Laplace transform inversion, Tables

This solution is then inversely transformed by use of a table of Laplace transforms (see Table 5.3) to obtain the desired solution ... 

The final step is to perform inverse Laplace transformation on c x, s) to obtain c x, t). Using the table of Laplace transformations, inverse transformation of equation (6.5.27) results in... 

The differential equation was solved only by transformation, which can be done using Laplace transform tables, the solution of an algebraic equation, and the inverse Laplace transform using tables. Tables of Laplace transforms can be easily... 

A short table (Table 3-1) of very common Laplace transforms and inverse transforms follows. The references include more detailed tables. NOTE F(/i -1- 1) = Iq x e dx (gamma function) /(f) = Bessel function of the first land of order n. 

Equation (8-14) shows that starts from 0 and builds up exponentially to a final concentration of Kcj. Note that to get Eq. (8-14), it was only necessaiy to solve the algebraic Eq. (8-12) and then find the inverse of C (s) in Table 8-1. The original differential equation was not solved directly. In general, techniques such as partial fraction expansion must be used to solve higher order differential equations with Laplace transforms. 

In practice, inverse transformation is most easily achieved by using partial fractions to break down solutions into standard components, and then use tables of Laplace transform pairs, as given in Table 3.1. 

To take the inverse Laplace transform means to reverse the process of taking the transform, and for this purpose a table of transforms is valuable. To illustrate, we consider a simple first-order reaction, whose differential rate equation is... 

The temperature 0 is then obtained from the tables of inverse Laplace transforms in the Appendix (Table 12, No 83) and is given by. 

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. 

In practice, the inverse Laplace transformations are obtained by reference to the rather extensive tables that are available. It is sometimes useful to develop the function in question in partial fractions, as employed in Section 3.3.3. The resulting sura of integrals can often be evaluated with the use of the tables. 

The pair of differential equations are solved either directly or with Laplace transform, with a table of inverses. 

Inversion with a table of Laplace transforms gives the results, A = Aq expC k1t) (12)... 

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. 

Only one other general solution exists. Two methods may be used to solve a partial differential equation such as the diffusion equation, or wave equation separation of variables or Laplace transformation (Carslaw and Jaeger [26] Crank [27]). The Laplace transformation route is often easier, especially if the inversion of the Laplace transform can be found in standard tables [28]. The Laplace transform of a function of time, (t), is defined as... 

Standard tables of inverse Laplace transforms enable this to be inverted very easily giving... 

Various sets of functions with their Laplace transforms have been compiled in tables f42, 74], which can also be used for inverse transformation -S 1 (F(s) = F(f), etc. 

When we have found a solution for the Laplace transformed function, then we need to make an inverse transformation to find the solution in terms of time and coordinates. There are elegant techniques for doing this based on the theory of complex functions, but often these are not necessary since there exist extensive tables in mathematical handbooks of functions and their Laplace transformed functions. Only in cases where the relevant functions have not been tabulated will it be necessary to carry out the inverse transformation using these techniques. 

We shall now determine the inverse Laplace transform of this function to find the concentration of B at time t. In tables of Laplace transforms the following pair has been found ... 

However, upgrading of the theory was not stopped. After the first attempt the MPK2 [247] and MPK3 [126] have arisen whose kernels are different. As can be seen from Table V, the kernel for MPK2 is actually defined as a solution of the transcendent equation. Another one, for MPK3, appears to be identical to that provided by MET. More recently a new theory named the self-consistent relaxation time approximation (SCRTA), was developed by Gopich and Szabo [180]. In the irreversible case the inverse relaxation time of SCRTA, kfc, is defined by the transcendent equation for kf shown in Table V. It is equivalent to the one expressed by the Laplace transformation of k(t) ck(kf) = 1 [177]. In spite of this complication, SCRTA is competitive in the concentration corrections with other theories. 

Table 5.5 shows the Laplace transforms of many functions encountered in many drug degradation and biotransformation reactions. Once the rate equations are transformed into the s-domain, the inverse Laplace transformations of this s-domain expression are carried out to obtain the time domain solution for the rate equations. 

Taking the inverse Laplace transforms of the Equation (5.118a), Equation (5.118b), Equation (5.118c), and Equation (5.118d) with the aid of Table 5.5 gives ... 

The probability density functions Exp( ), Erl(A, u), Gam(A,/r), and Rec(a,/3) are defined in Tables 9.1 and 9.2, and Chip) is the y2 distribution with v degrees of freedom. After modeling in frequency s-space, the solution in time f-space must be obtained by inverse Laplace transform. Nevertheless, given the complexity of the obtained model, the inverse transform may be rarely obtained from the above table. Usually, the numerical inverse Laplace transform is used [353,360]. 

From Table B.l we know that the inverse Laplace transform of... 

The most common approach to solution of partial differential equations of the type represented by (7) involves the use of Laplace transformation (Crank, 1957). The method involves transforming the partial differential equation into a total differential equation in a single independent variable. After solving the total differential equation inverse transformation of the solution can be carried out in order to reintroduce the second independent variable. Standard Laplace transforms are collected in tables. 

Just as there are tables of logarithms/ there are tables to aid the mathematical process of obtaining Laplace transforms (X) and inverse Laplace transforms ( ). Laplace transforms can also be calculated directly from the integral ... 

The table of inverse Laplace transforms shows that there are two solutions for this equation. Usually, a k and... 

If the Laplace transform of a function /(/) is f s), then f(t) is the inverse Laplace transform of f(s). Although an integral inversion formula can be used to obtain the inverse Laplace transform, in most cases it proves to be too complicated. Instead, a transform table (1), is used to find the image function f f). For more complicated functions, approximate methods are available. In many cases the inverse of a ratio of two polynomials must be... 

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