an:04201525
Zbl 0727.65076
Golichev, I. I.
Some iterative methods of solving problems for parabolic equations
EN
Sov. Math., Dokl. 37, No. 3, 714-717 (1988); translation from Dokl. Akad. Nauk SSSR 300, No. 4, 782-785 (1988).
00181310
1988
j
65M12 35K15
iterative methods; parabolic equations; convergence
The iterative process
\[
\frac{\partial u_{k+1}}{\partial t}-\alpha \Delta u_{k+1}=\frac{\partial}{\partial x_ i}[(A_{ij}(Pu_ k)- \alpha \delta_{ij})Q\frac{\partial u_ k}{\partial x_ i}]-A_ 0(Pu_ k,Qu_{k,k}),
\]
\(u_{k+1}(x,0)=\psi_ 0\), \(u_{k+1}|_ S=\psi_ 1\), is studied for the problem \(\partial u/\partial t-(\partial /\partial x_ i)(A_{ij}(u)\partial u/\partial x_ j)+A_ 0(u,u_ x)=0,\) \(u(x,0)=\psi_ 0\), \(u|_ S=\psi_ 1\), where \(\alpha\) is a parameter and P, Q are shear operators. Conditions for convergence are obtained. The author indicates that the iterative process can be applied to proving existence and evaluating some subset of solutions.