variable boundaries
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1—10 of 28 matching pages
1: 28.10 Integral Equations
2: 1.4 Calculus of One Variable
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►For the functions discussed in the following DLMF chapters these two integration measures are adequate, as these special functions are analytic functions of their variables, and thus , and well defined for all values of these variables; possible exceptions being at boundary points.
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3: 12.17 Physical Applications
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►By using instead coordinates of the parabolic cylinder , defined by
…Setting and separating variables, we obtain
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►Buchholz (1969) collects many results on boundary-value problems involving PCFs.
Miller (1974) treats separation of variables by group theoretic methods.
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►For this topic and other boundary-value problems see Boyd (1973), Hillion (1997), Magnus (1941), Morse and Feshbach (1953a, b), Müller (1988), Ott (1985), Rice (1954), and Shanmugam (1978).
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4: 1.13 Differential Equations
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§1.13(iv) Change of Variables
… ►Elimination of First Derivative by Change of Independent Variable
… ►Assuming that satisfies un-mixed boundary conditions of the form …or periodic boundary conditions … ►For a regular Sturm-Liouville system, equations (1.13.26) and (1.13.29) have: (i) identical eigenvalues, ; (ii) the corresponding (real) eigenfunctions, and , have the same number of zeros, also called nodes, for as for ; (iii) the eigenfunctions also satisfy the same type of boundary conditions, un-mixed or periodic, for both forms at the corresponding boundary points. …5: 28.32 Mathematical Applications
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►If the boundary conditions in a physical problem relate to the perimeter of an ellipse, then elliptical coordinates are convenient.
These are given by
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28.32.2
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►This leads to integral equations and an integral relation between the solutions of Mathieu’s equation (setting , in (28.32.3)).
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28.32.6
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6: 10.73 Physical Applications
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►and on separation of variables we obtain solutions of the form , from which a solution satisfying prescribed boundary conditions may be constructed.
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7: 12.15 Generalized Parabolic Cylinder Functions
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12.15.1
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►This equation arises in the study of non-self-adjoint elliptic boundary-value problems involving an indefinite weight function.
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8: 20.13 Physical Applications
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►For , with real, (20.13.1) takes the form of a real-time diffusion equation
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20.13.2
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20.13.3
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►These two apparently different solutions differ only in their normalization and boundary conditions.
…Theta-function solutions to the heat diffusion equation with simple boundary conditions are discussed in Lawden (1989, pp. 1–3), and with more general boundary conditions in Körner (1989, pp. 274–281).
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9: 28.33 Physical Applications
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