基于边界配置法的抛物方程零阶项系数反演及其唯一性分析

Inverse Problem for the Zeroth-Order Coefficient in a Parabolic Equation Based on the Boundary Collocation Method and Uniqueness Analysis

摘要: 本文研究一维抛物型偏微分方程中与时间相关的零阶项系数$p\left(t\right)$ 的反演问题。考虑如下模型$\frac{\partial u}{\partial t}-\frac{{\partial }^{2}u}{\partial x^2}=p\left(t\right)u,\left(x,t\right)\in \left(0,L\right)\times \left(0,T\right),$ 带有齐次Dirichlet边界条件与初始条件。本文的目的是通过两个固定时刻$T_1$ 和$T_2$ 的测量数据$u\left(x,T_1\right)=\tau \left(x\right)$ 和$u\left(x,T_2\right)=\psi \left(x\right)$ 来识别未知函数$p\left(t\right)$ 在区间$\left[0,T\right]$ 上的值，并在数值方法中通过分别假设$p\left(t\right)$ 为常数函数和线性函数来重构$p\left(t\right)$ 。本文首先通过变量替换将原问题转化为齐次热传导方程，并证明$p\left(t\right)$ 在区间$\left[0,T\right]$ 上具有唯一性。随后，采用边界配置法进行空间离散，结合Tikhonov正则化技术处理反问题的不适定性，并通过L-曲线准则选取正则化参数。数值实验表明，所提方法在噪声数据下仍能稳定重建$p\left(t\right)$ 。

Abstract: This paper investigates the inverse problem of recovering the time-dependent zero-order coefficient $p\left(t\right)$ in a one-dimensional parabolic partial differential equation. The model under consideration is $\frac{\partial u}{\partial t}-\frac{{\partial }^{2}u}{\partial x^2}=p\left(t\right)u,\left(x,t\right)\in \left(0,L\right)\times \left(0,T\right),$ subject to homogeneous Dirichlet boundary conditions and an initial condition. The objective of the inverse problem is to identify the integral value of the unknown function $p\left(t\right)$ over the interval $\left[0,T\right]$ , and reconstruct $p\left(t\right)$ assuming it to be a constant function and a linear function, using measurement data $u\left(x,T_1\right)=\tau \left(x\right)$ and $u\left(x,T_2\right)=\psi \left(x\right)$ at two fixed time instances $T_1$ and $T_2$ , respectively. By applying a variable transformation, the original problem is first converted into a homogeneous heat conduction equation. It is then proven that the integral value of $p\left(t\right)$ over the interval $\left[0,T\right]$ is uniquely determined. Subsequently, the boundary collocation method is employed for spatial discretization, combined with Tikhonov regularization to address the ill-posed nature of the inverse problem. The regularization parameter is selected using the L-curve criterion. Numerical experiments demonstrate that the proposed method can stably reconstruct $p\left(t\right)$ even in the presence of noisy data.