A problem of asymptotic stabilization by the right-hand side. (English) Zbl 1152.65064

Kornev, A. A.

Russ. J. Numer. Anal. Math. Model. 23, No. 4, 407-422 (2008).

Summary: We consider the following problem for an operator \(S\) acting in a Hilbert space \(H\) and specifying for any \(u_0\in H\) an evolutionary process \(u_{i+1} = S(u_i)\), \(i = 0, 1,\dots\): for given points \(z_0\), \(a_0\) construct corrections \(f_i\in{\mathcal F}\) from a fixed subset \({\mathcal F}\subset H\) so that positive semitrajectories \(\{a_{i+1}=S(a_i)+f_{i+1}\}^\infty_{i=0}\) and \(\{z_{i+1}=S(z_i)\}^\infty_{i=0}\) approach each other, i.e. \(\|a_i-z_i\|_2\to 0\) for \(i\to\infty\). Methods for construction of desired corrections \(f_i\) are proposed under some restrictions, and results of numerical calculations are presented for the one-dimensional Chafee-Infante equation.

MSC:

65J15	Numerical solutions to equations with nonlinear operators
35K55	Nonlinear parabolic equations
65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
47J25	Iterative procedures involving nonlinear operators
65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs

Keywords:

finite difference method; numerical examples; nonlinear operator equation; semidynamic system with discrete time; asymptotic stabilization