Laplace transform differentiation

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... 

Other applications of Laplace transforms are given under Differential Equations. ... 

The z -transform can also be used to solve difference equations, just like the Laplace transform can be used to solve differential equations. 

Equation (8-14) shows that starts from 0 and builds up exponentially to a final concentration of Kcj. Note that to get Eq. (8-14), it was only necessaiy to solve the algebraic Eq. (8-12) and then find the inverse of C (s) in Table 8-1. The original differential equation was not solved directly. In general, techniques such as partial fraction expansion must be used to solve higher order differential equations with Laplace transforms. 

For operation with an inert tracer, the material balances are conveniently handled as Laplace transforms. For a stirred tank, the differential equation... 

Differential equations and their solutions will be stated for the elementary models with the main lands of inputs. Since the ODEs are linear, solutions by Laplace transforms are feasible. 

The Laplace transform technique also allows the reduction of the partial differential equation in two variables to one of a single variable In the present case. 

A transfer function is the Laplace transform of a differential equation with zero initial conditions. It is a very easy way to transform from the time to the. v domain, and a powerful tool for the control engineer. 

Find the Laplace transform of the following differential equation given ... 

G(.v) is the transfer function, i.e. the Laplace transform of the differential equation for zero initial conditions. 

Linear differential equations with constant coefficients can be solved by a mathematical technique called the Laplace transformation . Systems of zero-order or first-order reactions give rise to differential rate equations of this type, and the Laplaee transformation often provides a simple solution. 

To take the inverse Laplace transform means to reverse the process of taking the transform, and for this purpose a table of transforms is valuable. To illustrate, we consider a simple first-order reaction, whose differential rate equation is... 

This is the procedure From the postulated kinetic scheme we write the differential rate equations. Take the Laplace transforms of the differential equations. Solve the resulting set of algebraie equations for the transforms of the concentrations. Then take the inverse transforms to obtain the coneentrations as funetions of time. 

Frost and Pearson treated Scheme XV by the eigenvalue method, and we have solved it by the method of Laplace transforms in the preceding subsection. The differential rate equations are... 

Equations (4-21) are linear first-order differential equations. We considered in detail the solution of such sets of rate equations in Section 3-2, so it is unnecessary to carry out the solutions here. In relaxation kinetics these equations are always solved by means of the secular equation, but the Laplace transformation can also be used. Let us write Eqs. (4-21) as... 

In mathematics, Laplace s name is most often associated with the Laplace transform, a technique for solving differential equations. Laplace transforms are an often-used mathematical tool of engineers and scientists. In probability theory he invented many techniques for calculating the probabilities of events, and he applied them not only to the usual problems of games but also to problems of civic interest such as population statistics, mortality, and annuities, as well as testimony and verdicts. 

Derivatives 35. Maxima and Minima 37. Differentials 38. Radius of Curvature 39. Indefinite Integrals 40. Definite Integrals 41. Improper and Multiple Integrals 44. Second Fundamental Theorem 45. Differential Equations 45. Laplace Transformation 48. 

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]. 

The Laplace transformation is based upon the Laplace integral which transforms a differential equation expressed in terms of time to an equation expressed in terms of a complex variable a + jco. The new equation may be manipulated algebraically to solve for the desired quantity as an explicit function of the complex variable. 

The above differential equation can be integrated through the use of the Laplace transformation, and, taking due consideration of the limits, this gives... 

This partial differential equation is most conveniently solved by the use of the Laplace transform of temperature with respect to time. As an illustration of the method of solution, the problem of the unidirectional flow of heat in a continuous medium will be considered. The basic differential equation for the X-direction is ... 

Thus, the fraction unreacted is the Laplace transform with respect to the transform parameter k of the differential distribution function. 

Given k fit) for nny reactor, you automatically have an expression for the fraction unreacted for a first-order reaction with rate constant k. Alternatively, given ttoutik), you also know the Laplace transform of the differential distribution of residence time (e.g., k[f(t)] = exp(—t/t) for a PER). This fact resolves what was long a mystery in chemical engineering science. What is f i) for an open system governed by the axial dispersion model Chapter 9 shows that the conversion in an open system is identical to that of a closed system. Thus, the residence time distributions must be the same. It cannot be directly measured in an open system because time spent outside the system boundaries does not count as residence but does affect the tracer measurements. 

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]. 

Since the integral is over time t, the resulting transform no longer depends on t, but instead is a function of the variable s which is introduced in the operand. Hence, the Laplace transform maps the function X(f) from the time domain into the s-domain. For this reason we will use the symbol when referring to Lap X t). To some extent, the variable s can be compared with the one which appears in the Fourier transform of periodic functions of time t (Section 40.3). While the Fourier domain can be associated with frequency, there is no obvious physical analogy for the Laplace domain. The Laplace transform plays an important role in the study of linear systems that often arise in mechanical, electrical and chemical kinetic systems. In particular, their interest lies in the transformation of linear differential equations with respect to time t into equations that only involve simple functions of s, such as polynomials, rational functions, etc. The latter are solved easily and the results can be transformed back to the original time domain. 

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

Complex systems can often be represented by linear time-dependent differential equations. These can conveniently be converted to algebraic form using Laplace transformation and have found use in the analysis of dynamic systems (e.g., Coughanowr and Koppel, 1965, Stephanopolous, 1984 and Luyben, 1990). 

The derivation of eqns. 3.80-3.84 by solving the differential equations using Laplace transformation and numerical methods has been clearly and comprehensively presented by Bard and Faulkner103. 

It may be useful to point out a few topics that go beyond a first course in control. With certain processes, we cannot take data continuously, but rather in certain selected slow intervals (c.f. titration in freshmen chemistry). These are called sampled-data systems. With computers, the analysis evolves into a new area of its own—discrete-time or digital control systems. Here, differential equations and Laplace transform do not work anymore. The mathematical techniques to handle discrete-time systems are difference equations and z-transform. Furthermore, there are multivariable and state space control, which we will encounter a brief introduction. Beyond the introductory level are optimal control, nonlinear control, adaptive control, stochastic control, and fuzzy logic control. Do not lose the perspective that control is an immense field. Classical control appears insignificant, but we have to start some where and onward we crawl. 

Classical process control builds on linear ordinary differential equations and the technique of Laplace transform. This is a topic that we no doubt have come across in an introductory course on differential equations—like two years ago Yes, we easily have forgotten the details. We will try to refresh the material necessary to solve control problems. Other details and steps will be skipped. We can always refer back to our old textbook if we want to answer long forgotten but not urgent questions. 

After Laplace transform, a differential equation of deviation variables can be thought of as an input-output model with transfer functions. The causal relationship of changes can be represented by block diagrams. 

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. 

Let us first state a few important points about the application of Laplace transform in solving differential equations (Fig. 2.1). After we have formulated a model in terms of a linear or linearized differential equation, dy/dt = f(y), we can solve for y(t). Alternatively, we can transform the equation into an algebraic problem as represented by the function G(s) in the Laplace domain and solve for Y(s). The time domain solution y(t) can be obtained with an inverse transform, but we rarely do so in control analysis. 

Another important quantity is the steady state gain.3 With reference to a general differential equation model (2-28) and its Laplace transform in (2-29), the steady state gain is defined as the... 

Since Laplace transform can only be applied to a linear differential equation, we must "fix" a nonlinear equation. The goal of control is to keep a process running at a specified condition (the steady state). For the most part, if we do a good job, the system should only be slightly perturbed from the steady state such that the dynamics of returning to the steady state is a first order decay, i.e., a linear process. This is the cornerstone of classical control theory. 

We now derive the time-domain solutions of first and second order differential equations. It is not that we want to do the inverse transform, but comparing the time-domain solution with its Laplace transform helps our learning process. What we hope to establish is a better feel between pole positions and dynamic characteristics. We also want to see how different parameters affect the time-domain solution. The results are useful in control analysis and in measuring model parameters. At the end of the chapter, dead time, reduced order model, and the effect of zeros will be discussed. 

Other integral transforms are obtained with the use of the kernels e" or xk among the infinite number of possibilities. The former yields the Laplace transform, which is of particular importance in the analysis of electrical circuits and the solution of certain differential equations. The latter was already introduced in the discussion of the gamma function (Section 5.5.4). 

It should be evident that the expressions for the Laplace transforms of derivatives of functions can facilitate the solution of differential equations. A trivial example is that of the classical harmonic oscillator. Its equation of motion is given by Eq. (5-33), namely,... 

Lagrange multipliers 255-256 Lagrange s moan-value theorem 30-32 Lagperre polynomials 140, 360 Lambert s law 11 Langevin function 61n Laplace transforms 279—286 convolution 283-284 delta function 285 derivative of a function 281-282 differential equation solutions 282-283... 

Extensive literature is available on general mathematical treatments of compartmental models [2], The compartmental system based on a set of differential equations may be solved by Laplace transform or integral calculus techniques. By far... 

Differentiating equation (II-8) with respect to time and using Leibnitz s rule, one can obtain the evolution of P(t) with time. A Laplace transform analysis will finally yield (58,59) ... 

---