By Martin, Edward Editor

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**Extra info for Elements of Mathematics Book B EM Problem Book Volume II: Answer Key**

**Sample text**

69) we have: yi ≥ 0, i = 1, 2, . . , N − 1 (yi ≤ 0, i = 1, 2, . . , N − 1). Proof. Let us follow the line of reasoning assuming the opposite. 51) with non-negative right-hand side and non-negative boundary conditions be not non-negative at all nodes of the grid. We designate as k the grid node at which the solution assumes the least negative value. If such a value is attained at several nodes, then we choose the node where yk−1 > yk . We write the difference equation at this node: −αk yk−1 + γk yk − βk yk+1 = ϕk .

Sin2 2 h 2l l Let us obtain such a lower estimate for A based on the following difference Friedrichs inequality. 1 For all mesh functions vanishing on ∂ω, the following inequality holds: y 2 ≤ M0 ( yx¯ + 2 ), w + ≡ ((w, w)+ )1/2 , M0 = l 2 /8. 2 29 Convergence of difference schemes Proof. 39) yx,i ¯ h. 39), we use the inequality 2 a k bk ≤ bk2 . 40) k=1 N yi2 ≤ (l − xi ) 2 (yx,k ¯ ) h. 41) k=i+1 Let us consider n = (N − 1)/2 if, for instance, N is an odd number. The case in which N is even will be considered separately.

We assign to evolutionary inverse problems inverse problems in which initial conditions (lacking in formulation of the problem as a direct problem) need to be identiﬁed. 48), a simplest evolutionary inverse problem can be formulated as follows. 48) are not speciﬁed; instead, the solution at the end time t = T is known: u(x, T ) = u T (x), 0 < x < l. 46) at preceding times (retrospective inverse problem). 50)). 1 On the function set v(0) = 0, v(l) = 0, we deﬁne the operator Au = − ∂ ∂u k(x) , ∂x ∂x 0 < x < l.

### Elements of Mathematics Book B EM Problem Book Volume II: Answer Key by Martin, Edward Editor

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