Regularized solution for backward heat equation in Sobolev space

Abstract

Inverse problems in partial deferential equations are severely ill posed in the sense of Hadamard. So the heat equation with a terminal condition problem is ill posed even in the sobolev space so regularization is needed. In this paper, we discuss about the convergence result of the approximation problem in the Sobolev space H 2 ðR n Þ, which is well posed. By using a small parameter, we construct an approximation problem and use a quasi-boundary value method to regularize nonlinear heat equation. Finally, we prove the approximation solution converges to the original solution whenever the parameter goes to zero in H 2 ðR n Þ. KEYWORDS backward heat equation, regularization MSC CLASSIFICATION 47A52

Recovering the initial value for a system of nonlocal diffusion equations with random noise on the measurements

“…In the case of constant or time-dependent diffusion coefficient, the classical backward heat conduction equation has been studied in many works, see for example, other studies. [8][9][10] However, to the best of our knowledge, there are not any result on (1)(2)(3). Our paper is the first study on this direction for parabolic systems with random model.…”

“…Some work related to the problem which is studied such as that of Hapuarachchi and Xu 1 constructed an approximation problem and use a quasi‐boundary value method to regularize nonlinear heat equation by using a small parameter. In addition, in 2019, the Liu et al 2 considered that the boundary value problem of the partial differential equations can be transformed into an equivalent system of nonlinear and ill‐posed integral equations for the unknown boundary.…”

“…{ t u 1 =  1 (u)Δu 1 + (x, t), (x, t) ∈  × (0, T), t u 2 =  2 (u)Δu 2 + g(x, t), (x, t) ∈  × (0, T), (1) subject to the final conditions u 1 (x, T) = (x), u 2 (x, T) = (x), x ∈ , (2) and homogeneous Dirichlet boundary condition u 1 (x, t) = u 2 (x, t) = 0, (x, t) ∈  × (0, T),…”