Numerical Scheme for the Solution to Laplace\u27s Equation using Local Conformal Mapping Techniques

This paper introduces a method to determine the pressure in a fixed thickness, smooth, periodic domain; namely a lead-over-pleat cartridge filter. Finding the pressure within the domain requires the numerical solution of Laplace\u27s equation, the first step of which is approximating, by interpolation, the curved portions of the filter to a circle in the xy plane.A conformal map is then applied to the filter, transforming the region into a rectangle in the uv plane. A finite difference method is introduced to numerically solve Laplace\u27s equation in the rectangular domain. There are currently methods in existence to solve partial differential equations on non- regular domains. In a method employed by Monchmeyer and Muller, a scheme is used to transform from cartesian to spherical polar coordinates. Monchmeyer and Muller stress that for non-linear domains, extrapolation of existing cartesian difference schemes may produce incorrect solutions, and therefore, a volume centered discretization is used. A difference scheme is then derived that relies on mean values. This method has second order accuracy.(Rosenfeld,Moshe, Kwak, Dochan, 1989) The method introduced in this paper is based on a 7-point stencil which takes into account the unequal spacing of the points. From all neighboring pairs, a linear system of equations is constructed, which takes into account the periodic domain.This method is solved by standard iterative methods. The solution is then mapped back to the original domain, with second order accuracy. The method is then tested to obtain a solution to a domain which satisfies $y=sin(x)$ at the center, a shape similar to that of a lead-over-pleat cartridge filter. As a result, a model for the pressure distribution within the filter is obtained

Status of the differential transformation method

Further to a recent controversy on whether the differential transformation method (DTM) for solving a differential equation is purely and solely the traditional Taylor series method, it is emphasized that the DTM is currently used, often only, as a technique for (analytically) calculating the power series of the solution (in terms of the initial value parameters). Sometimes, a piecewise analytic continuation process is implemented either in a numerical routine (e.g., within a shooting method) or in a semi-analytical procedure (e.g., to solve a boundary value problem). Emphasized also is the fact that, at the time of its invention, the currently-used basic ingredients of the DTM (that transform a differential equation into a difference equation of same order that is iteratively solvable) were already known for a long time by the "traditional"-Taylor-method users (notably in the elaboration of software packages --numerical routines-- for automatically solving ordinary differential equations). At now, the defenders of the DTM still ignore the, though much better developed, studies of the "traditional"-Taylor-method users who, in turn, seem to ignore similarly the existence of the DTM. The DTM has been given an apparent strong formalization (set on the same footing as the Fourier, Laplace or Mellin transformations). Though often used trivially, it is easily attainable and easily adaptable to different kinds of differentiation procedures. That has made it very attractive. Hence applications to various problems of the Taylor method, and more generally of the power series method (including noninteger powers) has been sketched. It seems that its potential has not been exploited as it could be. After a discussion on the reasons of the "misunderstandings" which have caused the controversy, the preceding topics are concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages, references and further considerations adde