By George Anastassiou
Comprises the lawsuits of the March 1991 annual convention of the Southeastern Approximation Theorists, in Memphis, Tenn. The 34 papers talk about subject matters of curiosity to graduate numerical analysts, utilized and commercial mathematicians, engineers, and different scientists equivalent to splines
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7. 51(∆x)2 (bottom). We shall brieﬂy justify this stability condition (a more through analysis will be carried out in the next chapter). Rewriting the explicit scheme in the form un+1 = j ν∆t n ν∆t u + 1−2 (∆x)2 j−1 (∆x)2 unj + ν∆t n u . 32) are positive and their sum is 1). In particular, if the initial data u0 is bounded by two constants m and M such that m ≤ u0j ≤ M for all j ∈ Z, then a recurrence easily shows that the same inequalities remain true for all time m ≤ unj ≤ M for all j ∈ Z and for all n ≥ 0.
The problem is therefore to ﬁnd u, the solution of A(u) = f. 39) is well-posed if for all data f it has a unique solution u, and if this solution u depends continuously on the data f . Let us examine Hadamard’s deﬁnition in detail: it contains, in fact, three conditions for the problem to be well-posed. First, a solution must at least exist: this is the least we can ask of a model supposed to represent reality! Second, the solution must be unique: this is more delicate since, while it is clear that, if we want to predict tomorrow’s weather, it is better to have ‘sun’ or ‘rain’ (with an exclusive REMARKS ON MATHEMATICAL MODELS 27 ‘or’) but not both with equal chance, there are other problems which ‘reasonably’ have several or an inﬁnity of solutions.
24) is a fourth order equation. We often distinguish between the order with respect to the time variable t and with respect to the space variable x. 18) is second order in space-time. In order to understand the vocabulary often used with PDEs, that is, elliptic, parabolic, or hyperbolic, we shall brieﬂy classify linear, second order PDEs acting on real functions of two real variables u(x, y) (we shall not carry out a systematic classiﬁcation for all PDEs). Such an equation is written a ∂2u ∂2u ∂u ∂2u ∂u +b +c 2 +d +e + f u = g.
Approximation Theory by George Anastassiou