Laplace transform complex systems

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 inverse transform Xp(t) in the time domain can be obtained by means of the method of indeterminate coefficients, which was presented above in Section 39.1.6. In this case the solution is the same as the one which was derived by conventional methods in Section 39.1.2 (eq. (39.16)). The solution of the two-compartment model in the Laplace domain (eq. (39.77)) can now be used in the analysis of more complex systems, as will be shown below. 

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). 

Equation (9.15) was written for macro-pore diffusion. Recognize that the diffusion of sorbates in the zeoHte crystals has a similar or even identical form. The substitution of an appropriate diffusion model can be made at either the macropore, the micro-pore or at both scales. The analytical solutions that can be derived can become so complex that they yield Httle understanding of the underlying phenomena. In a seminal work that sought to bridge the gap between tractabUity and clarity, the work of Haynes and Sarma [10] stands out They took the approach of formulating the equations of continuity for the column, the macro-pores of the sorbent and the specific sorption sites in the sorbent. Each formulation was a pde with its appropriate initial and boundary conditions. They used the method of moments to derive the contributions of the three distinct mass transfer mechanisms to the overall mass transfer coefficient. The method of moments employs the solutions to all relevant pde s by use of a Laplace transform. While the solutions in Laplace domain are actually easy to obtain, those same solutions cannot be readily inverted to obtain a complete description of the system. The moments of the solutions in the Laplace domain can however be derived with relative ease. 

All the system response curves in frequency and time domains were calculated numerically from equations that are much too involved to reproduce in detail here. Transfer functions in Laplace transform notation are easily defined for the potentiostat and cell of Figure 7.1. Appropriate combinations of these functions then yield system transfer functions that may be cast into time- or frequency-dependent equations by inverse Laplace transformation or by using complex number manipulation techniques. These methods have become rather common in electrochemical literature and are not described here. The interested reader will find several citations in the bibliography to be helpful in clarifying details. 

In chemical degradation kinetics and pharmacokinetics, the methods of eigenvalue and Laplace transform have been employed for complex systems, and a choice between two methods is up to the individual and dependent upon the algebraic steps required to obtain the final solution. The eigenvalue method and the Laplace transform method derive the general solution from various possible cases, and then the specific case is applied to the general solution. When the specific problem is complicated, the Laplace transform method is easy to use. The reversible and consecutive series reactions described in Section 5.6 can be easily solved by the Laplace transform method ... 

Once the transfer function is known for a particular complex system, then the response of the system to any known input is readily found by multiplying the transfer function by the Laplace transform of the input. 

When discussing diffusion, one inevitably needs to solve diffusion equations. The Laplace transform has proven to be the most effective solution for these differential equations, as it converts them to polynomial equations. The Laplace transform is also a powerful technique for both steady-state and transient analysis of linear time-invariant systems such as electric circuits. It dramatically reduces the complexity of the mathematical calculations required to solve integral and differential equations. Furthermore, it has many other important applications in areas such as physics, control engineering, signal processing, and probability theory. 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

Other problems of transient system response may be solved in a similar way. More complex examples are presented, for example, in Refs. 33-34. It should be added that an arbitrary signal may be applied to the system and if the Laplace transforms of the potential and current are determined, for example, by numerical transform calculations, the system impedance is determined. In the Laplace space the equations [e.g., Eqs. (9) and (11)] are much simpler than those in the time space [e.g. Eqs. (10) and (12)] and analysis in the frequency space 5 allows the determination of the system parameters. This analysis is especially important when an ideal potential step caimot be applied to the system because of the bandwidth limitations of the potentiostat. In this case it is sufficient to know i(t) and the real value of the potential applied to the electrodes by the potentiostat, E(t), which allows numerical Laplace transformations to be carried out and the system impedance obtained. 

It was shown in Section I. l(i) that the system impedance is defined as the ratio of Laplace transforms [Eq. (6)], of potential and current. In general, the transformation parameter is complex, 5 = v -i- jto. The imaginary Laplace transform... 

We can ask — why do we need the complex s-plane at all if we are going to use s = jco anyway at the end The answer to that is — we don t always. For example, at some later stage we may want to compute the exact response of the power supply to a specific disturbance (like a step change in line or load). Then we would need the s-plane and the Laplace transform. So, even though more often we end up just doing steady state analysis, by having already characterized the system in the framework of s, we retain the option to be able to conduct a more elaborate analysis of the system response to a more general stimulus if required. 

The Laplace transform, which is a linear operator, is frequently used as a mathematical tool when dealing with linear systems. The Laplace transform is often useful in dealing with more complex convolution relationships. For example, consider the following property of the Laplace transform operation L ... 

As mentioned above, the obvious step is to solve equation (3.17) sequentially, i.e. start with Pj, then substitute it into the equation for Pj, solve it and so on. For the case of (3.17) this can be done analytically and it becomes clear there is a general form of the solution for P,- (see example 3.1). In more complex cases this cannot be done analytically but can still be carried out numerically in the style of Liu and Amundson [15]. The simple system (3.17) is dso amenable to a direct analytical solution via the Laplace transform [16], Pj... 

It can be seen from the discussion earlier that NMR relaxation T2 can be treated as an indicator of molecular mobility. In some complex sample systems such as foodstuffs, the NMR relaxation mechanism is very complicated, and it is impossible to use a small number of discrete relaxation times to describe molecular mobility. A new approach has been explored to use a continuous distribution of T to study the thermodynamics of complex systems. An inverse Laplace transformation is applied to the echo train to resolve a number of different exponentials. The result is expressed in a form of T2 distribution as illustrated in Figure 3.72. 

While this book only uses Laplace transforms when the alternative would be overly complex, they are used in many text books and by control system vendors. So that they can be recognised, the transforms for the common types of process are listed here. 

As complex reactions follow a reaction mechanism involving various elementary steps, the determination of the corresponding kinetic law involves the solution of a system of differential equations, and the complete analytical solution of these systems is only possible for the simplest cases. In slightly more complicated cases it may stiU be possible to resolve the system of corresponding differential equations using methods such as Laplace transforms or matrix methods. However, there are systems which cannot be resolved analytically, or whose... 

The linearized and Laplace-transformed equations of the models described above are used to evaluate the various system transfer functions as functions of the Laplace variables s = cr + jco, where a is the real part and co is the imaginary part of the complex variable s. a refers to the damping constant (or damped exponential frequency) and co refers to the resonant oscillation frequency of the system. 

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