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1: 29.9 Stability
§29.9 Stability
2: 28.17 Stability as x ±
§28.17 Stability as x ±
See accompanying text
Figure 28.17.1: Stability chart for eigenvalues of Mathieu’s equation (28.2.1). Magnify
3: Bernard Deconinck
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. …
4: 9.16 Physical Applications
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. …
5: 11.13 Methods of Computation
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. … There are similar problems to those described in §11.13(iv) concerning stability. …
6: 28.33 Physical Applications
§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 cos ( 2 ω t ) . …
7: 10.74 Methods of Computation
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. … Similarly, to maintain stability in the interval 0 < x < the integration direction has to be forwards in the case of I ν ( x ) and backwards in the case of K ν ( x ) , with initial values obtained in an analogous manner to those for J ν ( x ) and Y ν ( x ) . … Then J n ( x ) and Y n ( x ) can be generated by either forward or backward recurrence on n when n < x , but if n > x then to maintain stability J n ( x ) has to be generated by backward recurrence on n , and Y n ( x ) has to be generated by forward recurrence on n . …
8: 9.17 Methods of Computation
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. …
9: 3.6 Linear Difference Equations
Stability can be restored, however, by backward recursion, provided that c n 0 , 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 , , w 0 . … … Thus in the inhomogeneous case it may sometimes be necessary to recur backwards to achieve stability. …
10: 3.10 Continued Fractions
This forward algorithm achieves efficiency and stability in the computation of the convergents C n = A n / B n , and is related to the forward series recurrence algorithm. …