inhomogeneous%20forms
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1: 1.13 Differential Equations
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§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
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►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.
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►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 and can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity.
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►Sequences of values of and , with fixed, can be computed by application of the inhomogeneous difference equations (11.4.23) and (11.4.25).
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3: 3.6 Linear Difference Equations
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►If , , then the difference equation is homogeneous; otherwise it is inhomogeneous.
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§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 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 first-order inhomogeneous equations, boundary-value methods are the rule rather than the exception. …4: 11.9 Lommel Functions
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►The inhomogeneous Bessel differential equation
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►the right-hand side being replaced by its limiting form when is an odd negative integer.
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►For uniform asymptotic expansions, for large and fixed , of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390).
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5: Bibliography K
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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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Methods of computing the Riemann zeta-function and some generalizations of it.
USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
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A vortex filament moving without change of form.
J. Fluid Mech. 112, pp. 397–409.
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Two new asymptotic methods in the theory of wave propagation in inhomogeneous media.
Sov. Phys. Acoust. 14, pp. 1–17.
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Quantum-Theoretical Formalism for Inhomogeneous Graded-Index Waveguides.
Akademie Verlag, Berlin-New York.
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6: Bibliography O
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On higher-order Stokes phenomena of an inhomogeneous linear ordinary differential equation.
J. Comput. Appl. Math. 169 (1), pp. 235–246.
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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series.
J. Inst. Math. Appl. 20 (3), pp. 379–391.
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A closed form solution of the -wave Bethe-Goldstone equation with an infinite repulsive core.
J. Math. Phys. 27 (4), pp. 1154–1158.
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7: Bibliography G
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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.
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On nonoscillating integrals for computing inhomogeneous Airy functions.
Math. Comp. 70 (235), pp. 1183–1194.
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Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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Mutual integrability, quadratic algebras, and dynamical symmetry.
Ann. Phys. 217 (1), pp. 1–20.
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General form of the quantum-defect theory.
Phys. Rev. A 19 (4), pp. 1485–1509.
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8: 3.7 Ordinary Differential Equations
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►If the differential equation is homogeneous, otherwise it is inhomogeneous.
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►(This can happen only for inhomogeneous equations.)
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►The remaining two equations are supplied by boundary conditions of the form
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►The latter is especially useful if the endpoint of is at , or if the differential equation is inhomogeneous.
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