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Difference–Differential Equations

Nature volume 162, page 334 (28 August 1948) | Download Citation

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Abstract

THE general linear homogeneous difference–differential equation with constant coefficients is where 0  μ m, 0νn, 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.

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Author information

Affiliations

  1. Department of Mathematics, University of Aberdeen. April 26.

    • E. M. WRIGHT

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DOI

https://doi.org/10.1038/162334a0

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