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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,0tT,u(x,T)=g(x),x·

It is assumed that the solution is given at final time t=T with some noise, i. e., g ε L 2 () with g ε -g 2 ε is available, where · 2 denotes the norm on L 2 ().

For the regularization of this problem, approximations of the form

v ε (x,t)=1 2π - e -ω 2 (F(t)+m) εω 2 +e -ω 2 (F(T)+m) g ε ^(ω)e iωx dω,x,0tT,

are considered, where m0 is arbitrary, but fixed, and F(t)= 0 t 1 a(s)ds, and g ε ^ denotes the Fourier transform of g ε , i. e., g ε ^(ω)=1 2π - g ε (x)e -iωx dx. It is shown that

v ε (·,t)-u(·,t) 2 c m ε F(t)+m F(T)+m [ln(F(T)/ε)] F(t)-F(T) F(T)+m ,

for ε small enough and a finite constant c m 0, provided that g,g ε L 2 () and u(·,t)L 2 () for each 0t<T, and - |ω 2 e ω 2 (F(T)+m) g ^(ω)| 2 dω< 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.

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