Partial Differential Equations of Parabolic Type

In the present chapter we consider the well-posedness of an abstract Cauchy problem for differential equations of parabolic type, $$v'(t) + A(t)v(t) = f(t)(0 \leqslant t \leqslant T),v(0) = {{v}_{0}}$$ in an arbitrary Banach space with the linear positive operators A(t). The high order of accuracy difference schemes generated by an exact difference scheme or by Taylor's decomposition on two points for the numerical solutions of this problem are presented. The well-posedness of these difference schemes in various Banach spaces are studied. The stability and coercive stability estimates in Holder norms for the solutions of the high order of accuracy difference schemes of mixed type boundary-value problems for parabolic equations are obtained.