In this paper, we consider the numerical solution for the reaction-diffusion equation. A finite difference scheme and the basic error equality are given. Then the error estimations are proved for the periodic problem with $v(x,t)\geq 0$, the first and second boundary value problems with $v(x,t)\geq v_0＞0$, and for $v(U)\geq v_0›0$. Under some conditions such estimations imply the stabilities and convergences of the schemes.

In this paper we use an iterative method to get an approximate solution $u^n$ and $\bar{u}^n$ which approximate the exact solution $u$ with the error estimates $\|u-u^n\|+ch\|u-u^n\|_1+\|u-\bar{u}^n\|_1\leq ch^{n+2}$.

A class of methods for solving the initial problem for ordinary differential equations are studied. We develop k-block implicit one step methods whose nodes in a block are nonequidistant. When the components of the node vector are related to the zeros of Jacobi's orthogonal polynomials, we can derive a subclass of formulas which are A or L-stable. The order can be arbitrarily high with A- or L-stability. We suggest a modified algorithm which avoids the inversion of a $km×km$ matrix during Newton-Raphson iterations, where $m$ is the number of differential equations. When k=4, for example, only a couple of $m×m$ matrices have to be inversed, but four values can be obtained at one time.

A class of exponentially fitted nonequidistant extrapolation methods based on an L-stable linear single-step formula are studied. Theorems about the extrapolation coefficients are given. These methods keep good numerical stability "quasi-L-stability", while raising the accuracy of the original formula.

This paper describes a method for investigating the analyticity and for obtaining perturbation expansions of eigenvalues and eigenvectors of a matrix dependent on several parameters. Some of results of this paper provide justification of the applications of the Newton method for inverse matrix eigenvalue problems.

The MGR[$v$] Algorithm of Ries, Trottenberg and Winter with v=0 and the Algorithm 2.1 of Braess are essentially the same multigrid algorithm for the discrete poisson equation. In this report we consider the extension to the general diffusion equation. In particular, we indicate the proof of the basic result $ρ≤\frac{1}{2}(1+Kh)$, thus extending the results of Braess and Ries Trottenberg and Winter. In addition to this theoretical result we present computational results which indicate that other constant coefficient estimates carry over to this case.