# initial-value problems

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## 6 matching pages

##### 1: 28.33 Physical Applications

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Initial-value problems, in which only one equation (28.2.1) or (28.20.1) is involved. See §28.33(iii).

###### §28.33(iii) Stability and Initial-Value Problems

… ►References for other initial-value problems include: …##### 2: 3.7 Ordinary Differential Equations

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###### §3.7(ii) Taylor-Series Method: Initial-Value Problems

… ►It will be observed that the present formulation of the Taylor-series method permits considerable parallelism in the computation, both for initial-value and boundary-value problems. …##### 3: Bibliography T

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Error bounds for the Liouville-Green approximation to initial-value problems.
Z. Angew. Math. Mech. 58 (12), pp. 529–537.
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##### 4: 28.34 Methods of Computation

##### 5: 6.18 Methods of Computation

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►For small or moderate values of $x$ and $|z|$, the expansion in power series (§6.6) or in series of spherical Bessel functions (§6.10(ii)) can be used.
…However, this problem is less severe for the series of spherical Bessel functions because of their more rapid rate of convergence, and also (except in the case of (6.10.6)) absence of cancellation when $z=x$ ($>0$).
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${A}_{0}$, ${B}_{0}$, and ${C}_{0}$ can be computed by Miller’s algorithm (§3.6(iii)), starting with initial values
$({A}_{N},{B}_{N},{C}_{N})=(1,0,0)$, say, where $N$ is an arbitrary large integer, and normalizing via ${C}_{0}=1/z$.
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►Zeros of $\mathrm{Ci}\left(x\right)$ and $\mathrm{si}\left(x\right)$ can be computed to high precision by Newton’s rule (§3.8(ii)), using values supplied by the asymptotic expansion (6.13.2) as initial approximations.
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##### 6: 3.6 Linear Difference Equations

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►In practice, however, problems of severe instability often arise and in §§3.6(ii)–3.6(vii) we show how these difficulties may be overcome.
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►We first compute, by forward recurrence, the solution ${p}_{n}$ of the homogeneous equation (3.6.3) with initial values
${p}_{0}=0$, ${p}_{1}=1$.
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►The least value of $N$ that satisfies (3.6.9) is found to be 16.
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►For a difference equation of order $k$ ($\ge 3$),
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