Difference–Differential Equations
Difference–Differential Equations
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差分-微分方程
作者：
E. M. Wright
一般的常系数线性齐次差分微分方程是其中0 ≤ μ m，0 ≤ v ≤ n，y（v）（t）是未知函数y（t）的v阶导数，0 = b 0 < b1 <...< bm.这个方程的具体例子出现在放射学1，2，经济学3，4和控制机制的理论5，6。最有用的“边界条件”也是从理论的角度来看最方便的;我们假设在初始区间0 ≤ t < bm中指定y（t）的值。根据这些给定的值，我们定义了一个函数（在特定情况下，这通常会简化为相当简单的东西）。显然，y = exp st是（1）的解，对于任何s满足。τ（S）的零点是无限多的;但它们的渐近行为是很容易计算的。在适当的条件下，（1）的解是其中s穿过τ（s）的所有零点。这里我假设τ（s）没有双零;如果有，则必须在相应项中做轻微的修改。（2）中的级数是收敛的，它的和是y（t）（i）对所有t，如果amn <$0和a0 n <$0，和（ii）对所有t > bm，如果amn =<$0。(2)最初是由Hilb 8给出的，但在排除大多数应用的条件下。在所述条件下其有效性的详细证明将在短期内公布9。
THE general linear homogeneous difference–differential equation with constant coefficients is where 0 ≤ μ m, 0 ≤ ν ≤ n, y(ν)(t) is the ν-th derivative of the unknown function y(t) and 0 = b0 < b1 < … < bm. Particular examples of this equation have appeared in radiology1,2, economics3,4 and the theory of control mechanisms5,6. The most useful ‘boundary conditions’ are also the most convenient from the theoretical point of view; We suppose assigned the values of y(t) in an initial interval 0 ≤ t < bm. In terms of these given values, we define a function (In particular cases this usually reduces to something fairly simple.) It is obvious that y = exp st is a solution of (1) for any s satisfying . The zeros of τ(S) are infinite in number; but their asymptotic behaviour is readily calculable7. Under suitable conditions, the solution of (1) is where s runs through all the zeros of τ(s). I here assume that τ(s) has no double zero; if it has, a slight modification must be made in the corresponding term. The series in (2) is convergent and its sum is y(t) (i) for all t, if amn ≠ 0 and a0n ≠ 0, and (ii) for all t > bm, if amn = ≠ 0. (2) Was first given by Hilb8, but under conditions which would exclude most of the applications. A detailed proof of its validity under the conditions stated will be published shortly9.