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A two-dimensional backward heat problem with statistical discrete data

2017; De Gruyter; Volume: 26; Issue: 1 Linguagem: Inglês

10.1515/jiip-2016-0038

1569-3945

Nguyen Dang Minh, Khanh To Duc, Nguyen Huy Tuan, Dang Duc Trong,

Nonlinear Partial Differential Equations

Abstract We focus on the nonhomogeneous backward heat problem of finding the initial temperature θ = θ ⁢ ( x , y ) = u ⁢ ( x , y , 0 ) {\theta=\theta(x,y)=u(x,y,0)} such that { u t - a ⁢ ( t ) ⁢ ( u x ⁢ x + u y ⁢ y ) = f ⁢ ( x , y , t ) , ( x , y , t ) ∈ Ω × ( 0 , T ) , u ⁢ ( x , y , t ) = 0 , ( x , y ) ∈ ∂ ⁡ Ω × ( 0 , T ) , u ⁢ ( x , y , T ) = h ⁢ ( x , y ) , ( x , y ) ∈ Ω ¯ , \left\{\begin{aligned} \displaystyle u_{t}-a(t)(u_{xx}+u_{yy})&\displaystyle=f% (x,y,t),&\hskip 10.0pt(x,y,t)&\displaystyle\in\Omega\times(0,T),\\ \displaystyle u(x,y,t)&\displaystyle=0,&\hskip 10.0pt(x,y)&\displaystyle\in% \partial\Omega\times(0,T),\\ \displaystyle u(x,y,T)&\displaystyle=h(x,y),&\hskip 10.0pt(x,y)&\displaystyle% \in\overline{\Omega},\end{aligned}\right.\vspace*{-0.5mm} where Ω = ( 0 , π ) × ( 0 , π ) {\Omega=(0,\pi)\times(0,\pi)} . In the problem, the source f = f ⁢ ( x , y , t ) {f=f(x,y,t)} and the final data h = h ⁢ ( x , y ) {h=h(x,y)} are determined through random noise data g i ⁢ j ⁢ ( t ) {g_{ij}(t)} and d i ⁢ j {d_{ij}} satisfying the regression models g i ⁢ j ⁢ ( t ) = f ⁢ ( X i , Y j , t ) + ϑ ⁢ ξ i ⁢ j ⁢ ( t ) , \displaystyle g_{ij}(t)=f(X_{i},Y_{j},t)+\vartheta\xi_{ij}(t), d i ⁢ j = h ⁢ ( X i , Y j ) + σ i ⁢ j ⁢ ε i ⁢ j , \displaystyle d_{ij}=h(X_{i},Y_{j})+\sigma_{ij}\varepsilon_{ij}, where ( X i , Y j ) {(X_{i},Y_{j})} are grid points of Ω. The problem is severely ill-posed. To regularize the instable solution of the problem, we use the trigonometric least squares method in nonparametric regression associated with the projection method. In addition, convergence rate is also investigated numerically.