Inversion theorem, Laplace

Once the Laplace transform u(x,s) of the temperature () (x, /,) which fits the initial and boundary conditions has been found, the back-transformation or so-called inverse transformation must be carried out. The easiest method for this is to use a table of correspondences, for example Table 2.3, from which the desired temperature distribution can be simply read off. However frequently u(x,s) is not present in such a table. In these cases the Laplace transformation theory gives an inversion theorem which can be used to find the required solution. The temperature distribution appears as a complex integral which can be evaluated using Cauchy s theorem. The required temperature distribution is yielded as an infinite series of exponential functions fading with time. We will not deal with the application of the inversion theorem, and so limit ourselves to cases where the inverse transformation is possible using the correspondence tables. Applications of... 

The application of the Laplace transformation delivers the same result. The inverse transformation from the frequency to the time region requires the use of the inversion theorem, see 2.3.2. In order to avoid this in this case the simple, classical product solution is applied. 

In example 8.9 the Laplace transform technique was used to solve a time dependent problem. Inversing the Laplace transform is not straightforward. For complicated time dependent boundary conditions the convolution theorem can be used to find the inverse Laplace transform efficiently. If H(s) is the solution obtained in the Laplace domain, H(s) is represented as a product of two functions ... 

All definitions are as before and F is a constant. The solution obtained by use of the Laplace transform and Inversion theorem is... 

The eigenenergies Si and S2 are complex (< i and (021 are the duals of 0i) and 102) [8]. The inverse Fourier-Laplace transformation (Appendix A) and the Cauchy theorem lead to the time-dependent wavefunction... 

It is important to stress that this curve encloses no singularities. In fact, it is the act of enclosing a singularity (or many singularities) that leads to a very simple, classical result for solving the Laplace Inversion theorem. 

The exploitation of Cauchy s First theorem requires us to test the theorem for exceptional behavior. This allows, as we shall see, direct applications to the Laplace Inversion theorem. 

At the beginning of this chapter, we quoted the Mellin-Fourier Inversion theorem for Laplace transforms, worth repeating here... 

Use the Inversion theorem to find fit) corresponding to the Laplace transform... 

We have derived the general Inversion theorem for pole singularities using Cauchy s Residue theory. This provides the fundamental basis (with a few exceptions, such as /s) for inverting Laplace transforms. However, the useful building blocks, along with a few practical observations, allow many functions to be inverted without undertaking the formality of the Residue theory. We shall discuss these practical, intuitive methods in the sections to follow. Two widely used practical approaches are (1) partial fractions, and (2) convolution. 

These are the two building blocks to prove the Fourier-Mellin inversion theorem for Laplace transforms. 

In order to make the meaning of this theorem more precise, let us consider the Laplace inverse of (Q (z) ky, which is defined by a formula similar to (65) ... 

B. PARTIAL-FRACTIONS EXPANSION. The linearity theorem [Eq. (18.36)] permits us to expand the function into a sum of simple terms and invert each individually. This is completely analogous to Laplace-transformation inversion. Let F, be a ratio of polynomials in z, Mth-order in the numerator and iVth-order in the denominator. We factor the denominator into its N roots pi, P2, Ps,... 

Benet, L.Z. Turi, J.S. Use of general partial fraction theorem for obtaining inverse laplace transforms in pharmacokinetic analysis. J. Pharm. Sci. 1971, 60, 1593-1594. 

A time periodic solution of the set of equations 15.17a to 15.17d with a period 2nr is easily derived by applying the residue theorem to the general inversion integral of the Laplace transform solution of these equations [15]. It can be written as ... 

The inverse Laplace transform of the matrix elements in equation (46b) can be found from Laplace transform tables or by using the Heaveside expansion theorem. For example, consider the entry in the rtjj position of equation (46b). We find from Table 8.1 of Varma and Morbidelli[3] that... 

In examples 8.3 and 8.4 Maple was used to invert from the Laplace domain to the time domain. Unfortunately, these two examples are very simple and, hence, we could invert to the time domain using Maple. For practical problems, inversion is not straightforward. The inversion to the time domain can be done in two different ways. In section 8.1.4, short time solutions will be obtained by converting the solution in Laplace domain to an infinite series. In section 8.1.5, a long time solution will be obtained by using the Heaviside expansion theorem. 

Often inversion to time domain solution is not trivial and the time domain involves an infinite series. In section 8.1.4 short time solution for parabolic partial differential equations was obtained by converting the solution obtained in the Laplace domain to an infinite series, in which each term can easily inverted to time domain. This short time solution is very useful for predicting the behavior at short time and medium times. For long times, a long term solution was obtained in section 8.1.5 using Heaviside expansion theorem. This solution is analogous to the separation of variables solution obtained in chapter 7. In section 8.1.6, the Heaviside expansion theorem was used for parabolic partial differential equations in which the solution obtained has multiple roots. In section 8.1.7, the Laplace transform technique was extended to parabolic partial differential equations in cylindrical coordinates. In section 8.1.8, the convolution theorem was used to solve the linear parabolic partial differential equations with complicated time dependent boundary conditions. For time dependent boundary conditions the Laplace transform technique was shown to be advantageous compared to the separation of variables technique. A total of fifteen examples were presented in this chapter. 

Here y = x/r[ = =C, /2/IkT is chosen as the inertial effects parameter (y = /2/y is effectively the inverse square root of the parameter y used earlier in Section I). Noting e initial condition, Eq. (134), all the < (()) in Eq. (136) will vanish with the exception of n = 0. Furthermore, Eq. (136) is an example of hoyv, using the Laplace integration theorem above, all recurrence relations associated with the Brownian motion may be generalized to fractional dynamics. The normalized complex susceptibility /(m) = x ( ) z"( ) is given by linear response theory as... 

These theorems can be used to construct the Laplace transforms of various functions, and to find inverse transforms without carrying out an integral in the complex plane. 

Integral transforms were discussed, including Fourier and Laplace transforms. Fourier transforms are the result of allowing the period of the function to be represented by a Fourier series to become larger and larger, so that the series approaches an integral in the limit. Fourier transforms are usually written with complex exponential basis functions, but sine and cosine transforms also occur. Laplace transforms are related to Fourier transforms, with real exponential basis functions. We presented several theorems that allow the determination of some kinds of inverse Laplace transforms and that allow later applications to the solution of differential equations. 

Some ordinary differential equations can be solved by using some theorems of Laplace transforms which transform a differential equation into an algebraic equation. If this equation can be solved for the transform of the unknown function, and if the inverse transform can be found, the equation is solved. 

A second important property of Laplace transforms is expressed in the integration theorem (Steinfeld et al., 1989 Forst, 1973). If p E) is the inverse Laplace transform of 2(P)> then the integral of p( ), given by N( ) is... 

The convolution theorem is useful to invert the final result for CAk(s) in the Laplace domain and recover CAt(time domain. The appropriate inverse Laplace transforms are (Wylie, 1975, p. 268, formula 5 p. 278, formula 3)... 

We take careful note that the last term in the summation defines the Laplace inversion, according to the Fourier-Mellin theorem in Eq. 9.3, since in the limit... 

The term-by-term inversion, based on the general expansion theorem for the Laplace transform [37], produces... 

To calculate the change of the temperature in the real space we must take inverse Laplace transform of Eq. (152). This can be accomplished with the use of the convolution theorem and the following result ... 

The solutions given in Table 8.1 were all obtained by Laplace transformation. To obtain the solution of the model equations in the Laplace domain is straightforward but inversion of the transform to obtain an analytic expression for the breakthrough curve or pulse response is difficult. Simple analytic expressions for the moments of the pulse response may, however, be derived rather easily directly from the solution in Laplace form by the application of van der Laan s theorem... 

The third standard method of inverting a Laplace transform is by making use of the Residue theorem [6,15,16,18,22]. The transform function F(s) is analytic, except for singularities. In this discussion, when F s) is analytic, the inverse transform of F(s) is given by... 

The previous example is very straightforward and could easily be inverted by use of partial fractions and a table of Laplace transforms. This next example may be tabulated. However, it is used here to demonstrate more clearly how the residue theorem may be useful for similar or more complicated inversions. Consider... 

The inversion step can be relatively easy if the terms of step 3 can be located in a table of Laplace transforms. Without such a craivenient table a more difficult technique involving the residue theorem has to be employed (see Example 6.11). 

The third chapter addresses linear second-order ordinary differential equations. A brief discourse, it reviews elementary differential equations, and the chapter serves as an important basis to the solution techniques of partial differential equations discussed in Chapter 6. An applications section is also included with ten worked-out examples covering heat transfer, fluid flow, and simultaneous diffusion and chemical reaction. In addition, the residue theorem as an alternative method for Laplace transform inversion is introduced. 

Even using the Convolution and Shift Theorems, however, the Laplace transform of the required solution to a problem may sometimes be of a form which cannot be inverted directly. Possible approaches to the solution of such difficult inversion include... 

Using the partial fractions (or Heaviside theorem), we can obtain the inverse Laplace transform. 

Here we need the convolution theorem which identifies the inverse Laplace transform of a product as being a convolution ... 

Upon Laplace transform inversion via the convolution theorem, one obtains... 

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