By A. A. Samarskii, Petr N. Vabishchevich
ISBN-10: 3110196662
ISBN-13: 9783110196665
The most periods of inverse difficulties for equations of mathematical physics and their numerical answer equipment are thought of during this booklet that's meant for graduate scholars and specialists in utilized arithmetic, computational arithmetic, and mathematical modelling.
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Additional resources for Numerical Methods for Solving Inverse Problems of Mathematical Physics
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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.
Numerical Methods for Solving Inverse Problems of Mathematical Physics by A. A. Samarskii, Petr N. Vabishchevich
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