Laplace object function

Figure 14.5. Comparison between experimental points and a Laplacian curve. The curve u = u s) is a theoretical profile based on the Laplace equation of capillarity, while points Vi, i = 1,2,N, are points selected from the meridian of an experimental drop profile. The deviation of the ith point from the Laplacian curve, di, can be calculated. The purpose of the objective function of ADSA is to minimize the sum of the square of the minimum distance, di. Since the coordinate systems of the experimental profile and the predicted Laplacian curve do not necessarily coincide, their offset xq, zo) and rotation angle (a) must be considered. Both of the latter are optimization parameters. Since the program does not require the coordinates of the drop apex as input, the drop can be measured from any convenient reference frame, and all measured points on a drop profile are equally important (from ref. (36))...

The objective of this study is to develop an analytical model for a soil column s response to a sinusoidally varying tracer loading function by applying the familiar Laplace transform method in which the convolution integral is used to obtain the inverse transformation. The solution methodology will use Laplace transforms and their inverses that are available in most introductory texts on Laplace transforms to develop both the quasi steady-state and unsteady-state solutions. Applications of the solutions will be listed and explained. 

In summary, Laplace s equation must be satisfied by the scalar velocity potential and the stream function for all two-dimensional planar flows that lack an axis of symmetry. The Laplacian operator is replaced by the operator to calculate the stream function for two-dimensional axisymmetric flows. For potential flow transverse to a long cylinder, vector algebra is required to determine the functional form of the stream function far from the submerged object. This is accomplished from a consideration of Vr and vg via equation (8-255) ... 

Determination of both the transmittance of the investigated object and the Laplace transform of the input functiony(r) furnishes the output function x(s)= y(s)- H(s). With the inverse transformation, we obtain y(t). Output functions x(t) of proportional, integrating and inertial objects for various input functions are collected in Table 2.1. 

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