Minzu University, Yinchuan 750021

 基金资助: 宁夏自然科学基金(2022AAC03234)国家自然科学基金(11761004)宁夏高等教育一流学科建设基金(NXYLXK2017B09)

 Fund supported: NSF of Ningxia(2022AAC03234)NSF of China(11761004)Construction Project of First-Class Disciplines in Ningxia Higher Education(NXYLXK2017B09)

Abstract

In this paper, we study an inverse boundary value problem for fractional elliptic equation of Tricomi-Gellerstedt-Keldysh-type. For this ill-posed problem, a conditional stability result is established. Based on the ill-posedness analysis, a fractional Tikhonov regularization method was constructed to recover the continuous dependence of the solution on the measurement data. Under the a-priori and a-posteriori selection rules for regularization parameter, the corresponding convergence results of Hölder type are derived and proved, respectively. Finally, the simulation effectiveness of the fractional Tikhonov method is verified by two numerical examples. The numerical results show that the method works stably and effectively in dealing with the inverse problem in the text.

Keywords： Inverse boundary value problem; Fractional elliptic equation; Fractional Tikhonov regularization; A-priori and a-posteriori convergence estimates; Numerical simulation

Zhang Xiao, Zhang Hongwu. Fractional Tikhonov Regularization Method for an Inverse Boundary Value Problem of the Fractional Elliptic Equation[J]. Acta Mathematica Scientia, 2024, 44(4): 978-993

1 引言

$$$\left\{\begin{array}{ll} D_{x}^{2 a}u(x, y)-x^{2\beta}L u(x, y)=0, &(x, y) \in[0, \infty) \times \Omega, \\ u(x, y)=0, &(x, y) \in[0, \infty)\times \partial \Omega, \\ u(0, y)=f(y), &y \in \Omega, \\ \lim\limits_{x \rightarrow \infty} u(x, y)=0, &y \in \Omega. \end{array}\right.$$$

(i) 当 $\alpha=1$, $\beta=0$, $L=-\Delta_{y}=-\sum\limits_{j=1}^{n} \frac{\partial^{2}}{\partial y_{j}^{2}}$ 时, 问题 (1.1) 的主方程为经典的拉普拉斯方程

$u_{x x}(x, y)+\Delta_{y} u(x, y)=0,\ x>0,\ y \in \Omega \subset \mathbb{R}^{N},$

(ii) 当 ${N}=1$, $\alpha=1$, $\beta=\frac{1}{2}$, $L=-\frac{\partial^{2}}{\partial y^{2}}$ 时, 问题 (1.1) 的主方程为 Tricomi 方程[17]

$u_{x x}(x, y)+ x u_{y y}(x, y)=0,\ x>0,\ y \in \Omega \subset \mathbb{R},$

(iii) 当 ${N}=1$, $\alpha=1$, $\beta=\frac{m}{2}>0$, $L=-\frac{\partial^{2}}{\partial y^{2}}$ 时, 问题 (1.1) 的主方程为 Gellerstedt 方程[18]

$u_{x x}(x, y)+x^{m} u_{y y}(x, y)=0,\ x>0,\ y \in \Omega \subset \mathbb{R},$

Gellerstedt 方程是 Tricomi 方程的推广, 常被用于描述流体力学中的流体运动现象.

(iv) 当 ${N}=1$, $\alpha=1$, $\beta=-\frac{k}{2}\in(-2, 0)$, $L=-\frac{\partial^{2}}{\partial y^{2}}$ 时, 问题 $(1.1)$ 的主方程为 Keldysh 方程[19]

$u_{x x}(x, y)+x^{-k} u_{y y}(x, y)=0,\ x>0,\ y \in \Omega \subset \mathbb{R},$

$$$\left\{\begin{array}{ll} D_{x}^{2 \alpha} u(x, y)-x^{2 \beta} L u(x, y)=0, &(x, y) \in[0, \infty) \times \Omega, \\ u(x, y)=0, &(x, y) \in[0, \infty) \times \partial \Omega, \\ u(T, y)=g(y), & y \in \Omega, \\ \lim\limits_{x \rightarrow \infty} u(x, y)=0, & y \in \Omega. \end{array}\right.$$$

$$$\frac{1}{1+\Gamma(1-\alpha) z} \leq E_{\alpha, m, m-1}(-z) \leq \frac{1}{1+\frac{\Gamma(1+(m-1) \alpha)}{\Gamma(1+m \alpha)} z}.$$$

$$$\begin{split} A(s)\leq A\left(s_{0}\right) \leq \frac{\eta_{2}^{\gamma} \gamma^{\frac{\gamma}{\gamma+1} } \mu^{-\frac{1}{\gamma+1}}}{(\gamma+1) \eta_{1}^{\gamma}}=c_{1}\left(r, \eta_{1}, \eta_{2}\right) \mu^{-\frac{1}{\gamma+1}}. \end{split} \nonumber$$$

$$$B(s)=\frac{\mu s^{\frac{\gamma+1-p}{2}}}{\eta_{1}^{ \gamma+1} +\mu s^{\frac{\gamma+1}{2}}} \leq\left\{\begin{array}{ll} c_{2} \mu^{\frac{p}{\gamma+1}}, & 0<p<\gamma+1, \\ c_{3} \mu, & p \geq \gamma+1, \end{array}\right.$$$

(1) $0<p<\gamma$. 因为 $\lim\limits_{s \rightarrow 0} C(s)=\lim\limits_{s \rightarrow \infty} C(s)=0$, 所以 $C(s)$ 有最大值. 设 $C^{\prime}\left(s_{0}\right)=0$, 有 $s_{0}=\left[\frac{\gamma-p}{(1+p) \mu}\right]^{\frac{2}{\gamma+1}} \eta_{1}^{2}>0$, 使得

$$$\begin{split} C(s) \leq C\left(s_{0}\right)=\frac{(1+p)^{\frac{1+p}{\gamma+1}}(r-p)^{\frac{\gamma-p}{\gamma+1}} \eta_{2}}{(1+\gamma) \eta_{1}^{p+1}} \mu^{\frac{p+1}{\gamma+1}}=c_{4}\left(\gamma, p, \eta_{1}, \eta_{2}\right) \mu^{\frac{p+1}{\gamma+1}}. \end{split}$$$

(2) $p \geq \gamma$. 注意到

$$$\begin{split} C(s) \leq \frac{\mu \eta_{2} s^{\frac{\gamma-p}{2}}}{\eta_{1}^{\gamma+1}}=\frac{\mu \eta_{2}}{\eta_{1}^{\gamma+1}s^{\frac{p-\gamma}{2}} }\leq \frac{\eta_{2}}{\eta_{1}^{\gamma+1} \lambda_{1}^{\frac{p-\gamma}{2}}} \mu=c_{5}\left(\gamma, p, \lambda_{1}, \eta_{1}, \eta_{2}\right) \mu. \end{split}$$$

3 反问题的不适定性和条件稳定性

$$$D\left(L^{q}\right)=\left\{\psi \in L^{2}(\Omega); \sum\limits_{n=1}^{\infty} \lambda_{n}^{2q}\lvert\left(\psi, \varphi_{n}\right)\rvert^{2}<\infty\right\},$$$

$$$\|\psi\|_{D\left(L^{q}\right)}=\left(\sum\limits_{n=1}^{\infty} \lambda_{n}^{2 q}\lvert\left(\psi, \varphi_{n}\right)\rvert^{2}\right)^{\frac{1}{2}}.$$$

$$$u(x, y)=\sum\limits_{n=1}^{\infty} f_{n} E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(-\sqrt{\lambda_{n}} x^{\alpha+\beta}\right) \varphi_{n}(y).$$$

$x=T$, 有

$$$g(y)=u(T, y)=\sum\limits_{n=1}^{\infty} f_{n} E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(-\sqrt{\lambda_{n}} T^{\alpha+\beta}\right) \varphi_{n}(y),$$$

$$$g_{n}=f_{n} E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(-\sqrt{\lambda_{n}} T^{\alpha+\beta}\right),$$$

$$$f(y)=\sum\limits_{n=1}^{\infty} \frac{1}{E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(-\sqrt{\lambda_{n}} T^{\alpha+\beta}\right)} g_{n} \varphi_{n}(y).$$$

$$$(K f)(y)=\int k(\xi, y) f(\xi) {\rm d}\xi=g(y),$$$

$$$k(\xi, y)=\sum\limits_{n=1}^{\infty} E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(-\sqrt{\lambda_{n}} T^{\alpha+\beta}\right) \varphi_{n}(y) \varphi_{n}(\xi).$$$

\begin{aligned} \left\|K_{m} f-K f\right\|_{L^{2}(\Omega)}^{2} &=\sum\limits_{n=m+1}^{\infty}\lvert E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}} \left(-\sqrt{\lambda_{n}} T^{\alpha+\beta}\right)\rvert^{2} f_{n}^{2}\\ &\leq \sum\limits_{n=m+1}^{\infty} \frac{\eta_{2}^{2}}{\lambda_{n}} f_{n}{ }^{2} \leq \frac{\eta_{2}{ }^{2}}{\lambda_{m+1}}\|f\|_{L^{2}(\Omega)}. \end{aligned} \nonumber

$$$\|f\|_{D\left(L^{\frac{p}{2}}\right)}=\left(\sum\limits_{n=1}^{\infty} \lambda_{n}^{p}\lvert\left(f, \varphi_{n}\right)\rvert^{2}\right)^{\frac{1}{2}} \leq E, p>0, E>0,$$$

$\|f\| \leq C E^{\frac{1}{p+1}}\|g\|^{\frac{p}{p+1}}, p>0,$

\begin{align*} \|f\|^{2} & =\sum\limits_{n=1}^{\infty} \frac{1}{\left(E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(-\sqrt{\lambda_{n}} T^{\alpha+\beta}\right)\right)^{2}} g_{n}^{2} \\ &=\sum\limits_{n=1}^{\infty} \frac{1}{\left(E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(-\sqrt{\lambda_{n}} T^{\alpha+\beta}\right)\right)^{2}} g_{n}^{\frac{2}{p+1}} g_{n}^{\frac{2 p}{p+1}}\\ &\leq\left(\sum\limits_{n=1}^{\infty} \frac{1}{\left(E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(-\sqrt{\lambda_{n}} T^{\alpha+\beta}\right)\right)^{2 p+2}} g_{n}{ }^{2}\right)^{\frac{1}{p+1}}\left(\sum\limits_{n=1}^{\infty} g_{n}{ }^{2}\right)^{\frac{p}{p+1}} \\ &\leq\left(\sum\limits_{n=1}^{\infty} \frac{1}{\left(E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(-\sqrt{\lambda_{n}} T^{\alpha+\beta}\right)\right)^{2 p+2}}\left(E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(-\sqrt{\lambda_{n}}T^{\alpha+\beta}\right)\right)^{2} f_{n}^{2}\right)^{\frac{1}{p+1}}\|g\|^{\frac{2 p}{p+1}} \\ &=\left(\sum\limits_{n=1}^{\infty} \frac{1}{\left(E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(-\sqrt{\lambda_{n}} T^{\alpha+\beta}\right)\right)^{2 p} \lambda_{n}^{p}} \lambda_{n}^{p} f_{n}^{2} \right)^{\frac{1}{p+1}}\|g\|^{\frac{2 p}{p+1}} \\ &\leq \sup_{n} \left(\frac{1}{\left(E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(-\sqrt{\lambda_{n}} T^{\alpha+\beta}\right)\right)^{2 p} \lambda_{n}{ }^{p}}\right)^{\frac{1}{p+1}}E^{\frac{2}{p+1}}\|g\|^{\frac{2 p}{p+1}}\\ &\leq\left(\frac{\lambda_{n}^{p}}{\eta_{1}^{2 p}} \cdot \frac{1}{\lambda_{n}^{p}}\right)^{\frac{1}{p+1}} E^{\frac{2}{p+1}}\|g\|^{\frac{2p}{p+1}}\\ &=\eta_{1}{ }^{-\frac{2 p}{p+1}} E^{\frac{2}{p+1}}\|g\|^{\frac{2 p}{p+1}}. \end{align*}

4 分数 Tikhonov 正则化方法

$$$J_{\mu}(f)=\underset{f \in L^{2}(\Omega)}{\operatorname{min}}\left\{\left\|K f-g^{\delta}\right\|_{W}^{2}+\mu\|f\|^{2}\right\},$$$

$$$\left[\left(K^{*} K\right)^{\frac{\gamma+1}{2}}+\mu I\right]^{-1}\left(K^{*} K\right)^{\frac{\gamma-1}{2}} K^{*} g^{\delta}=f_{\mu}^{\delta},$$$

$$$f_{\mu}^{\delta}(y)=\sum\limits_{n=1}^{\infty} \frac{\left(E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(-\sqrt{\lambda_{n}} T^{\alpha+\beta}\right)\right)^{\gamma}}{\left(E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(-\sqrt{\lambda_{n}} T^{\alpha+\beta}\right)\right)^{\gamma+1}+\mu} g_{n}^{\delta} \varphi_{n}(y), 0<\gamma \leq 1,$$$

$$$f_{\mu}(y)=\sum\limits_{n=1}^{\infty} \frac{\left(E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(-\sqrt{\lambda_{n}} T^{\alpha+\beta}\right)\right)^{\gamma}}{\left(E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(-\sqrt{\lambda_{n}} T^{\alpha+\beta}\right)\right)^{\gamma+1}+\mu} g_{n} \varphi_{n}(y), 0<\gamma \leq 1,$$$

5.1 先验选取规则下的收敛性估计

(1) 当 $0<p<\gamma+1$, $\mu=\left(\frac{\delta}{E}\right)^{\frac{\gamma+1}{p+1}}$ 时, 有收敛性估计

$$$\left\|f_{\mu}^{\delta}(y)-f(y)\right\| \leq\left(c_{1}+c_{2}\right) \delta^{\frac{p}{p+1}} E^{\frac{1}{p+1}}.$$$

(2) 当 $p \geq \gamma+1$, $\mu=\left(\frac{\delta}{E}\right)^{\frac{\gamma+1}{\gamma+2}}$ 时, 有收敛性估计

$$$\left\|f_{\mu}^{\delta}(y)-f(y)\right\| \leq\left(c_{1}+c_{3}\right) \delta^{\frac{\gamma+1}{\gamma+2}} E^{\frac{1}{\gamma+2}}.$$$

$$$\left\|f_{\mu}^{\delta}(y)-f(y)\right\| \leq\left\|f_{\mu}^{\delta}(y)-f_{\mu}(y)\right\|+\left\|f_{\mu}(y)-f(y)\right\|=I_{1}+I_{2}.$$$

\begin{align*} I_{1}&=\left\|f_{\mu}^{\delta}(y)-f_{\mu}(y)\right\| \nonumber \\ \nonumber &=\left\|\sum\limits_{n=1}^{\infty} \frac{\left(E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(\sqrt{\lambda_{n}}T^{\alpha+\beta}\right)\right)^{\gamma}}{\left(E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(-\sqrt{\lambda_{n}} T^{\alpha+\beta}\right)\right)^{\gamma+1}+\mu}\left(g_{n}^{\delta}-g_{n}\right) \varphi_{n}(y)\right\| \\ \nonumber &\leq \delta \sup _{n} \frac{\left(E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(-\sqrt{\lambda_{n}}T^{\alpha+\beta}\right)\right)^{\gamma}}{\left(E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(-\sqrt{\lambda_{n}} T^{\alpha+\beta}\right)\right)^{\gamma+1}+\mu} \nonumber\\ &\leq \delta \frac{\eta_{2}^{\gamma} \sqrt{\lambda_{n}}}{\eta_{1}^{\gamma+1}+\mu \lambda_{n}{ }^{\frac{\gamma+1}{2}}} \leq \delta c_{1}\left(\gamma, \eta_{1}, \eta_{2}\right) \mu^{-\frac{1}{\gamma+1}} =c_{1} \delta \mu^{-\frac{1}{\gamma+1}}. \end{align*}

\begin{align*} \nonumber I_{2}&=\left\|f_{\mu}(y)-f(y)\right\|\\ \nonumber &=\left\|\sum\limits_{n=1}^{\infty}\left(\frac{\left(E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(-\sqrt{\lambda_{n}} T^{\alpha+\beta}\right)\right)^{\gamma}}{\left(E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(-\sqrt{\lambda_{n}} T^{\alpha+\beta}\right)\right)^{\gamma+1}+\mu} g_{n}\right. \left. -\frac{1}{E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(-\sqrt{\lambda_{n}} T^{\alpha+\beta}\right)} g_{n}\right) \varphi_{n}(y) \right\|\\ \nonumber &=\left\|\sum\limits_{n=1}^{\infty} \frac{\mu \lambda_{n}^{-\frac{p}{2}}}{\left(E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(-\sqrt{\lambda_{n}} T^{\alpha+\beta}\right)\right)^{\gamma+1}+\mu} \cdot \frac{\lambda_{n}^{\frac{p}{2}} g_{n}}{E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(-\sqrt{\lambda_{n}} T^{\alpha+\beta}\right)} \varphi_{n}(y)\right\| \\ \nonumber &\leq \operatorname{E}\sup_{n} \frac{\mu \lambda_{n}^{-\frac{p}{2}}}{\left(E_{\alpha, 1+\frac{\beta}{\alpha}, \frac{\beta}{\alpha}}\left(-\sqrt{\lambda_{n}} T^{\alpha+\beta}\right)\right)^{\gamma+1}+\mu} \leq E \sup_{n} \frac{\mu \lambda_{n} ^{\frac{\gamma+1-p}{2}}} {\eta_{1}^{\gamma+1}+\mu \lambda_{n}^{\frac{\gamma+1}{2}}} \\ &\leq E\left\{\begin{array}{ll} c_{2} \mu^{\frac{p}{\gamma+1}}, & 0<p<\gamma+1, \\ c_{3} \mu, & p \geq \gamma+1. \end{array}\right. \end{align*}

$$$\left\|f_{\mu}^{\delta}(y)-f(y)\right\| \leq c_{1} \delta \mu^{-\frac{1}{\gamma+1}}+E\left\{\begin{array}{ll} c_{2} \mu^{\frac{p}{\gamma+1}}, & 0<p<\gamma+1, \\ c_{3} \mu, & p \geq \gamma+1. \end{array}\right. \\$$$

$$$\mu=\left\{\begin{array}{ll} \left(\frac{\delta}{E}\right)^{\frac{\gamma+1}{p+1}}, & 0<p<\gamma+1, \\[3mm] \left(\frac{\delta}{E}\right)^{\frac{\gamma+1}{\gamma+2}}, &p \geq \gamma+1, \end{array}\right.$$$

$$$\left\|f_{\mu}^{\delta}(y)-f(y)\right\| \leq\left\{\begin{array}{ll} \left(c_{1}+c_{2}\right) \delta^{\frac{p}{p+1}} E^{\frac{1}{p+1}}, & 0<p<\gamma+1, \\ \left(c_{1}+c_{3}\right) \delta^{\frac{\gamma+1}{\gamma+2}} E^{\frac{1}{\gamma+2}}, & p \geq \gamma+1. \end{array}\right.$$$

5.2 后验选取规则下的收敛性估计

$0<\tau \delta<\left\|\mathrm{g}^{\delta}\right\|$, 我们通过下述方程选取正则化参数 $\mu$

$$$\left\|K f_{\mu}^{\delta}(y)-g^{\delta}(y)\right\|=\tau \delta,$$$

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