TY - JOUR
T1 - The truncation method for a two-dimensional nonhomogeneous backward heat problem
AU - Phan, Thanh Nam
AU - Dang, Duc Trong
AU - Nguyen, Huy Tuan
PY - 2010/8/15
Y1 - 2010/8/15
N2 - We consider the backward heat problem(ut - uxx - uyy = f (x, y, t), (x, y, t) ∈ Ω × (0, T),; u (x, y, T) = g (x, y), (x, y) ∈ Ω,)with the homogeneous Dirichlet condition on the rectangle Ω = (0, π) × (0, π), where the data f and g are given approximately. The problem is severely ill-posed. Using the truncation method for Fourier series we propose a simple regularized solution which not only works on a very weak condition on the exact data but also attains, due to the smoothness of the exact solution, explicit error estimates which include the approximation (ln (ε{lunate}- 1))3 / 2 sqrt(ε{lunate}) in H2(Ω). Some numerical examples are given to illuminate the effect of our method.
AB - We consider the backward heat problem(ut - uxx - uyy = f (x, y, t), (x, y, t) ∈ Ω × (0, T),; u (x, y, T) = g (x, y), (x, y) ∈ Ω,)with the homogeneous Dirichlet condition on the rectangle Ω = (0, π) × (0, π), where the data f and g are given approximately. The problem is severely ill-posed. Using the truncation method for Fourier series we propose a simple regularized solution which not only works on a very weak condition on the exact data but also attains, due to the smoothness of the exact solution, explicit error estimates which include the approximation (ln (ε{lunate}- 1))3 / 2 sqrt(ε{lunate}) in H2(Ω). Some numerical examples are given to illuminate the effect of our method.
M3 - Journal article
SN - 0096-3003
VL - 216
SP - 3423
EP - 3432
JO - Applied Mathematics and Computation
JF - Applied Mathematics and Computation
IS - 12
ER -