Approximate solution of the backward problem for Kirchhoff's model of parabolic type with discrete random noise (Q2197859)

In this paper, the final boundary value problem for non-local Kirchhoff's model of parabolic type with discrete random noise is considered. Namely, let \(T\) be a given positive real number. The paper is aimed to study the problem of reconstructing the initial data for a non-local Kirchhoff's model of parabolic type \[ u_t = K( \Vert \nabla u(\cdot, t) \Vert_{L_2(\Omega)}) \Delta u+ f(x,t), \] with the Dirichlet condition and the final condition \[ \begin{aligned} u(x, t) &= 0, \ \ (x, t) \in \partial \Omega \times (0,T), \\ u(x, T ) &= g(x), \ \ x \in \Omega, \end{aligned} \] where \(\Omega = (0, \pi)\). The above problem is often called the backward problem. Such problems are often called non-local because of the presence of the integral over the entire domain \(\Omega\). First, the instability of solutions is discussed. Then the authors construct the regularized solution by the trigonometric method in non-parametric regression associated with the truncated expansion method. In addition, under a prior assumption on the exact solution, the convergence rate is obtained. At last, the effectiveness and suitability of the theoretical results are justified via numerical experiments.