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1: 28.33 Physical Applications
§28.33(iii) Stability and Initial-Value Problems
2: 9.16 Physical Applications
These examples of transitions to turbulence are presented in detail in Drazin and Reid (1981) with the problem of hydrodynamic stability. …
3: 11.13 Methods of Computation
There are similar problems to those described in §11.13(iv) concerning stability. …
4: 3.7 Ordinary Differential Equations
§3.7(ii) Taylor-Series Method: Initial-Value Problems
§3.7(iii) Taylor-Series Method: Boundary-Value Problems
§3.7(iv) Sturm–Liouville Eigenvalue Problems
This converts the problem into a tridiagonal matrix problem in which the elements of the matrix are polynomials in λ ; compare §3.2(vi). …
5: 3.6 Linear Difference Equations
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. … Thus in the inhomogeneous case it may sometimes be necessary to recur backwards to achieve stability. … For a difference equation of order k ( 3 ), …
6: 3.10 Continued Fractions
In contrast to the preceding algorithms in this subsection no scaling problems arise and no a priori information is needed. … 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. Again, no scaling problems arise and no a priori information is needed. …
7: Bibliography H
  • E. Hairer, S. P. Nørsett, and G. Wanner (1993) Solving Ordinary Differential Equations. I. Nonstiff Problems. 2nd edition, Springer Series in Computational Mathematics, Vol. 8, Springer-Verlag, Berlin.
  • E. Hairer, S. P. Nørsett, and G. Wanner (2000) Solving Ordinary Differential Equations. I. Nonstiff Problems. 2nd edition, Springer-Verlag, Berlin.
  • E. Hairer and G. Wanner (1996) Solving Ordinary Differential Equations. II. Stiff and Differential-Algebraic Problems. 2nd edition, Springer Series in Computational Mathematics, Vol. 14, Springer-Verlag, Berlin.
  • G. H. Hardy and J. E. Littlewood (1925) Some problems of “Partitio Numerorum” (VI): Further researches in Waring’s Problem. Math. Z. 23, pp. 1–37.
  • H. Hochstadt (1963) Estimates of the stability intervals for Hill’s equation. Proc. Amer. Math. Soc. 14 (6), pp. 930–932.
  • 8: Bibliography J
  • W. B. Jones and W. J. Thron (1974) Numerical stability in evaluating continued fractions. Math. Comp. 28 (127), pp. 795–810.
  • N. Joshi and A. V. Kitaev (2005) The Dirichlet boundary value problem for real solutions of the first Painlevé equation on segments in non-positive semi-axis. J. Reine Angew. Math. 583, pp. 29–86.
  • N. Joshi and M. D. Kruskal (1992) The Painlevé connection problem: An asymptotic approach. I. Stud. Appl. Math. 86 (4), pp. 315–376.
  • 9: Bibliography D
  • D. Dai, M. E. H. Ismail, and X. Wang (2014) Plancherel-Rotach asymptotic expansion for some polynomials from indeterminate moment problems. Constr. Approx. 40 (1), pp. 61–104.
  • K. Dekker and J. G. Verwer (1984) Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations. CWI Monographs, Vol. 2, North-Holland Publishing Co., Amsterdam.
  • R. C. Desai and M. Nelkin (1966) Atomic motions in a rigid sphere gas as a problem in neutron transport. Nucl. Sci. Eng. 24 (2), pp. 142–152.
  • P. G. Drazin and W. H. Reid (1981) Hydrodynamic Stability. Cambridge University Press, Cambridge.
  • 10: Bibliography B
  • P. Baldwin (1991) Coefficient functions for an inhomogeneous turning-point problem. Mathematika 38 (2), pp. 217–238.
  • W. G. Bickley (1935) Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.
  • Å. Björck (1996) Numerical Methods for Least Squares Problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
  • P. Bleher and A. Its (1999) Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model. Ann. of Math. (2) 150 (1), pp. 185–266.
  • J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow (2001) Stability of repulsive Bose-Einstein condensates in a periodic potential. Phys. Rev. E (3) 63 (036612), pp. 1–11.