The Galerkin method for parabolic equations with operators of local type (in Russian)

Abstract

The existence and the uniqueness of solution are investigated for the Cauchy problem $u’(t)+A(t)u(t)+(Mu)(t)=f(t),u(0)=u_0$, with a family of linear operators $A(t)$ and a local typo operator $M$. In the applications $M$ represents, in particular, the retardation of the argument or an integral operator. It is shown that this Cauchy problem produces an isomorphism between some natural spaces of given data and solutions. The convergence of the Galerkin method takes place in strong norms containing derivatives by t. The rate of convergence is characterized by two-sided estimates. The influence of perturbations, arising in the course of calculating the coefficients of the approximate problem as scalar products, on the approximate solutions is given also by two-sided estimates in the same norms in which the convergence is proved.