By Allaberen Ashyralyev, Pavel E. Sobolevskii

The current monograph is dedicated to the development and research of the recent excessive order of accuracy distinction schemes of approximating the recommendations of normal and singular perturbation boundary worth difficulties for partial differential equations. the development is predicated at the unique distinction scheme and Taylor's decomposition at the or 3 issues. This technique approved basically to increase to a category of difficulties the place the speculation of distinction equipment is appropriate. specifically, now it truly is attainable to enquire the differential equations with variable coefficients and common and singular perturbation boundary worth difficulties. The research is predicated on new coercivity inequalities.

The publication could be of worth to expert mathematicians, in addition to complicated scholars within the fields of numerical research, sensible research, and usual and partial differential equations.

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22). 26)) >. ) + Ei=l i at Xa\ia;J(t, y(t))fi(t, y(t)), 0::; 1, >. ::; p - 1. 2. 30). 3jFj (tk_d rj - 1, 1 :::; k :::; N, Uo = Yo· Note that using Taylor's decomposition on two points, we can extend our discussion to construct the difference schemes of an arbitrary high order of accuracy for approximate solutions of the initial-value problem for the first order nonlinear system of ordinary differential equations y'(t) + A(t)y(t) = f(t, y(t)), 0 < t :::; T, y(O) = Yo. It is left to the reader. Finally, in this chapter we consider two types of arbitrary high order of accuracy difference schemes generated by an exact difference scheme or Taylor's decomposition on two points for the numerical solutions of an initial-value problem for differential equations and systems of differential equations.

F (tk+l) (m + I)! p+q-m-3 X ~ f='o n n (-1) T d (t )Bn,p+q-m-n(t ) (n + I)! \ k+1 48 Chapter 3. Difference Schemes for Second-Order Differential Equations + p+q-3 L L m ( ~ m=l A=O X( T p+q-3 )1 p+q-m-1. + X( T l; {; p+q-3 m p+q-2 )' p+q-m. d- ) f(A)(tk_d (m d- ( )B-p+q-m-22( p+q-m-2 tk ( 1 + I)! ~ m-A ' f(A)(tk-d (m + I)! 1. Two-Step Exact Difference Scheme and Its Applications + E P+q-l(p + q - 1) A f (A) r P+-2 q (tk-l) (p + q)! where I;+Ol,k 1, I;+\'k = R",l-,,( -rib(tk)), K = 0,1, 2 n,2 = Ra,2-a ( .

1). 3). Therefore for the smooth a(t) there exist smooth solutions of this system defined on the segment [0, T]. We will choose one of these smooth solutions b(t) = br(t) + ibi(t), t ::; T. Now, we will consider the applications of this exact difference scheme. 1. Two-Step Exact Difference Scheme and Its Applications 43 and is sufficiently simple. The choice formula