variable%20boundaries
(0.001 seconds)
21—30 of 682 matching pages
21: 1.13 Differential Equations
…
►
§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. …22: 25.12 Polylogarithms
23: 18.39 Applications in the Physical Sciences
…
►The solutions of (18.39.8) are subject to boundary conditions at and .
…
►The solutions (18.39.8) are called the stationary states as separation of variables in (18.39.9) yields solutions of form
…
►Namely the th eigenfunction, listed in order of increasing eigenvalues, starting at , has precisely nodes, as real zeros of wave-functions away from boundaries are often referred to.
…
►
§18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom
… ►Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. …24: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
…
►Ignoring the boundary value terms it follows that
…
►Other choices of boundary conditions, identical for and , and which also lead to the vanishing of the boundary terms in (1.18.26), each lead to a distinct self adjoint extension of .
…
►
Self-adjoint extensions of (1.18.28) and the Weyl alternative
… ►Boundary values and boundary conditions for the end point are defined in a similar way. If then there are no nonzero boundary values at ; if then the above boundary values at form a two-dimensional class. …25: 25.3 Graphics
…
►
…
26: 16.25 Methods of Computation
…
►Instead a boundary-value problem needs to be formulated and solved.
…
27: 10.3 Graphics
…
►
§10.3(i) Real Order and Variable
… ►§10.3(ii) Real Order, Complex Variable
… ► … ► ►§10.3(iii) Imaginary Order, Real Variable
…28: 28.10 Integral Equations
…
►
28.10.1
►
28.10.2
►
28.10.3
…
►
§28.10(iii) Further Equations
… ►For relations with variable boundaries see Volkmer (1983).29: 9.18 Tables
…
►
•
…
►
•
►
•
…
§9.18(ii) Real Variables
… ►Zhang and Jin (1996, p. 337) tabulates , , , for to 8S and for to 9D.
§9.18(iii) Complex Variables
… ►Sherry (1959) tabulates , , , , ; 20S.
30: 28.32 Mathematical Applications
…
►If the boundary conditions in a physical problem relate to the perimeter of an ellipse, then elliptical coordinates are convenient.
These are given by
…
►
28.32.2
…
►This leads to integral equations and an integral relation between the solutions of Mathieu’s equation (setting , in (28.32.3)).
…
►
28.32.6
…