A filter method for inverse nonlinear sideways heat equation

In this paper, we study a sideways heat equation with a nonlinear source in a bounded domain, in which the Cauchy data at x =X are given and the solution in 0 ≤ x <X is sought. The problem is severely ill-posed in the sense of Hadamard. Based on the fundamental solution to the sideways heat equation, we propose to solve this problem by the filter method of degree α, which generates a well-posed integral equation. Moreover, we show that its solution converges to the exact solution uniformly and strongly in Lp(ω,X ;L2(R)), ω ∈ [0,X ) under a priori assumptions on the exact solution. The proposed regularized method is illustrated by numerical results in the final section.


Introduction
In this paper, we determine the surface temperature u(x, t) for 0 ≤ x < X from the known temperature measurements u(X , t) = φ(t) and heat-flux measurement ∂u ∂x (X , t) = ψ(t) when u(x, t) satisfies the following system: ∂x 2 + f (u)G(x, t; u), (x, t) ∈ (0, X ) × R, u(x, t)| t→±∞ = 0, (x, t) ∈ (0, X ) × R, where φ, ψ ∈ L 2 (R) are given functions. The source terms f (u), G(u) are globally Lipschitz functions satisfying (2.15a) and (2.15b), respectively. The problem called the inverse nonlinear sideways heat equation (INSHE for short) is a model of a problem where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. In many dynamic heat transfer situations, one wishes to determine the temperature on the surface of a body, where the surface itself is inaccessible for measurements. The physical situation at the surface may be unsuitable for attaching a sensor, or the accuracy of a surface measurement may be seriously impaired by the presence of the sensor. Typical practical applications are the estimation of the heat flux and the temperature at the surface of the body under investigation, e.g., re-entry vehicles, calorimeter-type instrumentation, and combustion chambers [1,2,5,11,13,15,16,19,20]. In such cases, one is restricted to interior measurements, and from these one wishes to compute the surface temperature. Cannon (1984) [4] considered the direct problem for the homogeneous heat equation in the quarter plane (x ≥ 0, t ≥ 0): (1. 2) The functions φ(·) and u(x, ·) are to be in L 2 (R) (φ and u vanish for t < 0). The author proved that, for each φ ∈ L 2 (R), (1.2) has a unique solution u with u(x, ·) ∈ L 2 (R) for each x ≥ 0. Fredrik Berntsson (1999) [3] considered the sideways heat equation The author used the spectral method to solve problem (1.3). Error estimates for the regularized solution were derived, and a procedure for selecting an appropriate regularization parameter was given. In recent years, linear homogeneous problem (1.1), i.e., f (u)G(x, t; u) = 0, has been researched by many authors, and various numerical methods have been proposed, e.g., the boundary element Tikhonov regularization method (Lesnic et al. (1996) [9]), the conjugate gradient method (Hao (2012) [8]), the difference regularization method (Xiong et al. (2006a) [17]), the "optimal filtering" method (Seidman & Elden (1990) [14]), the Fourier method (Xiong et al. 2006b [18]), the quasi-reversibility method (Elden (1987) [6], Liu & Wei (2013) [10]), the wavelet, wavelet-Galerkin, and the spectral regularization methods (Elden et al. (2000) [7], Reginska & Elden (1997) [12]), to mention only a few.
The more important but challenging semilinear sideways heat equation with the heat source depends nonlinearly on the temperature, which occurs in many applications related to reaction-diffusion. The function f (u)G(u) is known as a special type of locally Lipschitz function. For example, if we choose f (u) := u, G(x, t; u) := sin u (individually they are globally Lipschitz), then Although there are some works on the nonlinear case, the literature on the case of locally Lipschitz sources f (u)G(x, t; u) is quite scarce. Our results extend problem (1.3), and we propose a new filter method to establish regularized solutions of problem (1.1) in the case of the locally Lipschitz function f (u)G(x, t; u). The paper is organized as follows. In Sect. 2, the formulation of problem and regularization methods is given. In Sect. 3, a stability estimate in L p (ω, X ; L 2 (R)), ω ∈ [0, X ) is proved under a priori condition of the exact solution and the locally Lipschitz source term. Finally, we present a numerical result to illustrate the proposed regularized method in Sect. 4.

Mathematical problem and mild solution of (INSHE)
For w ∈ L 2 (R), we have the Fourier transform and the L 2 norm of w is Suppose that the solution of problem (1.1) is represented as a Fourier transform Throughout this paper, we let W(x, t; u) = f (u)G(x, t; u), ∀(x, t) ∈ (0, X ) × R. From (1.1), we have the following systems of second order ordinary equation: We thus have after some direct calculation Moreover, for ξ = 0, we have From (2.5), the exact form of u is given by We say that u is a mild solution of problem (1.1) if u satisfies integral (2.6). We know that the three functions are unbounded as a function of the variable ξ . Consequently, small errors in high frequency components can blow up and completely destroy the solution for 0 < x < z < X . A natural idea to stabilize the problem is to replace them by a bounded approximation. In a natural way, we can replace the terms in (2.7) by (respectively), with δ > 0 is a small positive number representing the level of noise and the parameter γ (δ) > 0 is small (regularization parameter). We introduce the first regularized solution U δ γ (δ) obtained by are defined for all 0 ≤ y ≤ X and ξ ∈ R in the following: We introduce some notations and assumptions that are needed for our analysis.
Definition 2.1 (Gevrey space) The Gevrey class of functions of order θ ≥ 0 defined as is equipped with the norm defined by the Banach spaces of measurable (respectively, continuous functions) functions w : We assume the following: (H 1 ) The data φ, ψ ∈ L 2 (R) are noisy and are represented by the observation data here δ > 0 is a small positive number representing the level of noise.
3 Error estimate in L p (ω, X ; L 2 (R)), 0 ≤ ω < X First, we have the following lemmas which will be useful.
Our result is in the next theorem.
2nd part. Error estimate U δ γ (δ)u L p (ω,X ;L 2 (R)) . Note From (2.6) and Lemma 3.2, we have From Lemma 3.1, we get Hence, we get Moreover, like in (3.15), we obtain Thus, we have the following inequality: From Gronwall's inequality, we conclude that Hence, we deduce that Next, we estimate U δ γ (δ) -V γ (δ) L p (ω,X ;L 2 (R)) . Using the basic inequality (a + b + c) 2 ≤ 3(a 2 + b 2 + c 2 ) and Hölder's inequality, we obtain Similar calculations as in (3.15) yield From Lemma 3.1 and using the Lipschitzian property of W, we get the following inequality: Based on the Fourier transform, for F ∈ L 2 (R), we have The exact solution of problem (4.1)-(4.3) is given by The data φ, ψ ∈ L 2 (R) are noisy and are represented by the observation data φ δ , ψ δ ∈ L 2 (R) satisfying here δ > 0 is a small positive number representing the level of noise (δ → 0 + ). We recall the regularized solution U δ γ (δ) obtained by Here cosh γ (δ) (y √ iξ ), sinh γ (δ) (y √ iξ ) are defined for all 0 ≤ y ≤ X and ξ ∈ R in the following: where Next, we consider the problem of computing the Fourier transform as follows: Let m, n ∈ R, m < n and assume that Put h t = n-m N t , t i = ih t + n, i = 1, N t , respectively. Noting that ξ k = 2π (k-N t
From them, we observe that the errors at δ = 0.001 are greater than those at δ = 0.0001 and smaller than those at δ = 0.01. Furthermore, with the smaller errors of input data, the results obtained are more accurate, which verifies the theoretical results.