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boundary-value methods


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1: 3.6 Linear Difference Equations
However, w n can be computed successfully in these circumstances by boundary-value methods, as follows. … For a difference equation of order k ( 3 ), …or for systems of k first-order inhomogeneous equations, boundary-value methods are the rule rather than the exception. …
2: 28.34 Methods of Computation
  • (c)

    Solution of (28.2.1) by boundary-value methods; see §3.7(iii). This can be combined with §28.34(ii)(c).

  • (d)

    Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

  • 3: 11.13 Methods of Computation
    For M ν ( x ) both forward and backward integration are unstable, and boundary-value methods are required (§3.7(iii)). … In consequence forward recurrence, backward recurrence, or boundary-value methods may be necessary. …
    4: 12.17 Physical Applications
    By using instead coordinates of the parabolic cylinder ξ , η , ζ , defined by …
    5: 9.17 Methods of Computation
    In these cases boundary-value methods need to be used instead; see §3.7(iii). …
    6: 3.7 Ordinary Differential Equations
    §3.7(iii) Taylor-Series Method: Boundary-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. … General methods for boundary-value problems for ordinary differential equations are given in Ascher et al. (1995). …
    7: 16.25 Methods of Computation
    §16.25 Methods of Computation
    Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. …Instead a boundary-value problem needs to be formulated and solved. …
    8: Bibliography
  • G. E. Andrews (1996) Pfaff’s method II: Diverse applications. J. Comput. Appl. Math. 68 (1-2), pp. 15–23.
  • T. M. Apostol and H. S. Zuckerman (1951) On magic squares constructed by the uniform step method. Proc. Amer. Math. Soc. 2 (4), pp. 557–565.
  • G. B. Arfken and H. J. Weber (2005) Mathematical Methods for Physicists. 6th edition, Elsevier, Oxford.
  • V. I. Arnol d (1997) Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, Vol. 60, Springer-Verlag, New York.
  • U. M. Ascher, R. M. M. Mattheij, and R. D. Russell (1995) Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Classics in Applied Mathematics, Vol. 13, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
  • 9: Bibliography J
  • H. Jeffreys and B. S. Jeffreys (1956) Methods of Mathematical Physics. 3rd edition, Cambridge University Press, Cambridge.
  • D. S. Jones (2001) Asymptotics of the hypergeometric function. Math. Methods Appl. Sci. 24 (6), pp. 369–389.
  • 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.
  • 10: Bibliography S
  • D. Schmidt and G. Wolf (1979) A method of generating integral relations by the simultaneous separability of generalized Schrödinger equations. SIAM J. Math. Anal. 10 (4), pp. 823–838.
  • J. Segura (1998) A global Newton method for the zeros of cylinder functions. Numer. Algorithms 18 (3-4), pp. 259–276.
  • A. Sidi (2003) Practical Extrapolation Methods: Theory and Applications. Cambridge Monographs on Applied and Computational Mathematics, Vol. 10, Cambridge University Press, Cambridge.
  • B. D. Sleeman (1966a) Some Boundary Value Problems Associated with the Heun Equation. Ph.D. Thesis, London University.
  • I. N. Sneddon (1966) Mixed Boundary Value Problems in Potential Theory. North-Holland Publishing Co., Amsterdam.