The truncation method for a two-dimensional nonhomogeneous backward heat problem

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 -