Laplace transform function

The Laplace transform of a convolution integral is simply the product of the Laplace transformed functions within the integral. Consequently, the Laplace transform of the equation aoove is... 

When we have found a solution for the Laplace transformed function, then we need to make an inverse transformation to find the solution in terms of time and coordinates. There are elegant techniques for doing this based on the theory of complex functions, but often these are not necessary since there exist extensive tables in mathematical handbooks of functions and their Laplace transformed functions. Only in cases where the relevant functions have not been tabulated will it be necessary to carry out the inverse transformation using these techniques. 

TABLE 19-4 Seme Common Laplace Transform Functions... 

When this is done, the reader can see that Laplace inversion is formally equivalent to contour integration in the complex plane. We shall see that exceptional behavior arises occasionally (singularities owing to multivaluedness, for example) and these special cases will be treated in the sections to follow. Our primary efforts will be directed toward the usual case, that is, pole and multiple pole singularities occurring in the Laplace transform function Fis). 

Previous sections dealt with the analytical development of Laplace transform and the inversion process. The method of residues is popular in the inversion of Laplace transforms for many applications in chemical engineering. However, there are cases where the Laplace transform functions are very complicated and for these cases the inversion of Laplace transforms can be more effectively done via a numerical procedure. This section will deal with two numerical methods of inverting Laplace transforms. One was developed by Zakian (1969), and the other method uses a Fourier series approximation (Crump 1976). Interested readers may also wish to perform transforms using a symbolic algebra language such as Maple (Heck 1993). 

Sufficient Conditions for the Existence of Laplace Transform Suppose/ is a function which is (1) piecewise continuous on eveiy finite intei val 0 < t exponential growth at infinity, and (3) Jo l/t)l dt exist (finite) for every finite 6 > 0. Then the Laplace transform of/exists for all complex numbers. s with sufficiently large real part. 

A short table (Table 3-1) of very common Laplace transforms and inverse transforms follows. The references include more detailed tables. NOTE F(/i -1- 1) = Iq x e dx (gamma function) /(f) = Bessel function of the first land of order n. 

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 ... 

The term in parentheses in Eq. (8-17) is zero at steady state and thus it can be dropped. Next the Laplace transform is taken, and the resulting algebraic equation solved. Denoting X s) as the Laplace transform of and X,(.s) as the transform of 4, the final transfer Function can be written as ... 

The combination of reac tor elements is facihtated by the concept of transfer functions. By this means the Laplace transform can be found for the overall model, and the residence time distribution can be found after inversion. Finally, the chemical conversion in the model can be developed with the segregation and maximum mixed models. 

Table 3.1 gives further Laplace transforms of common functions (called Laplace transform pairs). 

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. 

JLo(.v) is the Laplace transform of the output function, or system response. 

Fig. 4.1 Block diagram of a closed-loop control system. R s) = Laplace transform of reference input r(t) C(s) = Laplace transform of controlled output c(t) B s) = Primary feedback signal, of value H(s)C(s) E s) = Actuating or error signal, of value R s) - B s), G s) = Product of all transfer functions along the forward path H s) = Product of all transfer functions along the feedback path G s)H s) = Open-loop transfer function = summing point symbol, used to denote algebraic summation = Signal take-off point Direction of information flow.

In this expression, z is the distance from the surface into the sample, a(z) is the absorption coefficient, and S, the depth of penetration, is given by Eq. 2. A depth profile can be obtained for a given functional group by determining a(z), which is the inverse Laplace transform of A(S), for an absorption band characteristic of that functional group. 

Experimentally, the absorbance A(5) of a band is measured as a function of the angle of incidence B and thus of S. Two techniques can be used to determine a(z). A functional form can be assumed for a(z) and Eqs. 2 and 3 used to calculate the Laplace transform A(5) as a function of 8 [4]. Variable parameters in the assumed form of a(z) are adjusted to obtain the best fit of A(5) to the experimental data. Another approach is to directly compute the inverse Laplace transform of A(5) [3,5]. Programs to compute inverse Laplace transforms are available [6]. 

The Laplace transformation converts a function of t, F(t), into a function of s, f s), where s is the transform variable. The quantity/(s) is called the Laplace transform of F(t). Equation (3-66) shows several equivalent symbolic representations of the Laplace transform of the function y = F(t). 

Then Eqs. (3-130) are substituted into Eqs. (3-128), giving Ca. and cc as functions of time. The final expressions are not written here because we have already derived them by the Laplace transform method they are Eqs. (3-99), (3-101), and (3-103), with X2 and X3 replacing a and p. 

In the context of chemical kinetics, the eigenvalue technique and the method of Laplace transforms have similar capabilities, and a choice between them is largely dependent upon the amount of algebraic labor required to reach the final result. Carpenter discusses matrix operations that can reduce the manipulations required to proceed from the eigenvalues to the concentration-time functions. When dealing with complex reactions that include irreversible steps by the eigenvalue method, the system should be treated as an equilibrium system, and then the desired special case derived from the general result. For such problems the Laplace transform method is more efficient. 

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 first expression on the right is the Laplace transform of the generating function of the now classical case of c = 1, except that p is everywhere replaced by 2p. 

This equation cannot be integrated directly since the temperature 9 is expressed as a function of two independent variables, distance jc and time t. The method of solution involves transforming the equation so that the Laplace transform of 6 with respect to time is used in place of 9. The equation then involves only the Laplace transform 0 and the distance jc. The Laplace transform of 9 is defined by the relation ... 

Thus, a single-valued connection is established between the kernel of this equation R(t) (a memory function ) and KM(t). Their Laplace transforms R and Km are related by... 

Without resorting to the impact approximation, perturbation theory is able to describe in the lowest order in both the dynamics of free rotation and its distortion produced by collisions. An additional advantage of the integral version of the theory is the simplicity of the relation following from Eq. (2.24) for the Laplace transforms of orientational and angular momentum correlation functions [107] ... 

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