Uniform Difference Schemes for Perturbation Problems

Abstract

In the present chapter we consider two abstract Cauchy problems

$$\varepsilon \upsilon '\left( t \right) + A\upsilon \left( t \right) = f\left( t \right),\left( {0 \leqslant t \leqslant T} \right),\upsilon \left( 0 \right) = {{\upsilon }_{0}}$$

(1)

$${{\varepsilon }^{2}}\upsilon ''\left( t \right) + A\upsilon \left( t \right) = f\left( t \right)\left( {0 \leqslant t \leqslant T} \right),\upsilon \left( 0 \right) = {{\upsilon }_{0}},\upsilon '\left( 0 \right) = {{\upsilon '}_{0}}$$

(2)

and the boundary-value problem

$$- {{\varepsilon }^{2}}\upsilon ''\left( t \right) + A\upsilon \left( t \right) = f\left( t \right)\left( {0 \leqslant t \leqslant T} \right),\upsilon \left( 0 \right) = \upsilon \left( T \right) = {{\upsilon }_{T}}$$

(3)

for differential equations in an arbitrary Banach space E with the positive operator A and an arbitrary E parameter multiplying the derivative term. The high order of accuracy single-step uniform difference schemes generated by an exact difference scheme of the approximate solutions for differential equations of parabolic type with an arbitrary parameter e on the highest derivative are presented. The high order of accuracy two-step uniform difference schemes generated by an exact difference scheme of the approximate solutions for differential equations of the elliptic and hyperbolic types with an arbitrary parameter E on the highest derivative are presented. The well-posedness of these difference schemes for parabolic and elliptic equations in various Banach spaces is established. The stability estimates of solutions of a high order of accuracy difference schemes for hyperbolic equations with an arbitrary e parameter on the highest derivative are obtained.