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A modified quasi-boundary value method for regularizing of a backward problem with time-dependent coefficient. (English) Zbl 1227.65089
The main subject of this paper is the regularization of the following homogeneous one-dimensional backwards heat equation with time-dependent coefficient: $$u_{xx}(x,t) = a(t) u_t(x,t), \ x \in \mathbb{R}, \ 0 \le t \le T, \quad u(x,T) = g(x), \ x \in \mathbb{R}.$$ It is assumed that the solution is given at final time $ t = T $ with some noise, i. e., $ g_\varepsilon \in L^2(\mathbb{R}) $ with $ \Vert g_\varepsilon - g \Vert_2 \le \varepsilon $ is available, where $ \Vert \cdot \Vert_2 $ denotes the norm on $ L^2(\mathbb{R}) $. For the regularization of this problem, approximations of the form $$v_\varepsilon(x,t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \dfrac{e^{-\omega^2( F(t) + m)}} {\varepsilon \omega^2 + e^{-\omega^2 (F(T)+m)}} \widehat{g_\varepsilon}(\omega) e^{i \omega x} d \omega, \ x \in \mathbb{R}, \ 0 \le t \le T,$$ are considered, where $ m \ge 0 $ is arbitrary, but fixed, and $ F(t) = \int_0^t \frac{1}{a(s)} ds $, and $\widehat{g_\varepsilon} $ denotes the Fourier transform of $ g_\varepsilon $, i. e., $\widehat{g_\varepsilon}(\omega) = \tfrac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} g_\varepsilon(x) e^{- i \omega x} d x $. It is shown that $$ \Vert v_\varepsilon(\cdot,t) - u(\cdot,t) \Vert_2 \le c_m \varepsilon^{\tfrac{F(t)+m}{F(T)+m}} [ \ln (F(T)/\varepsilon)]^{\tfrac{F(t)-F(T)}{F(T)+m}},$$ for $ \varepsilon $ small enough and a finite constant $ c_m \ge 0 $, provided that $ g, \, g_\varepsilon \in L^2(\mathbb{R}) $ and $ u(\cdot, t) \in L^2(\mathbb{R}) $ for each $ 0 \le t < T $, and $ \int_{-\infty}^{\infty} \vert \omega^2 e^{\omega^2( F(T)+ m)} \widehat{g}(\omega) \vert^2 d \omega < \infty $ is also required here. A similar approach is presented for the regularization of a final value problem for the inhomogeneous backwards heat equation $ u_{xx}(x,t) = a(t) u_t(x,t) + f(x,t) $. The paper concludes with the presentation of some numerical results.

65M30Improperly posed problems (IVP of PDE, numerical methods)
35K05Heat equation
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