Laplace transformation and inversion

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. 

Equations (1) - (3) and (6) were simultaneously solved using the above boundary conditions by Laplace transformation and inversion by the method of residues to obtain the following analytical equations ... 

Maple can be used to obtain Laplace transforms and inverse Laplace transforms of functions symbolically. For this purpose, the package with(inttrans) is used ... 

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

It is readily shown that several simple mathematical operations such as differentiation, integration, and linear transformations (scaling and translation), as well as more complex operations such as convolution, deconvolution, and Laplace transformations (and inverse Laplace transformation) have the above linear operator property. 

Figure 5.26 shows a flow diagram for the Laplace transformation and inverse transformation. It is clear that the main function of the Laplace transformation is to put the differential equation (in the time domain) into an algebraic form (in the 5-domain). These 5-domain algebraic equations can be easily manipulated as input-output relations. 

Using these three properties and tables containing both the Laplace transforms and inverse transforms, it is possible to solve many differential kinetic equations analytically. 

J. G. McWhirter and E. R. Pike, On the numerical inversion of the Laplace transform and similar Fredholm integral equations of the first kind, J. Phys. A Math. Gen. 11, 1729-1745 (1978). 

On the other hand, the first convolution integral in Eq. (3.22) can be found by Laplace transformation and its inversion, as... 

This expression is obtained by reporting (9.16) and (9.19) into (9.10). Using the numerical inverse Laplace transform and k = 2h-1, A = lh-1, u>e = 0.8, and f.i= 1,4, 6, Figure 9.14 illustrates the p (t) time profiles for a 6-h constant-rate infusion. This figure takes into account the infusion duration, whereas Figure 9.10 considers that all molecules are in compartment 1 at initial times. ... 

Here is an example solving a non-steady diffusion equation using the Laplace transform and the inverse Laplace transform. According to Fick s second law, the diffusion equation can be expressed as... 

Hence using the Bromwich integral (complex inversion formula for the Laplace transform) and the inverse Fourier transform we have... 

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. 

Suppose we suddenly increase the load current of a converter from 4 A to 5 A. This is a step load and is essentially a nonrepetitive stimulus. But by writing all the transfer functions in terms of s rather than just as a function of jco, we have created the framework for analyzing the response to such disturbances too. We will need to map the stimulus into the s-plane with the help of the Laplace transform, multiply it by the appropriate transfer function, and that will give us the response in the s-plane. We then apply the inverse Laplace transform and get the response with respect to time. This was the procedure symbolically indicated in Figure 7-3, and that is what we need to follow here too. However, we will not perform the detailed analysis for arbitrary load transients here, but simply provide the key equations required to do so. 

The Laplace transform and its inverse are often denoted in the following way ... 

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. 

T (t) given by eq. (8.4) is the solution to our initial differential equation (5.3). Indeed, taking the Laplace of eq. (8.4), it yields eq. (8.3). The procedure by which we find the time function when its Laplace transform is known is called the inverse Laplace transformation and is the most critical step while solving linear differential equations using Laplace transforms. To summarize the solution procedure described in the example above, we can identify the following steps ... 

The Laplace transform and its inverse are well known for simple functions (Stein-feld et al., 1989 Forst, 1973). Consider, for instance, the case of a single harmonic oscillator in the classical limit, hv< kT). If the exponential in Eq. (6.47) is expanded in a power series and the higher-order terms dropped, we find that (y fclassical) = k TIhv, or using the symbol in the Laplace transform, = (3/Jv)". The density of states of a single classical harmonic oscillator is then given as follows ... 

Previous sections dealt with the analytical development of Laplace transform and the inversion process. The method of residues is popular in the inversion of Laplace transforms for many applications in chemical engineering. However, there are cases where the Laplace transform functions are very complicated and for these cases the inversion of Laplace transforms can be more effectively done via a numerical procedure. This section will deal with two numerical methods of inverting Laplace transforms. One was developed by Zakian (1969), and the other method uses a Fourier series approximation (Crump 1976). Interested readers may also wish to perform transforms using a symbolic algebra language such as Maple (Heck 1993). 

The electrostatic energy is the dominant term for many inorganic materials, particularly oxides, and therefore it is important to evaluate it accurately. For small- to moderate-sized systems this is most efficiently achieved through the Ewald summation (Ewald 1921) in which the inverse distance is rewritten as its Laplace transform and then split into two rapidly convergent series, one in reciprocal-space and one in real-space. The distribution of the summation between real- and reciprocal-space is controlled by a parameter t]. The resulting expression for the energy is ... 

Kipp (1985) transformed the above equations to the dimensionless form by dimensionless factors and variable parameters and solved the equations by the method of Laplace transformation, and finally got the solution through Laplace inverse transformation principle. 

What one does in solving an equation with Laplace transforms is to transform the equation by applying eq. (10.11), solve the resulting algebraic equation for the transform F(u), and then invert the transform to obtain the solution f t). Calculating inverse Laplace transforms is not easy it requires facility with contour integrals in the complex plane. On the other hand, many reference books, like the Handbook of Chemistry and Physics, contain extensive tables of Laplace transforms and their inverses. We recommend their use. It is easy to see from eq. (10.11) that the Laplace transform of f(t — r) is just... 

Whether DLS, DWS, Mie scattering, or other applications in which unfractionated samples are analyzed, the resulting distributions produced by modern instruments, while frequently facile to obtain and neat in appearance, must be treated with caution, as there is usually a large amount of data smoothing, fitting, and assumptions applied in using inverse Laplace transform and several other commonly employed methods. The best means of finding distributions of size and mass continue to be fractionation methods, such as SEC [32-34], field flow fractionation (FEE) [35-37], capillary electrophoresis [38], capillary hydrodynamic fractionation [39], and so on. 

Thus, provided the functions F(/3, t) are known, one can perform the inverse Laplace transform and obtain functions 6(P,t)... 

The frequency responses I(s) and V(s) are sent to subroutine FLT given in Figure 2.62. Then, the FLT carries out the inverse Laplace transform and the time solutions are obtained. Figure 2.63 shows an example of a calculated result v(t) in comparison with the accurate solution. Note that the accuracy of the FLT is quite high. 

One of the important properties of the Laplace transform and the inverse Laplace transform is that they are linear operators a linear operator satisfies the superposition principle ... 

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