A Nonlinear Case of the 1-D Backward Heat Problem: Regularization and Error Estimate

Abstract

We consider the problem of finding, from the final data u(x, T ) = ϕ(x), the temperature function u(x, t), x ∈ (0, π), t ∈ [0, T ] satisfies the following nonlinear systemThe nonlinear problem is severely ill-posed. We shall improve the quasi-boundary value method to regularize the problem and to get some error estimates. The approximation solution is calculated by the contraction principle. A numerical experiment is given.

The Backward Problem for Ginzurg-Landau-Type Equation

“…The quasi-boundary value method and the quasi-reversibility method have been used to regularize this problem. Note that the function f ∈ L ∞ ([0, π] × [0, T ] × R) in [12] has to satisfy a globally Lipschitz condition with respect to the quantity u.…”

“…In [12], the authors considered a problem of finding the function of temperature distribution u such that u t = cu xx + f (x, t, u(x, t)) , (x, t) ∈ (0, π) × (0, T ) , u (0, t) = u (π, t) = 0, t ∈ (0, T ) ,…”

“…In our problem, the function f (u) = au − bu 3 is one of the source. It is not easy to solve directly by the methods given in [12][13][14]. Hence, an improved regularization method for the case is needed.…”

Determination temperature of a backward heat equation with time-dependent coefficients

“…The problem is called the backward heat problem with time-dependent coefficient. In the simple case a(t) = 1, the problem (1.1) is investigated in many papers, such as Clark and Oppenheimer [3], Denche and Bessila [5], Tautenhahn et al [24] Melnikova et al [15,16], ChuLiFu [4,10,9], Tautenhahn [24], Trong et al [21,22], B. Yildiz et al [25,26]. Although there are many papers on the backward heat equation with the constant coefficient, there are rarely works considered the backward heat with the time-dependent coefficient, such as (1.1).…”

Regularized solutions for terminal problems of parabolic equations.

“…Here we use a modified quasi-boundary value method to regularize. A quasi boundary value method is the most common method to regularize nonlinear parabolic equations, see in [8,9,10,15,16,17].…”