On convergence of difference schemes for delay parabolic equations

The convergence of difference schemes for the approximate solutions of the initial–boundary value problem for the delay parabolic differential equation {dv(t)dt+Av(t)=B(t)v(t−ω),t≥0,v(t)=g(t)(−ω≤t≤0) in an arbitrary Banach space E with the unbounded linear operators A and B(t) in E with dense domains D(A)⊆D(B(t)) is investigated. Theorems on convergence estimates for the solutions of the first and the second order of accuracy difference schemes in fractional spaces Eα are established. In practice, the convergence estimates in Hölder norms for the solutions of difference schemes of the first and the second order of approximation in t of the approximate solutions of multi-dimensional delay parabolic equations are obtained. The theoretical statements for the solution of these difference schemes are supported by the numerical example.