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By J. A. Richards

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First fractional exponentiation is required in the case of a Bessel function of fractional order. This is a time-consuming operation, computationally. Secondly it is difficult, with the series definition, to obtain an estimate of the error made by truncating the series at a particular term. Both of these difficulties are circumvented fortunately by the use of Chebyshev series techniques for functionally approximating the required Bessel functions. This method is outlined in the Appendix. 24) As an illustration, Eq.

5. In this chapter equations which are tractable are treated in depth using the matrix approach laid down in Chap. 2. Methods for handling homogeneous equations are dealt with first whilst particular integrals are considered in a later section. 17) describe a particularly important property of periodically timevarying equations, viz. that once the solution to a homogeneous form is known over one period (presumably 0 ::; t < e) then it can be found easily for all time, simply by matrix multiplication with the discrete transition matrix C =

The characteristic exponents are just the eigenvalues of the matrix r, whilst the eigenvalues of the discrete transition matrix C = ¢(e, 0) are sometimes called characteristic multipliers. 22) Ai = exp (Jlie). 23b) = det{¢(e, O)}. 23c) In Eq. 23b) is the average value of gv-1 (t), over a period e. It is important to realise that the v entries in Eq. 21) are the v linearly independent basis solutions. Should a number of Jli be degenerate, t-multiplied forms are adopted, in the usual manner.

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