Use of Laplace transformation

Renyi (1953) gave a recursive formula for the Laplace transform of the 

---

The preceding two equations are examples of linear differential equations with constant coefficients and their solutions are often found most simply by the use of Laplace transforms [1]. 

In classical control theory, we make extensive use of Laplace transform to analyze the dynamics of a system. The key point (and at this moment the trick) is that we will try to predict the time response without doing the inverse transformation. Later, we will see that the answer lies in the roots of the characteristic equation. This is the basis of classical control analyses. Hence, in going through Laplace transform again, it is not so much that we need a remedial course. Your old differential equation textbook would do fine. The key task here is to pitch this mathematical technique in light that may help us to apply it to control problems. 

The analytical methods for solving the Fourier equation, in which Q and T are functions of the spatial co-ordinate and time, include a change of variables by combination, and in the more general case the use of Laplace transforms. 

The basic principles are taken from Zwillinger (1989). Duhamel s principle enables solutions for surface conditions being functions of time to be calculated from solutions with permanent surface conditions. Although this principle is most easily derived through the use of Laplace transforms, more conventional demonstrations, not repeated here, can be found in Sneddon (1957) or Carslaw and Jaeger (1959). 

As you will see, several different approaches are used in this book to analyze the dynamics of systems. Direct solution of the differential equations to give functions of time is a time domain teehnique. Use of Laplace transforms to characterize the dynamics of systems is a Laplace domain technique. Frequency response methods provide another approaeh to the problem. 

Our primary use of Laplace transformations in process control involves representing the dynamics of the process in terms of "transfer functions." These are output-input relationships and are obtained by Laplace-transfonning algebraic... 

Laplace transformation is particularly useful in pharmacokinetics where a number of series first-order reactions are used to model the kinetics of drug absorption, distribution, metabolism, and excretion. Likewise, the relaxation kinetics of certain multistep chemical and physical processes are well suited for the use of Laplace transforms. 

There are no source terms in Eq. (31), since the injection point is upstream from the sections over which the equations are to be used. The boundary conditions are of the same type used by van der Laan (V4), except for Eq. (33b). This merely states that we are going to measure the concentration at X = Xo, which we call Co. The solution of Eqs. (30), (31), (32), and (33) is again accomplished by the use of Laplace transforms following the same scheme as given in (A8, Bll, B14). The Laplace transform of the concentration at x = x , is... 

Use of Computer Simulation to Solve Differential Equations Pertaining to Diffusion Problems. As shown earlier (Section 4.2.11), differential equations used in the solutions of Fick s second law can often be solved analytically by the use of Laplace transform techniques. However, there are some cases in which the equations can be solved more quickly by using an approximate technique known as the finite-difference method (Feldberg, 1968). 

As in previous problems, these equations may be solved by the use of Laplace transforms. For y > 0 ... 

In electrochemistry, the most frequent use of Laplace transformation is in solving problems involving -> diffusion to an electrode. -> Fick s second law, a partial differential equation, becomes an ordinary differential equation on Laplace transformation and may thereby be solved more easily. 

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. 

The solution of Pick s second law is facilitated by the use of Laplace transforms, which convert the partial differential equation into an easily integrable total differential equation. By utilizing Laplace transforms, the concentration of diffusing species as a function of time and distance from the diffusion sink when a constant normalized current, or flux, is switched on at f = 0 was shown to be... 

The response of the burning rate to pressure variations is needed for use in equation (83). If the mechanism of Section 9.1.5.4 is considered, then either by use of Laplace transforms or (more simply) by seeking solutions proportional to it is readily found from equation (83) that... 

We have selected a simple example to illustrate the use of Laplace transform methods. A more advanced application is given in the next chapter, in which equations are derived for a two-compartment model. It will be shown subsequently that Laplace transform methods also are helpful in pharmacokinetics when convolution/deconvolution methods are used to characterize drug absorption processes. 

There are a number of different techniques that can be used to solve this set of equations, such as use of Laplace transforms, generating functions, statistical methods, and numerical and analytical techniques. We can obtain an analytical solution by using the following transformation. Let... 

While mathematical insight is gained by use of Laplace transformations, Fourier transformation is used for gaining physical insight in terms of spectra. Theorems for Laplace transforms and the transforms of common functions are tabulated in the literature [Spil]. 

We now introduce the use of Laplace transforms to illustrate how we can use the solute concentration and various derivatives in the Laplace domain to obtain equations for the various moments. 

Central to the use of Laplace transforms for iden-tifiability analysis is the concept of the transfer function H(s) defined as... 

We had no difficulty in formulating the problem. However, the solution process is considerably involved and remains beyond the scope of this text. For example, the use of Laplace transforms (see Chapter 7 of Conduction Heat Transfer by Arpaci) conveniently leads to... 

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. 

A differential equation contains one or more derivatives of an unknown function, and solving a differential equation means finding what that function is. One important class of differential equations consists of classical equations of motion, which come from Newton s second law of motion. We will discuss the solution of several kinds of differential equations, including linear differential equations, in which the unknown function and its derivatives enter only to the first power, and exact differential equations, which can be solved by a line integration. We will also introduce partial differential equations, in which partial derivatives occur and in which there are two or more independent variables. We will also discuss the solution of differential equations by use of Laplace transformations. Some differential equations can be solved either symbolically or numerically using Mathematica. 

The use of Laplace transforms offers a very simple and elegant method of solving linear or linearized differential equations which result from the mathematical modeling of chemical processes. 

Part III (Chapters 6 through 12) is devoted to the analysis of static and dynamic behavior of processing systems. The emphasis here is on identifying those process characteristics which shape the dynamic response for a variety of processing units. The results of such analysis are used later to design effective controllers. Input-output models have been employed through the use of Laplace transforms. 

The use of Laplace transforms is not limited to the solution of simple differential equations, like the second-order equation of Example 8.1. It extends to the solution of sets of differential equations. Con-... 

The use of Laplace transforms allows us to form a very simple, convenient, and meaningful representation of chemical process dynamics. It is simple because it uses only algebraic equations (not differential equations, as we have seen in Part II). It is convenient because it allows a quick analysis of process dynamics and finally, it is meaningful because it provides directly the relationship between the inputs (disturbances, manipulated variables) and the outputs (controlled variables) of a process. 

For the use of Laplace transforms to the solution of differential equations (ordinary, partial, or sets of), the book by Jenson and Jeffreys can be very valuable ... 

The use of Laplace transformations yields some very useful simplifications in notation and computation. Laplace-transforming the linear ordinary differential equations describing our processes in terms of the independent variable t converts them into algebraic equations in the Laplace transform variable s. This provides a very convenient representation of system dynamics. 

There are various methods of solving the equations for a three-phase short circuit on the basic that the set of equations are linear and where the use of Laplace transforms, or the Heaviside calculus, is appropriate. See References 3, 5, 6 and 8 for examples. These methods are complicated and appropriate assumptions concerning the relative magnitudes of resistances, inductances and time constants need to be made in order to obtain practical solution. The relative magnitudes of the parameters are derived from typical machinery data. Adkins in Reference 3 gives a solution of the following form. 

Prosperetti [6] applied an alternate technique, based on the use of Laplace transforms, to the initial value problem of infinitesimal-amplitude oscillations of viscous drops. His results show that the motion consists of modulated oscillations with varying frequency and damping parameter. The frequency of oscillations for small viscosity is given by ... 

In the first situation, a carrier fluid (which is usually an inert fluid but this is not necessary) is passed through the column, and once this is stabilised a tracer is injected into the column with a concentration of Co(t) at the inlet. The concentration is chosen such that the adsorption isotherm of this tracer towards the solid packing is linear. This results in a set of linear equations which permit the use of Laplace transform to obtain solution analytically. Knowing the solution in the Laplace domain, the solution in real time can be in principle obtained by some inversion procedure whether it be analytically or numerically. However, the moment method illustrated in Chapter 13 can be utilised to obtain moments from the Laplace solutions directly without the tedious process of inversion. 

Alternatively, one could have used Laplace transform methods to solve Example 3.24. In order to demonstrate the use of Laplace transform, we will examine a less cumbersome example, that is, consider... 

A very valuable technique, useful in the solution of ordinary and partial differential equations as well as differential delay equations, is the use of Laplace transforms. Laplace transforms (Churchill, 1972), though less familiar and somewhat more difficult to invert than their cousins, Fourier transforms, are broadly applicable and often enable us to convert differential equations to algebraic equations. For rate equations based on mass action kinetics, taking the Laplace transform affords sets of polynomial algebraic equations. For DDEs, we obtain transcendental equations. 

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... 

Another method for solving differential equations, which is favoured by some investigators, deserves attention. This involves the use of Laplace transforms. Physiologists, who often think of rate processes in terms of electrical circuits (see section 2.1), tend to introduce the transform method into their derivations, since this is the frequent choice of electrical engineers. 

---