1549301742-The_Theory_of_Difference_Schemes__Samarskii

16 Preliminaries Corollary 2 Under conditions (22) the problem (23) [, [Yi] = 0 , i = 1, 2, ... , N - l; Yo = 0, YN = 0, has the ...

Difference equations 17 The problem of auxiliary character such as ,C [ y;] = 0, O<i<N· , will complernent our stucliea, f ...

18 and, consequently, I IF; 0 I lyio I = O<i<N max IYi < - -=--D· < - io F D c Preliminaries Before going furt ...

Difference equations 19 The solution of problem (32) can be 1nost readily evaluated with the aid of the relation llYc < llYc, ...

20 Preliminaries 0 Proof Recall that the difference u; = Yi - Yi is the solution of proble1n 0 (29) with the right part F; = D; ...

Difference equations 21 b) if i 0 = N, that is, max; y, = yN = M 0 > 0, but YN-l < M 0 , then for 0 < X 2 < 1. In bo ...

22 Preliminaries can be made self-adjoint: Ayi = aitl (Yitl - Yi) - ai (Yi - Yi-i) - di Yi= -<pi, i = 1, 2, ... , N - 1. Inde ...

Difference equations 23 Since la 1 + 1 I< l under conditions (39), formulae ( 40) imply that yielding, in turn, N ( 41) IYil& ...

24 Preliminaries The second-order difference equations with constant coefficients. If the coefficients of the difference equati ...

Difference equations 25 A; .0..i,i+I = - B; .0..i,i-I. Due to this fact the condition .0..; 1 , ; 1 + 1 -::/= 0 for some i = i 1 ...

26 whose roots are c ± J c^2 - 4 ab 2b Preliminaries Three possibilities of interest in accordance with the discriminant sign ar ...

Difference equations 27 3) If D c^2 - 4ab < 0, equation (46) possesses two complex-conjugate roots c+JIDTi. q 1 = 2 b = p(cos ...

28 Preliminaries with discriminant D = 4 ( cos^2 CY - 1) = -4 sin^2 CY < 0 and e ±i a. Then q~. 2 = e ±i k a, particular solu ...

Difference equations^29 assuming 'P to be arbitrary fixed. Indeed, the chain of the identities occur: [cos ( k + 1) Vi - cos ( k ...

30 Preliminaries with the well-established notations of the right and left differences: 6. Yi = Yi+i - Yi and VYi =Yi -Yi-1, so ...

Difference equations where i^1 = i + l. For Yi = Yo+ (y 1 - y 0 ) =Yo + \7 Yi we arrive at N (y, 6. v) = YN VN - Yo Vi - L Vi V ...

32 Preliminaries which are called the first and the second Green formulae. Usually the first formula can be modified into a more ...

Difference equations 33 yields the first Green formula ( 5 6) ( y, A u) = - (a \J u, \J y ] - (du, y) + (a y \J u) N - Yo (a \J ...

34 Prelirninaries 1.2 SOME VARIANTS OF THE ELIMINATION METHOD The flow variant of the elimination method for difference problem ...

Some variants of the elimination method 35 making it possible to rearrange the problem statement and boundary con- ditions (61)- ...