Properties of Laplace Transforms

The properties of Laplace transform and the transforms of some common functions. 

This approach works because of the linear property of Laplace transform. 

One of the most important properties of Laplace transformation is that it is linear. 

Two important properties of Laplace transforms are their scaling and linearity as ... 

One approach to this question is to set up the new diffusion problem with the initial and boundary conditions characteristic of the sinusoidally varying flux and to obtain a solution. There is, however, a simpler approach. Using the property of Laplace transforms, one can use the solution (4.65) of the constant-flux diffusion problem to generate solutions for other problems. 

Moment analysis. A technique for obtaining analytical information on the elution curve described by Equation 25 is to make use of the moment generating property of Laplace transforms (41). It is readily shown that the various moments of the real-time concentration profile are related to the transform solution by the following ... 

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... 

In this section, the method of Laplace transform will be used. The properties of Laplace transforms, and especially Theorem 3.7 (Section 3.6.1), will be applicable. The Laplace transform was introduced earlier for use in solving ordinary differential equations. Now we will emphasize its use in solving PDEs. 

A further useful property of Laplace Transforms is that of converting functions with discontinuities in f-space into continuous functions in s-space. Let us consider the function... 

We usually use tables and some properties to get the Laplaee transforms. A table of the most important Laplace transforms is given in Appendix D. In order to cover a wider range of functions using a limited table like the one in Appendix D, we must utilize some important properties of the Laplace transformation. The main properties of Laplace transform are as follows ... 

Some of the properties of Laplace transform that are useful, are the following ... 

Useful Properties of Laplace Transform limit functions... 

These properties allow one to solve the integration problem in transform space and then back-transform the result into real space. There are extensive tables of Laplace transforms, some of which are given in Table 2.9. 

Thus, it becomes apparent the output and the impulse response are one-sided in the time domain and this property can be exploited in such studies. Solving linear system problems by Fourier transform is a convenient method. Unfortunately, there are many instances of input/ output functions for which the Fourier transform does not exist. This necessitates developing a general transform procedure that would apply to a wider class of functions than the Fourier transform does. This is the subject area of one-sided Laplace transform that is being discussed here as well. The idea used here is to multiply the function by an exponentially convergent factor and then using Fourier transform technique on this altered function. For causal functions that are zero for t < 0, an appropriate factor turns out to be where a > 0. This is how Laplace transform is constructed and is discussed. However, there is another reason for which we use another variant of Laplace transform, namely the bi-lateral Laplace transform. 

The Laplace transform is similar to a one-sided Fourier transform, except that it has a real exponential instead of the complex exponential of the Fourier transform. If we consider complex values of the variables, the two transforms become different versions of the same transform, and their properties are related. The integral that is carried out to invert the Laplace transform is carried out in the complex plane, and we do not discuss it. Fortunately, it is often possible to apply Laplace transforms without carrying out such an integral. We will discuss the use of Laplace transforms in solving differential equations in Chapter 8. 

Laplace transformation is a simple mathematical technique that transforms differential equations from the time domain (t) to the Laplace domain (.s). Some detail about Laplace transformation is given later in this chapter. One of the main characteristics of Laplace transformation is to transform derivatives in the time domain (t) to algebraic form in the Laplace domain [i.e., dy t)/dt X )]- Using this property, we find that the Laplace transformation of Eq. (5.2) gives... 

As discussed in Chapter 7 real material properties extend over many decades of time and for realistic solutions of boundary value problems it is necessary to have methods to incorporate these real measured properties. When material properties can be represented by a Prony series composed of a number of terms, it is possible to obtain solutions for more practical representation of polymers. Examples of the use of Laplace transforms for... 

In this book we are more concerned with operational aspects of Laplace transforms—that is, using them to obtain solutions or the properties of solutions of linear differential equations. For more details on mathematical aspects of the Laplace transform, the texts by Churchill (1971) and Dyke (1999) are recommended. 

In this chapter we have considered the application of Laplace transform techniques to solve linear differential equations. Although this material may be a review for some readers, an attempt has been made to concentrate on the important properties of the Laplace transform and its inverse, and to point out the techniques that make manipulation of transforms easier and less prone to error. 

We can use z-transforms in a similar way to Laplace transforms and ultimately express a transfer function for discrete time that corresponds to a difference equation. First we need to derive some properties of z-transforms. Using (17-23), we develop the real translation theorem as follows ... 

These two categories are still found for the methods of resolution. The method of separation of the variables is used for the first, whereas the resolution of the second rests on the properties of the transform of Laplace. People can find a great number of solutions in specialized books [ADD 66, JOS 60] and in particular in J. Crank s... 

Application of the definition shows that the Laplace transform is a linear oper-ator " this property is represented in Eqs. (3-67) and (3-68). 

In this section, we will outline only those properties of the Laplace transform that are directly relevant to the solution of systems of linear differential equations with constant coefficients. A more extensive coverage can be found, for example, in the text book by Franklin [6]. 

When we apply these two properties of the Laplace transform to differential equations of our pharmacokinetic model in eq. (39.46), we obtain ... 

An important property of the Laplace transform is that it is a linear operator, and contribution of individual terms can simply be added together (superimposed) ... 

The convolution and general properties of the Fourier transform, as presented in Section 11.1, are equally applicable to the Laplace transform. Thus,... 

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