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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:

${u}_{xx}\left(x,t\right)=a\left(t\right){u}_{t}\left(x,t\right),\phantom{\rule{4pt}{0ex}}x\in ℝ,\phantom{\rule{4pt}{0ex}}0\le t\le T,\phantom{\rule{1.em}{0ex}}u\left(x,T\right)=g\left(x\right),\phantom{\rule{4pt}{0ex}}x\in ℝ·$

It is assumed that the solution is given at final time $t=T$ with some noise, i. e., ${g}_{\epsilon }\in {L}^{2}\left(ℝ\right)$ with $\parallel {g}_{\epsilon }{-g\parallel }_{2}\le \epsilon$ is available, where ${\parallel ·\parallel }_{2}$ denotes the norm on ${L}^{2}\left(ℝ\right)$.

For the regularization of this problem, approximations of the form

${v}_{\epsilon }\left(x,t\right)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }\frac{{e}^{-{\omega }^{2}\left(F\left(t\right)+m\right)}}{\epsilon {\omega }^{2}+{e}^{-{\omega }^{2}\left(F\left(T\right)+m\right)}}\stackrel{^}{{g}_{\epsilon }}\left(\omega \right){e}^{i\omega x}d\omega ,\phantom{\rule{4pt}{0ex}}x\in ℝ,\phantom{\rule{4pt}{0ex}}0\le t\le T,$

are considered, where $m\ge 0$ is arbitrary, but fixed, and $F\left(t\right)={\int }_{0}^{t}\frac{1}{a\left(s\right)}ds$, and $\stackrel{^}{{g}_{\epsilon }}$ denotes the Fourier transform of ${g}_{\epsilon }$, i. e., $\stackrel{^}{{g}_{\epsilon }}\left(\omega \right)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }{g}_{\epsilon }\left(x\right){e}^{-i\omega x}dx$. It is shown that

$\parallel {v}_{\epsilon }{\left(·,t\right)-u\left(·,t\right)\parallel }_{2}\le {c}_{m}{\epsilon }^{\frac{F\left(t\right)+m}{F\left(T\right)+m}}{\left[ln\left(F\left(T\right)/\epsilon \right)\right]}^{\frac{F\left(t\right)-F\left(T\right)}{F\left(T\right)+m}},$

for $\epsilon$ small enough and a finite constant ${c}_{m}\ge 0$, provided that $g,\phantom{\rule{0.166667em}{0ex}}{g}_{\epsilon }\in {L}^{2}\left(ℝ\right)$ and $u\left(·,t\right)\in {L}^{2}\left(ℝ\right)$ for each $0\le t, and ${\int }_{-\infty }^{\infty }{|{\omega }^{2}{e}^{{\omega }^{2}\left(F\left(T\right)+m\right)}\stackrel{^}{g}\left(\omega \right)|}^{2}d\omega <\infty$ is also required here.

A similar approach is presented for the regularization of a final value problem for the inhomogeneous backwards heat equation ${u}_{xx}\left(x,t\right)=a\left(t\right){u}_{t}\left(x,t\right)+f\left(x,t\right)$. The paper concludes with the presentation of some numerical results.

##### MSC:
 65M30 Improperly posed problems (IVP of PDE, numerical methods) 35K05 Heat equation