# stability

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## 1—10 of 17 matching pages

##### 1: 29.9 Stability

###### §29.9 Stability

…##### 2: 28.17 Stability as $x\to \pm \mathrm{\infty}$

###### §28.17 Stability as $x\to \pm \mathrm{\infty}$

… ►##### 3: Bernard Deconinck

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►He has worked on integrable systems, algorithms for computations with Riemann surfaces, Bose-Einstein condensates, and methods to investigate the stability of solutions of nonlinear wave equations.
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##### 4: 9.16 Physical Applications

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►In the study of the stability of a two-dimensional viscous fluid, the flow is governed by the Orr–Sommerfeld equation (a fourth-order differential equation).
…These examples of transitions to turbulence are presented in detail in Drazin and Reid (1981) with the problem of hydrodynamic stability.
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##### 5: 11.13 Methods of Computation

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►To insure stability the integration path must be chosen so that as we proceed along it the wanted solution grows in magnitude at least as rapidly as the complementary solutions.
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►There are similar problems to those described in §11.13(iv) concerning stability.
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##### 6: 28.33 Physical Applications

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

►If the parameters of a physical system vary periodically with time, then the question of stability arises, for example, a mathematical pendulum whose length varies as $\mathrm{cos}\left(2\omega t\right)$. …##### 7: 10.74 Methods of Computation

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►As described in §3.7(ii), to insure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows in magnitude at least as fast as all other solutions of the differential equation.
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►Similarly, to maintain stability in the interval $$ the integration direction has to be forwards in the case of ${I}_{\nu}\left(x\right)$ and backwards in the case of ${K}_{\nu}\left(x\right)$, with initial values obtained in an analogous manner to those for ${J}_{\nu}\left(x\right)$ and ${Y}_{\nu}\left(x\right)$.
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►Then ${J}_{n}\left(x\right)$ and ${Y}_{n}\left(x\right)$ can be generated by either forward or backward recurrence on $n$ when $$, but if $n>x$ then to maintain stability
${J}_{n}\left(x\right)$ has to be generated by backward recurrence on $n$, and ${Y}_{n}\left(x\right)$ has to be generated by forward recurrence on $n$.
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##### 8: 9.17 Methods of Computation

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►As described in §3.7(ii), to ensure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows at least as fast as all other solutions of the differential equation.
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##### 9: 3.6 Linear Difference Equations

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►Stability can be restored, however, by

*backward recursion*, provided that ${c}_{n}\ne 0$, $\forall n$: starting from ${w}_{N}$ and ${w}_{N+1}$, with $N$ large, equation (3.6.3) is applied to generate in succession ${w}_{N-1},{w}_{N-2},\mathrm{\dots},{w}_{0}$. … … ►Thus in the inhomogeneous case it may sometimes be necessary to recur backwards to achieve stability. …