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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: Brian D. Sleeman
     thesis was Some Boundary Value Problems Associated with the Heun Equation. … Sleeman was elected a Fellow of the Royal Society of Edinburgh in 1976 and is the founding editor of the journal Computational and Mathematical Methods in Medicine. …
    9: 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.
  • 10: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
    Ignoring the boundary value terms it follows that
    Self-adjoint extensions of (1.18.28) and the Weyl alternative
    A boundary value for the end point a is a linear form on 𝒟 ( * ) of the formwhere α and β are given functions on X , and where the limit has to exist for all f . Then, if the linear form is nonzero, the condition ( f ) = 0 is called a boundary condition at a . Boundary values and boundary conditions for the end point b are defined in a similar way. If n 1 = 1 then there are no nonzero boundary values at a ; if n 1 = 2 then the above boundary values at a form a two-dimensional class. Similarly at b . Any self-adjoint extension of can be obtained by restricting * to those f 𝒟 ( * ) for which, if n 1 = 2 , 1 ( f ) = 0 for a chosen 1 at a and, if n 2 = 2 , 2 ( f ) = 0 for a chosen 2 at b .The above results, especially the discussions of deficiency indices and limit point and limit circle boundary conditions, lay the basis for further applications. The reader is referred to Coddington and Levinson (1955), Friedman (1990, Ch. 3), Titchmarsh (1962a), and Everitt (2005b, pp. 45–74) and Everitt (2005a, pp. 272–331), for detailed methods and results.