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

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 system ut − uxx = f(x, t, u(x, t)), (x, t) ∈ (0, π)× (0, T ) u(0, t) = u(π, t) = 0, t ∈ (0, T ). The 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.