An approximate solution for nonlinear backward parabolic equations. (English) Zbl 1194.35494
Summary: We consider the backward parabolic equation
\[ \begin{cases} u_t+Au=f(t,u(t)),\quad & 0<t<T,\\ u(t)=g\end{cases} \]
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 \(t\in [0,T]\). An example on heat equations and numerical experiments are given.
\[ \begin{cases} u_t+Au=f(t,u(t)),\quad & 0<t<T,\\ u(t)=g\end{cases} \]
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 \(t\in [0,T]\). An example on heat equations and numerical experiments are given.
MSC:
| 35R25 | Ill-posed problems for PDEs |
| 35K05 | Heat equation |
| 65M30 | Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
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