An approximate solution for nonlinear backward parabolic equations

TY - JOUR

T1 - An approximate solution for nonlinear backward parabolic equations

AU - Phan, Thanh Nam

PY - 2010/7/15

Y1 - 2010/7/15

N2 - We consider the backward parabolic equation{(ut + A u = f (t, u (t)), 0 < t < T,; u (T) = g,) where A is a positive unbounded operator and f is a nonlinear function satisfying a Lipschitz condition, with an approximate datum g. The problem is severely ill-posed. Using the truncation method we propose a regularized solution which is the solution of a system of differential equations in finite dimensional subspaces. According to some a priori assumptions on the regularity of the exact solution we obtain several explicit error estimates including an error estimate of Hölder type for all t ∈ [0, T]. An example on heat equations and numerical experiments are given.

AB - We consider the backward parabolic equation{(ut + A u = f (t, u (t)), 0 < t < T,; u (T) = g,) where A is a positive unbounded operator and f is a nonlinear function satisfying a Lipschitz condition, with an approximate datum g. The problem is severely ill-posed. Using the truncation method we propose a regularized solution which is the solution of a system of differential equations in finite dimensional subspaces. According to some a priori assumptions on the regularity of the exact solution we obtain several explicit error estimates including an error estimate of Hölder type for all t ∈ [0, T]. An example on heat equations and numerical experiments are given.

M3 - Journal article

SN - 0022-247X

VL - 367

SP - 337

EP - 349

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

ER -