New theorems of association of variables in multiple dimensional Laplace transform

Abstract

In non-linear systems analysis, multiple dimensional Laplace transform is often applied in solving the voltcrra model. Tho special technique for the inverse Laplaco transform solution is called thc association ol variables. Throe new theorems are developed for the theory of association. Compared with the inspection method of Brilliant, the pair listing method of Lubbock, the new approach is much simpler and easier and more systematic. Several illusLrative examples are included.

Two-dimensional analog signal processing and its implications on circuit theory

“…A system is said to be linear autonomous if it satisfies both properties (4) and (5). Clearly, for autonomous systems, the origin of time variables can always be taken to be zero.…”

“…Using definition (22), the general initial condition of a function h 2 t; s ð Þ; is obtained as: Nevertheless, obtaining the result of (23) is not, in general, an easy task. The conversion of H 2 s 1 ; s 2 ð Þ into H 1 s ð Þ in which H 1 s ð Þ denotes the common one-dimensional Laplace transform of h 1 s ð Þ can be performed using techniques of association of variables, as described in the literature [5]. A number of tables have been produced for this purpose and can be easily used.…”

“…Under these assumptions, the operator T, in essence, is a set of integrodifferential equations relating the output to the input and an auxiliary equation that satisfies the initial conditions. In system theory, the procedure of associating a number of independent variables together is called association of variables [5]. If it is possible to associate t and s in the t; s ð Þ-domain, the system operator T can be considered to be a Volterra functional operator [6], which, in general, represents a nonlinear dynamic system.…”

Numerical inversion of multidimensional Laplace transforms via block-pulse functions

“…The block-pulse coefficients of the exact solution obtained from the associationvariable method [20,21] 1 1 G ( 5 ) ~ s + a + b s + 2…”

Associated transforms for solution of nonlinear equations

“…This special method of computing the inverse transform is said to be the association of variables. The function G(s) is called the associated transform of F(sl s2, Recently, Chert and Chiu [6] and Koh [7] …”