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1: 1.13 Differential Equations
§1.13(iii) Inhomogeneous Equations
The inhomogeneous (or nonhomogeneous) equation …
Variation of Parameters
§1.13(vii) Closed-Form Solutions
§1.13(viii) Eigenvalues and Eigenfunctions: Sturm-Liouville and Liouville forms
2: 11.13 Methods of Computation
A comprehensive approach is to integrate the defining inhomogeneous differential equations (11.2.7) and (11.2.9) numerically, using methods described in §3.7. … Then from the limiting forms for small argument (§§11.2(i), 10.7(i), 10.30(i)), limiting forms for large argument (§§11.6(i), 10.7(ii), 10.30(ii)), and the connection formulas (11.2.5) and (11.2.6), it is seen that 𝐇 ν ( x ) and 𝐋 ν ( x ) can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity. … Sequences of values of 𝐇 ν ( z ) and 𝐋 ν ( z ) , with z fixed, can be computed by application of the inhomogeneous difference equations (11.4.23) and (11.4.25). …
3: 3.6 Linear Difference Equations
If d n = 0 , n , then the difference equation is homogeneous; otherwise it is inhomogeneous. …
§3.6(iv) Inhomogeneous Equations
It is applicable equally to the computation of the recessive solution of the homogeneous equation (3.6.3) or the computation of any solution w n of the inhomogeneous equation (3.6.1) for which the conditions of §3.6(iv) are satisfied. … Thus in the inhomogeneous case it may sometimes be necessary to recur backwards to achieve stability. … or for systems of k first-order inhomogeneous equations, boundary-value methods are the rule rather than the exception. …
4: 11.9 Lommel Functions
The inhomogeneous Bessel differential equation … the right-hand side being replaced by its limiting form when μ ± ν is an odd negative integer. … For uniform asymptotic expansions, for large ν and fixed μ = 1 , 0 , 1 , 2 , , of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390). … …
5: Bibliography K
  • R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.
  • M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
  • S. Kida (1981) A vortex filament moving without change of form. J. Fluid Mech. 112, pp. 397–409.
  • Y. A. Kravtsov (1968) Two new asymptotic methods in the theory of wave propagation in inhomogeneous media. Sov. Phys. Acoust. 14, pp. 1–17.
  • S. G. Krivoshlykov (1994) Quantum-Theoretical Formalism for Inhomogeneous Graded-Index Waveguides. Akademie Verlag, Berlin-New York.
  • 6: Bibliography O
  • A. B. Olde Daalhuis (2004b) On higher-order Stokes phenomena of an inhomogeneous linear ordinary differential equation. J. Comput. Appl. Math. 169 (1), pp. 235–246.
  • J. Oliver (1977) An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.
  • M. K. Ong (1986) A closed form solution of the s -wave Bethe-Goldstone equation with an infinite repulsive core. J. Math. Phys. 27 (4), pp. 1154–1158.
  • 7: Bibliography G
  • W. Gautschi (1994) Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.
  • A. Gil, J. Segura, and N. M. Temme (2001) On nonoscillating integrals for computing inhomogeneous Airy functions. Math. Comp. 70 (235), pp. 1183–1194.
  • A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
  • Ya. I. Granovskiĭ, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.
  • C. H. Greene, U. Fano, and G. Strinati (1979) General form of the quantum-defect theory. Phys. Rev. A 19 (4), pp. 1485–1509.
  • 8: 3.7 Ordinary Differential Equations
    If h = 0 the differential equation is homogeneous, otherwise it is inhomogeneous. … … (This can happen only for inhomogeneous equations.) … The remaining two equations are supplied by boundary conditions of the formThe latter is especially useful if the endpoint b of 𝒫 is at , or if the differential equation is inhomogeneous. …
    9: 11.2 Definitions
    §11.2(ii) Differential Equations
    Modified Struve’s Equation
    10: 9.1 Special Notation
    k nonnegative integer, except in §9.9(iii).
    The main functions treated in this chapter are the Airy functions Ai ( z ) and Bi ( z ) , and the Scorer functions Gi ( z ) and Hi ( z ) (also known as inhomogeneous Airy functions). …