Laplace transform operations, Table

Since we are doing inverse transform using a look-up table, we need to break down any given transfer functions into smaller parts which match what the table has—what is called partial fractions. The time-domain function is the sum of the inverse transform of the individual terms, making use of the fact that Laplace transform is a linear operator. 

Having Laplace transformed a function or equation and then carried out certain manipulations in the Laplace domain, it is frequently desired to invert that Laplace domain expression back into the time domain so as to obtain the time domain solution to the problem under investigation. This operation, symbolically represented as Sf [f(s)], can usually be performed using the tables of functions and tremsforms referred to above as will be seen later, the problem of inversion can sometimes be circumvented. 

Laplace transforms can be used to transform this system of linear differential equations in the time domain into a system of linear equations in the Laplace domain. From the table of Laplace operations (Appendix I) we obtain... 

At this point, the inverse Laplace transformation of equation (6.5.19) is carried out. In order to do this, one must consult a table of Laplace transforms such as the one given in appendix A. Operating on each term individually, one obtains... 

Often PK equations for common inputs such as first-order absorption zero-order input, or constant-rate infusions are derived from the differential equations describing the kinetics. LSA offers a very attractive alternative to such derivations that is more direct and does not require the use of differential equations or Laplace transforms. The LSA derivations can be done simply by elementary convolution operations (see Tables 16.1 and 16.2) in conjunction with the input-response convolution relationship between concentration, c(t), and the rate of input,/(t) ... 

R. V. Churchill, Operational Mathematics, 3rd edn., McGraw-Hill, New York, 1972 G. E. Roberts and H. Kaufman, Tables of Laplace Transforms, Saunders, Philadelphia, 1966. 

The nature of the Laplace transform has now been sufficiently studied so that direct applications to solution to physicochemical problems are possible. Because the Laplace transform is a linear operator, it is not suitable for nonlinear problems. Moreover, it is a suitable technique only for initial-value problems. We have seen (Table 9.1) that certain classes of variable coefficient ODE can also be treated by Laplace transforms, so we are not constrained by the constant coefficient restriction required using Heaviside operators. 

The use of the Laplace transform is relatively simple using either Laplace transform tables or programs that make it possible to perform symbolic operations such as Maple or Mathematica. Application of the Laplace transform to solve current-voltage relations in electrical circuits will be illustrated in Sect. 2.8 on the impedance of electrical circuits. 

In the precomputer age the above-stated sequence of operations was performed, as a rule, using special tables of original functions and Laplace transforms. Modem mathematical packages, including Mathcad and Maple, are equipped with corresponding tools to perform the direct and inverse Laplace transforms. 

In this appendix some important mathematical methods are briefly outlined. These include Laplace and Fourier transformations which are often used in the solution of ordinary and partial differential equations. Some basic operations with complex numbers and functions are also outlined. Power series, which are useful in making approximations, are summarized. Vector calculus, a subject which is important in electricity and magnetism, is dealt with in appendix B. The material given here is intended to provide only a brief introduction. The interested reader is referred to the monograph by Kreyszig [1] for further details. Extensive tables relevant to these topics are available in the handbook by Abramowitz and Stegun [2]. 

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