Uniform Schemes

Abstract

In this course of lectures we consider primarily the system of differential equations of the type

$$ \frac{{\partial {\mkern 1mu} u\left( {x,t} \right)}}{{\partial t}} = {\mkern 1mu} L\left( D \right){\mkern 1mu} u\left( {x,t} \right) + {\mkern 1mu} f\left( {x,t} \right),$$

(1.1.1)

where

$$ u\left( {x,t} \right) = {\mkern 1mu} \left\{ {{u_1}\left( {{x_1}, \ldots ,{x_m},t} \right),{u_2}{\mkern 1mu} \left( {{x_1}, \ldots ,{x_m},t} \right), \ldots ,{u_n}{\mkern 1mu} \left( {{x_1}, \ldots ,{x_m},t} \right)} \right\}, $$

$$ f\left( {x,t} \right) = \left\{ {{f_1}\left( {{x_1}, \ldots ,{x_m},t} \right),{f_2}{\mkern 1mu} \left( {{x_1}, \ldots ,{x_m},t} \right), \ldots ,{f_n}{\mkern 1mu} \left( {{x_1}, \ldots ,{x_m},t} \right)} \right\} $$

are vector functions of the vector space variable x = (x₁,…, x_m) and of time t; L(D) is a matrix linear differential operator with variable coefficients, D = {D _i }, D _i = ∂/∂x _i , i = 1,…, m.