A spectral regularization method for solving surface heat flux on a general sideways parabolic

Abstract

We consider a general sideways heat equation, which is a well known ill-posed problem: a small noise in the exact data may cause dramatically large errors in the solution. The literature available are mainly devoted to determining the temperature with constant coefficients equation. In this paper, a Fourier regularization method for determining surface heat flux is given. An error estimate for a general sideways parabolic equation is established.

Introduction

In some engineering context, one needs to determine the temperature or heat flux on both sides of a thick wall, but one side is inaccessible to measurements [1], [2]. The following sideways parabolic equation in a quarter plane sometimes is a model of the above mentioned problem [3]:

$$\begin{array}{cc}\hfill & u_t=a(x)u_{\mathit{xx}}+b(x)u_x+c(x)u\text{,}x>0\text{,}t>0\text{,}\hfill \\ \hfill & u(1\text{,}t)=g(t)\text{,}t>0\text{,}\hfill \\ \hfill & u(x\text{,}0)=0\text{,}x>0\text{.}\hfill \end{array}$$

Here, a( · ), b( · ) and c( · ) are given functions and satisfy: for some λ, Λ > 0,

$$\lambda ⩽a(x)⩽\Lambda \text{,}x\in (0\text{,}\infty )\text{,}$$

and

$$c(x)⩽0\text{.}$$

We assume that a( · ) ∈ C²(0, ∞), b( · ) ∈ C¹(0, ∞), c( · ) ∈ C(0, ∞). We want to seek u_x(x, t) (0 ⩽ x < 1) based on the measured data g_δ( · ) which is inexact. This is a severely ill-posed problem: a small error between g_δ( · ) and g( · ) can cause large errors in the solution u_x(x, t) for x ∈ [0, 1). There are many methods for determining u(x, t) for x ∈ (0, 1) in a stable way (see [4], [5], [6], [7], [8]). But the corresponding result on heat flux is very scarce [2]. As J.V. Beck in his book [1] mentioned “the computation of heat flux is more difficult than that of temperature”.

Fourier method applied to sideways heat equation u_t = u_xx has been provided by Eldén et al. [9], but the convergence at x = 0 for temperature was not obtained in [9]. In this paper, by introducing the Sobolev space $(H_p{)}_{p\in {\mathcal{R}}^{+}}≔\{v(t)\in L^2(\mathcal{R}):\parallel v{\parallel }_p<\infty \}$, we can get the convergence at x = 0 for heat flux when $p>\frac{1}{2}$, where ∥v∥_p is defined by

$$\Vert v{\Vert }_p≔{\int }_{-\infty }^{\infty }(1+{\xi }^2{)}^p|\hat{v}(\xi ){|}^2\mathrm{d}\xi \text{,}$$

and the Fourier transform of v(t) is given by

$$\hat{v}(\xi )=\frac{1}{\sqrt{2\mathrm{\pi }}}{\int }_{-\infty }^{\infty }v(t){\mathrm{e}}^{-i\xi t}\mathrm{d}t\text{,}\xi \in \mathcal{R}\text{.}$$

The corresponding direct problem with (1.1) is an initial boundary value problem in the quarter plane

$$\begin{array}{cc}\hfill & u_t=a(x)u_{\mathit{xx}}+b(x)u_x+c(x)u\text{,}x>0\text{,}t>0\text{,}\hfill \\ \hfill & u(0\text{,}t)=g(t)\text{,}t>0\text{,}\hfill \\ \hfill & u(x\text{,}0)=0\text{,}x>0\text{.}\hfill \end{array}$$

We assume

$$f(·)\in L^2(0\text{,}\infty )\text{.}$$

As a solution of problem (1.6), we understand a function u(x, t) satisfying (1.6) in the classical sense. For the uniqueness of the solution, we require that ∥u(x, ·)∥ be bounded [4], where the notation ∥ · ∥ denotes L²-norm. In order to use Fourier transform technique, we extend the domain of definition of the functions u(x, ·), g( · ): = u(1, ·), f( · ): = u(0, ·) and other functions appearing in the paper to whole t-axis by defining them to be zero for t < 0.

Throughout this paper, let $g(·)\in L^2(\mathcal{R})$ and $g_{\delta }(·)\in L^2(\mathcal{R})$ denote the exact and measured data, respectively, which satisfy

$$\Vert g_{\delta }(·)-g(·)\Vert ⩽\delta \text{,}$$

where δ is the noise level.

In addition, we suppose that there exists a priori bound for f(t) ≔ u(0, t):

$$\Vert f{\Vert }_p⩽E\text{,}$$

where ∥ · ∥_p is given by (1.4).

The following conclusions can be found in [4].

Lemma 1.1

Let v(x, ξ) be the solution of the following boundary value problem for ordinary differential equation

$$\begin{array}{cc}\hfill & \mathrm{i}\xi v(x\text{,}\xi )=a(x)v_{\mathit{xx}}+b(x)v_x+c(x)v\text{,}x>0\text{,}\xi \in \mathcal{R}\text{,}\hfill \\ \hfill & v(0\text{,}\xi )=1\text{,}\hfill \\ \hfill & \underset{x\to \infty }{\lim }v(x\text{,}\xi )=0\text{,}\xi \ne 0\text{,}\hfill \end{array}$$

for ξ = 0, we require v(x, 0) be bounded as x → ∞. Suppose that the problem (1.6) has a solution u, then

$$u(x\text{,}t)=\frac{1}{\sqrt{2\mathrm{\pi }}}{\int }_{-\infty }^{\infty }{\mathrm{e}}^{\mathrm{i}\xi t}v(x\text{,}\xi )\hat{f}(\xi )\mathrm{d}t\text{,}x>0\text{.}$$

and

$$\hat{u}(x\text{,}\xi )=v(x\text{,}\xi )\hat{f}(\xi )\text{.}$$

If the problem (1.6) has a solution, the following can be derived from (1.12)

$$\hat{g}(\xi )=v(1\text{,}\xi )\hat{f}(\xi )\text{,}$$

$$\hat{u}(x\text{,}\xi )=\frac{v(x\text{,}\xi )}{v(1\text{,}\xi )}\hat{g}(\xi )\text{,}$$

$${\hat{u}}_x(x\text{,}\xi )=\frac{v_x(x\text{,}\xi )}{v(1\text{,}\xi )}\hat{g}(\xi )\text{.}$$

Lemma 1.2

There exist constant c₁, c₂, such that for x ∈ [0, 1] and ∣ξ∣ large enough, says ∣ξ∣ ⩾ ξ₀, then

$$c_1\sqrt{|\xi |}{\mathrm{e}}^{-A(x)\sqrt{|\xi |/2}}⩽|v_x(x\text{,}\xi )|⩽c_2\sqrt{|\xi |}{\mathrm{e}}^{-A(x)\sqrt{|\xi |/2}}\text{,}$$

where $A(x)={\int }_0^x\frac{1}{\sqrt{a(s)}}\mathrm{d}s$. Moreover, for x ∈ [0, 1], the right-hand side inequality in (1.16) is valid for all $\xi \in \mathcal{R}$ with another constant c₂.

Lemma 1.3

If the boundary value problem

$$\begin{array}{cc}\hfill & a(x)v_{\mathit{xx}}+b(x)v_x+c(x)v=0\text{,}x>0\text{,}\hfill \\ \hfill & v(0)=1\text{,}\underset{x\to \infty }{\lim }v(x)\mathit{bounded}\text{,}\hfill \end{array}$$

has a unique solution, then there exists constant c₃, c₄, such that

$$c_3{\mathrm{e}}^{-A(1)\sqrt{|\xi |/2}}⩽|v(1\text{,}\xi )|⩽c_4{\mathrm{e}}^{-A(1)\sqrt{|\xi |/2}}\text{,}\forall \xi \in \mathcal{R}\text{.}$$