One-sided Laplace transform

Integral-Transform Method A number of integral transforms are used in the solution of differential equations. Only one, the Laplace transform, will be discussed here [for others, see Integral Transforms (Operational Methods) ]. The one-sided Laplace transform indicated by L[f t)] is defined by the equation L[f t)] = /(O dt. It has... 

The mathematical basis for the exponential series method is Eq. (5.3), the use of which has recently been criticized by Phillips and Lyke.(19) Based on their analysis of the one-sided Laplace transform of model excited-state distribution functions, it is concluded that a small, finite series of decay constants cannot be used to represent a continuous distribution. Livesey and Brouchon(20) described a method of analysis using pulse fluorometry which determines a distribution using a maximum entropy method. Similarly to Phillips and Lyke, they viewed the determination of the distribution function as a problem related to the inversion of the Laplace transform of the distribution function convoluted with the excitation pulse. Since Laplace transform inversion is very sensitive to errors in experimental data,(21) physically and nonphysically realistic distributions can result from the same data. The latter technique provides for the exclusion of nonrealistic trial solutions and the determination of a physically realistic solution. These authors noted that this technique should be easily extendable to data from phase-modulation fluorometry. 

The one-sided Laplace transforms of commonly used functions and the main properties of the one-sided Laplace transform are listed in Tables B.l and B.2, respectively. The two-sided Laplace transforms and their main properties can be found in Tables B.3 and B.4, respectively [1-4],... 

Table B.2. Main properties of the one-sided Laplace transform...

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. 

If, now, one takes Laplace transforms of both sides and uses Eq. (4.38) to evaluate the right-hand side, one has... 

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. 

To illustrate how Laplace transforms work, consider the problem of solving Eq. (8-2), subjec t to the initial condition that = 0 at t = 0, and Cj is constant. If were not initially zero, one would define a deviation variable between and its initial value (c — Cq). Then the transfer function would be developed using this deviation variable. Taking the Laplace transform of both sides of Eq. (8-2) gives ... 

To obtain the characteristic function of a one-sided stable law, one calculates the Laplace transform. 

Generally, the default Laplace transform is unilateral or one-sided. 

Laplace transformation to the constant-flux boundary condition (4.50). Laplace transformation on the left-hand side of the boundary condition leads to (dc /dx), and the same operation performed on the right-hand side, to - l/Dp (Appendix 4.2). Thus, from the boundary condition (4.50) one gets... 

The variable 5 now contains the information about the time dependence of the problem. Applying Laplace transformation to the derivative on the right-hand side of equation (6.5.1), one obtains... 

One can take the Laplace transform of the right-hand side of (2.5.5), which transforms from the variables T, L, N into the variables T, P, N, where P is the one-dimensional pressure. Denoting by... 

An interesting property of the convolution product is that it can be transformed into an ordinary product of transformed functions. The transformation adapted to the time range of the convolution is not the Fourier transform, which works on a full range to +°o (two-sided transform), but the Laplace transform, which is analogous but working on a half range from 0 to infinite (one-sided transform). 

The complex dielectric function is the one-sided Fourier or pure imaginary Laplace transform of the correlation function of the polarization fluctuations

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To go up a level, one applies either a Laplace transform or a one-sided complex Fourier... 

Taking Laplace transforms of the both sides of Eq. 55, solving the obtained algebraic equation for the transform of the rth order moment, and taking the inverse transform, one obtains the following recursive expression (Takacs 1956) ... 

Therefore f( ) is convex and strictly increasing for / > 0. In this particular case one can compute exactly the Laplace transform of K -), that is one can make the left-hand side of (1.6) explicit as a function of f(/ ), and invert the expression to find P(/ ). However, on a more abstract ground, the left-hand side of (1.6), for / > 0, and hence f(/ ) > 0, is a real anal3dic function of f(/3) > 0 and therefore its inverse is real analytic too. So f( ) is real analytic on the positive semi-axis. We conclude that we are dealing with a smooth function, except at 0, where of course it cannot be anal d ic. But which derivative (if any) is discontinuous ... 

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