Standard Laplace transforms

Process engineering and design using Visual Basic 

---

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. 

For some complicated problems. Maple cannot find the inverse Laplace transform. In these cases, one can split use standard Laplace transform formulae to simplify the expressions. By manipulating the expressions. Maple can be used to find the inverse Laplace transform. This is best illustrated with the following examples. 

In this chapter, analytical solutions were obtained for parabolic and elliptic partial differential equations in semi-infinite domains. In section 4.2, the given linear parabolic partial differential equations were converted to an ordinary differential equation boundary value problem in the Laplace domain. The dependent variable was then solved in the Laplace domain using Maple s dsolve command. The solution obtained in the Laplace domain was then converted to the time domain using Maple s inverse Laplace transform technique. Maple is not capable of inverting complicated functions. Two such examples were illustrated in section 4.3. As shown in section 4.3, even when Maple fails, one can arrive at the transient solution by simplifying the integrals using standard Laplace transform formulae. 

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. 

Substituting the reaction rate, Q, into Eq. 2 and after Laplace transformation, the equations obtained can be solved again by standard methods. The mass transfer rate for the heterogeneous part of the interface can be given as follows ... 

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... 

Equivalently, integration of eqn. (169) with respect to f0 over — to t leads to eqn. (170). Before discussing the solution of eqn. (170) and its physical significance, it is very rewarding to take the Laplace Transform of eqn. (146). Using the standard methods discussed in Chap. 2, Sect. 3.6 and the initial condition (131) together with the simplification t0 = 0 (for convenience), it leads to... 

Expression (24) reduces to the standard 1 = 0 for q —> n, due to the divergence of the gamma function T(z) for nonpositive integers. The fractional Riemann-Liouville integral operator oDJq fulfills the generalized integration theorem of the Laplace transformation ... 

In descriptions of this problem, the names of Randles [460] and Sevclk [505] are prominent. They both worked on the problem and reported their work in 1948. Randles was in fact the first to do electrochemical simulation, as he solved this system by explicit finite differences (and using a three-point current approximation), referring to Emmons [218]. Sevclk attempted to solve the system analytically, using two different methods. The second of these was by Laplace transformation, which today is the standard method. He arrived at (9.116) and then applied a series approximation for the current. Galus writes [257] that there was an error in a constant. Other analytical solutions were described (see Galus and Bard and Faulkner for references), all in the form of series, which themselves require quite some computation to evaluate. 

Risken, Vollmer, and Mdrsch studied the Kramers equation, that is, the Fokker-Planck equation (1.9), by expanding the distribution function p(x, o /) in Hermitian polynomials (velocity part) and in another complete set satisfying boundary conditions (position part). The Laplace transform of the initial value problem was obtained in terms of continued fractions. An inverse friction expansion of the matrix continued fraction was then used to show that the first Hermitian expansion coefficient may be determined by a generalized Smoluchowski equation. This provides results correcting the standard Smoluchowski equation with terms of increasing power in 1/y. They evaluated explicit expressions up to order y . ... 

By applying a Laplace transform to a three-element standard solid, the transitory response can also be obtained. In this case, Eq. (PIO.1.7) can be written as... 

The response of a standard solid (a spring with shear rigidity given by G in parallel with a Maxwell element with shear rigidity Gi and viscosity q) to a sudden force Pq can be found from Eq. (16.219) by taking, as usual, Laplace transforms. After the pertinent calculations [see Eqs. (16.127) and (16.184)] we obtain... 

The ILT technique may be used to calculate microcanonical rate coefficients from experimental Arrhenius expressions of the high pressure canonical rate coefficient. The idea was first proposed by Slater [38] who observed that the standard expression that relates the microcanonical and canonical rate coefficients can be written in terms of a Laplace transform... 

The flow under consideration is mathematically more simple in a plane duct than in an axisymmetric round tube. However, no readily solution for smooth walls, i.e. in the case where the EPR is absent, is given in the literature. The formulations and solutions of the problem of a pulsating viscous flow to be taken for a comparison have been presented only for round tubes. The known solutions employ mainly the Laplace transformation and small-parameter methods. The present investigation follows the standard technique of mathematical physics of finding the periodic solutions described particularly by Schlichting [566] with references to some original sources. Their solution for smooth walls will serve for a comparison. 

From the differentiation theorem of Laplace transform, J f /(t) = uP u) —P t = 0), we infer that the left-hand side in (x,t) space corresponds to 0P(x, t)/dt, with initial condition P(x. 0) = 8(x). Similarly in the Gaussian limit a = 2, the right-hand side is Dd2P(x, f)/0x2, so that we recover the standard diffusion equation. For general a, the right-hand side defines a fractional differential operator in the Riesz-Weyl sense (see below) and we find the fractional diffusion equation [52-56]... 

The problem posed by (3-119) with the boundary and initial conditions, (3-120b), is very simple to solve by either Fourier or Laplace transform methods.14 Further, because it is linear, an exact solution is possible, and nondimensionalization need not play a significant role in the solution process. Nevertheless, we pursue the solution by use of a so-called similarity transformation, whose existence is suggested by an attempt to nondimensionalize the equation and boundary conditions. Although it may seem redundant to introduce a new solution technique when standard transform methods could be used, the use of similarity transformations is not limited to linear problems (as are the Fourier and Laplace transform methods), and we shall find the method to be extremely useful in the solution of certain nonlinear DEs later in this book. 

The second-order partial differential equations given in the text of the paper contain time derivatives of concentration (dC/dt) and terms containing dC/dz and C. The solutions of these equations, unless referred to a literature source, were obtained by the method of Laplace transformation with the aid of standard tables of Laplace transforms. Good working summaries of the Laplace transformation method as applied to... 

The last term in Eq. (A-3) is the Laplace transform of E(a) Aweq(a), where a plays the role of t in Eq. (A-4) and t plays the role of s in Eq. (A-4). Thus the problem is reduced to this experimental observations give us the Laplace transform of the desired distribution function [modified by multiplication by Aweq(A)], We want the distribution function. The required calculation is a numerical inverse Laplace transform. This is clearly feasible, and numerical techniques are discussed in the literature 56). It is not a simple matter to carry out, however, and the accuracy requirements on the data are likely to be stringent. No direct method of computing the distribution from frequency response is known, although the step response can be computed from the frequency response by standard techniques. In view of the foregoing discussion, it appears that in principle, at least, the distribution can be computed from the frequency response. 

If eq. (11.2) is in terms of deviation variables, the initial conditions are zero and its Laplace transformation yields the following standard transfer function for a second-order system ... 

The inverse Laplace transforms of (s) in Eq. (A.35) are calculated using standard methods of complex analysis to yield ifi(T) ... 

---