Laplace transform examples

Example The equation 3c/3f = D(3 c/3a. ) represents the diffusion in a semi-infinite medium, a. > 0. Under the boundary conditions c(0, t) = Cq, c(x, 0) = 0 find a solution of the diffusion equation. By taking the Laplace transform of both sides with respect to t,... 

Equations readily solvable by Laplace transforms. For example ... 

Sets of first-order rate equations are solvable by Laplace transform (Rodiguin and Rodiguina, Consecutive Chemical Reactions, Van Nostrand, 1964). The methods of linear algebra are applied to large sets of coupled first-order reactions by Wei and Prater Adv. Catal., 1.3, 203 [1962]). Reactions of petroleum fractions are examples of this type. 

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]. 

Example 15.2 ETse Laplace transform techniques to apply a delta function input to a CSTR to determine f i). 

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]. 

If X (0 and Xjit) are the input and output functions in the time domain (for example, the contents in the reservoir and in the plasma compartment), then XJj) is the convolution of Xj(r) with G(t), the inverse Laplace transform of the transfer function between input and output ... 

Now that we can do Laplace transform, let us return to our very first example. The Laplace transform of Eq. (2-2) with its zero initial condition is (xs + l)C (s) = C in(s), which we rewrite as... 

Example 2.16. Derive the closed-loop transfer function X,/U for the block diagram in Fig. E2.16a. We will see this one again in Chapter 4 on state space models. With the integrator 1/s, X2 is the Laplace transform of the time derivative of x,(t), and X3 is the second order derivative of x,(t). 

A brief review is in order Recall that Laplace transform is a linear operator. The effects of individual inputs can be superimposed to form the output. In other words, an observed output change can be attributed to the individual effects of the inputs. From the stirred-tank heater example in Section 2.8.2 (p. 2-23), we found ... 

From Eq. (2-18) on page 2-7, the Laplace transform of a time delay is an exponential function. For example, first and second order models with dead time will appear as... 

From the last example, we may see why the primary mathematical tools in modem control are based on linear system theories and time domain analysis. Part of the confusion in learning these more advanced techniques is that the umbilical cord to Laplace transform is not entirely severed, and we need to appreciate the link between the two approaches. On the bright side, if we can convert a state space model to transfer function form, we can still make use of classical control techniques. A couple of examples in Chapter 9 will illustrate how classical and state space techniques can work together. 

We have shown how the state transition matrix can be derived in a relatively simple problem in Example 4.7. For complex problems, there are numerical techniques that we can use to compute (t), or even the Laplace transform (s), but which of course, we shall skip. 

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,... 

This relation can be employed to obtain other Laplace transforms. For example, with the use of Eqs. (49) and (50),... 

Figure 23 Calculation of the shape of the actively compensated pulse can be carried out on the software. (A) shows the real (red line) and the imaginary (green line) component of an example of the target pulse shape t>,(f). Its leading and the trailing edges have a cosine shape with a transition time of 1.25 xs in 50 steps, and the width of the plateau is 5 ps. (B) Laplace transformation B(s) multiplied by the Laplace transformed step function U(s). (C) It was then divided by the Laplace transformation Y(s) of the measured step response y(t) of the proton channel of a 3.2-mm Varian T3 probe tuned at 400.244 MHz to obtain V(s). (D) Finally, inverse Laplace transformation was performed on V(s) to obtain the compensated pulse that results in the RF pulse with the target shape. Time resolution was 25 ns, and o = 20 was used for the Laplace and inverse Laplace transformations.

Next, bi(t) was Laplace transformed into B(s), and then multiplied by the Laplace transformation U(s) of the step function u(t). The result B(s)U(s) is displayed in Figure 23B. In this example, the step response y(t) was measured for the 1H channel of a Varian 3.2 mm T3 probe tuned at 400.244 MHz with a time resolution of 25 ns, and Laplace transformed into Y(s). By dividing B(s)U(s) by Y(s), the function plotted in Figure 23C was obtained, from which, by performing inverse Laplace transformation, the programming pulse shape v(t) was finally obtained, as shown in Figure 23D. The amplitude and the phase of the complex function v(t) give the intensity and the phase of the transient-compensated shaped pulse. 

Using deterministic kinetics, one can force-fit the time evolution of one species—for example, eh but then those of other yields (e.g., OH) will be inconsistent. Stochastic kinetics can predict the evolutions of radicals correctly and relate these to scavenging yields via Laplace transforms. 

Tables of Laplace transforms for a large number of functions have been calculated, and can be obtained from published data. In the present example, the transformed equation is...

For multiple reactions, material balances must be made for each stoichiometry. An example is the consecutive reactions, A = B = C, for which problem P4.04.52 develops a closed form solution. Other cases of sets of first order reactions are solvable by Laplace Transform, and of course numerically. 

Inputs + Sources = Outputs + Sinks + Accumulation Formulation of differential equations in general is described in Chapter 1. Usually the ODE is of the first or second order and is readily solvable directly or by aid of the Laplace Transform. For example, for the special case of initial equilibrium or dead state (All derivatives zero at time zero), the preceding equation has the transform... 

A particular vessel behavior sometimes can be modelled as a series or parallel arrangement of simpler elements, for example, some combination of a PFR and a CSTR. Such elements can be combined mathematically through their transfer functions which relate the Laplace transforms of input and output signals. In the simplest case the transfer function is obtained by transforming the linear differential equation of the process. The transfer function relation is... 

If the excitation electric field is an s-polanzed evanescent field instead of the above p-polarized example, then wH 11 [ = wHJI(z)] does not depend upon p. Therefore, an approximate C(z) can be calculated from the observed fluorescence (P) (obtained experimentally by varying 0) by ignoring the z dependence in the bracketed term in Eq. (7.45) and by inverse Laplace transforming Eq. (7.44) after the ,(0, /J) 2 term has been factored out.,37 39)... 

There are several methods for finding particular solutions. Laplace transform methods are probably the most convenient, and we will use them in Part IV. Here we will present the method of undetermined co cients. It consists of assuming a particular solution that has the same form as the forcing function. It is illustrated in the examples below. 

If the same input as used in Example 6.6 is imposed on the system, we should be able to use Laplace transforms to find the response of Ca to a step change of magnitude. ... 

This example can be generalized to any case where Af Af to show that differentiation would be required. Therefore N must always be greater than or equal to M. Laplace-transforming Eq. (9.96) gives... 

Use Laplace transform techniques to solve Example 6.7 where a ramp disturbance drives a first-order system. 

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