Wronskian determinant

The denominators in Eqs. 2.358 and 2.359 represent the negative of the so-called Wronskian determinant... 

Wronskian determinant, defined in Eq. 2.359 mass transfer rate of the species A, Eq. 2.120 coordinate axial coordinate... 

Because our interest is with second-order differential equations, two linearly independent solutions always arise (the Wronskian of solutions is non-zero [490], see Sect. 5) and requires two arbitrary constants to be fixed from the two boundary conditions imposed on p0(r, t) by the physics of the problem being modelled. These boundary conditions determine how much of each of the two linearly independent solutions of the homogeneous equation (317) must be added to the particular integral to ensure that the solution of eqn. (316) is consistent with the boundary conditions. In the next three sections, the method of deriving the particular integral from the two linearly independent solutions of the homogeneous equation are discussed. 

To determine the interplay between the spectral properties, both boundary conditions, we return to Weyl s theory [32]. The key quantity in Weyl s extension of the Sturm-Liouville problem to the singular case is the m-function or ra-matrix [32-36]. To define this quantity, we need the so-called Green s formula that essentially relates the volume integral over the product of two general solutions of Eq. (1), u and v with eigenvalue X and the Wronskian between the two solutions for more details, see Appendix C. The formulas are derived so that it immediately conforms to appropriate coordinate separation into the... 

The variational equations imply [r — t- =a 0 on each cell boundary ct/2. Given independent expansions ijr = E/. within each atomic cell, and (ct) = J2/1.L Ni(To)Pl on the global matching surface, the coefficients are determined by the implied variational equations. The surface matching theorem implies two independent Wronskian integral conditions for each atomic cell,... 

Other asymptotic forms consistent with unit Wronskian define different but equally valid Green functions, with different values of the asymptotic coefficient of u>i. In particular, if w k 2 exp i(kr — ln), this determines the outgoing-wave Green function, and the asymptotic coefficient of w is the single-channel F-matrix, F sin ij. This is the basis of the T-matrix method [342, 344], which has been used for electron-molecule scattering calculations [126], It is assumed that Avf is regular at the origin and that Ad vanishes more rapidly than r 2 for r — oo. 

The conditicm described by Equatirai 3.11 is called the Wronskian and is commonly written in the determinant form as [1]... 

Step 4 Determine if the solutions form a fundamental set by examining the Wronskian (Section 3.2) of these solutions... 

We use Eq. (30-9) for the transverse-field components and determine the constants from continuity of e j, h j, e j and h j at the interface. With the aid of the Wronskian of Eq. (37-77) this leads to the expressions in Table 25-3 for the ITE and ITM modes. The orthogonality and normalization of each radiation mode is identical to the corresponding free-space normalization of Table 25-2 for reasons given above. Alternatively, we can parallel the derivation of Nj(Q) in Section 25-7 using the radiation-mode fields. 

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