Trail weighting function

The notation is that of Butler, Gordon and Malcolm. The reader unfamiliar with the power of the cascade approach to polymer statistics will gain a better insight into its fundamentals from a recent paper (cf. also v/hich generalises the earlier formulae to embrace, e.g., the particle scattering functions through the notion of a trail-weighting function, which is of wide utility. ... 

The finite approximations to be used in the discretization process have to be selected. In a finite difference method, approximations for the derivatives at the grid points have to be selected. In a finite volume method, one has to select the methods of approximating surface and volume integrals. In a weighted residual method, one has to select appropriate trail - and weighting functions. A compromise between simplicity, ease of implementation, accuracy and computational efficiency has to be made. For the low order finite difference- and finite volume methods, at least second order discretization schemes (both in time and space) are recommended. For the WRMs, high order approximations are normally employed. 

The key elements of the MWR are the expansion functions (also called the trail-, basis- or approximating functions) and the weight functions (also known as test functions). The trial functions are used as the basis functions for a truncated series expansion of the solution, which, when substituted into the differential equation, produces the residual. The test functions are used to ensure that the differential equation is satisfied as closely as possible by the truncated series expansion. This is achieved by minimizing the residual, i.e., the error in the differential equation produced by using the truncated expansion instead of the exact solution, with respect to a suitable norm. An equivalent requirement is that the residual satisfy a suitable orthogonality condition with respect to each of the test functions. 

In one view the choice of trial functions is one of the features which distinguishes the spectral methods (SMs) from the spectral element Methods (SEMs). The finite element methods (FEMs) can thus be regarded as SEMs with linear expansion- and weight functions. The trial functions for spectral methods are infinitely differentiable global functions. In the case of spectral element methods, the domain is divided into small elements, and the trail function is specified in each element. The trial and test functions are thus local in character, and well suited for handling complex geometries. 

Fig. 28 The gel partition coefficient as a function of palmitoyl CoA concentration. The circles indicate the weight average partition coefficients as determined from the movement of the trailing edge of a large zone of palmitoyl-CoA as measured by direct optical gel scanning chromatography. The solid was estimated by using aggregation number, m, as 35 and K = 1.52-10 [78]. (Redrawn with changes)...

---