The Method of Laplace Transforms

Laplace transforms are particularly suitable for obtaining analytical solutions for certain forms of population balance equations. In aggregating systems, the population balance equation in particle mass (or volume) features a convolution integral in the source term which makes it amenable to solution by Laplace transforms. We shall illustrate the solution of the aggregation problem represented by Eq. (3.3.5), for suitably selected aggregation frequencies. We recall the population balance equation (3.3.5) as 

It is convenient to nondimensionalize the population balance equation using dimensionless variables. Further, we introduce the dimensionless quantities 

Because of (4.3.3) and the conservation of mass, it readily emerges that 

We shall develop the solution for (4.3.4) and (4.3.5) by defining the Laplace transform with respect to the size variable 

By modifying the limits of integration in the first term on the right-hand side of the preceding equation as done just below Eq. (3.3.6) in Section 3.3, we obtain 

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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... 

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 equation is most conveniently solved by the method of Laplace transforms, used for the solution of the unsteady state thermal conduction problem in Chapter 9. 

The solution of Equations 7 and 8 evaluated at the column exit yields the chromatogram. Since these equations cannot be solved analytically, statistical moments were obtained using the method of Laplace transforms (29),... 

First of all, the mathematical background will be developed for the case of a simple electrode reaction O + n e = R. In this treatment, contrasts like potential versus current perturbation, large amplitude versus small amplitude, and single step versus periodical perturbation are emphasized. While discussing these principles, the most common methods derived from them will be briefly mentioned. On the other hand, it will be shown that, by virtue of the method of Laplace transformation, these methods have much in common and contain, in principle, the same information if the detected cell response is of the same order. 

The rigorous solution of this set of equations has been given by Berzins and Delahay [31] using the method of Laplace transformation. We will... 

The method of Laplace transforms can be used to solve equations (7.17)-(7.24), Appendix 1. Re-arrangement of (7.A14) yields ... 

However, one useful technique is the method of Laplace transforms. An excellent tutorial is presented in the two papers by Mayersohn and Gibaldi. Benet and Turi, and Benet present more advanced techniques, such as the input and output disposition functions and the fingerprint technique for the solution of differential equations. 

The second-order partial differential equations given in the text of the paper contain time derivatives of concentration (dC/dt) and terms containing dC/dz and C. The solutions of these equations, unless referred to a literature source, were obtained by the method of Laplace transformation with the aid of standard tables of Laplace transforms. Good working summaries of the Laplace transformation method as applied to... 

In subsequent chapters we will study the method of Laplace transforms, which allows a much simpler development of input-output models from the corresponding state models. 

Solving these linear equations by the method of Laplace transforms and alternately finding the scalar products of the solutions with Ai(q, 0) and A2(q, 0) gives... 

We still have to integrate over the ordered times. This is most simply done by the method of Laplace transforms. We shall merely state the final result ... 

The determination of the density of states for s classical oscillators by the method of Laplace transforms is of limited value because this can be obtained by other methods as well. Of much greater interest is the fact that the product of the quantum oscillators in Eq. (6.48) can be inverted by the Laplace transform method. However, it requires solving the inverse Laplace transform integral (Forst, 1971, 1973 Hoare and Ruijgrok, 1970) ... 

The multiple-trapping model can also be solved analytically for the transit time by the method of Laplace transforms. The one-dimensional transport equations for the fiee-electron density n(x, t) in a semiconductor with a distribution of discrete trapping levels are... 

When the system equations are linear (that is when the adsorption isotherm is linear or the system is perturbed incrementally), we can apply the method of Laplace transform to solve the set of equations and obtain the inverse by either the method of residues or a numerical inversion scheme. For the two types of input, the impulse and the square input, the inversion is not necessary if we are interested in using the response to extract the system parameters. If this is our goal which is the case for the diffusion cell method, then the method of moment can be useful for this purpose. 

Solving the mass balance equations for the case of linear isotherm subject to the boundary conditions (13.2-17) by the method of Laplace transform and from the solution we obtain the following moments when the input is an impulse (Dogu and Ercan, 1983) ... 

The set of governing linear equations are solved by the method of Laplace transform. The transform is defined as... 

Solving these equations subject to the entrance condition (14.2-9) by the method of Laplace transform yields the solution for the exit concentration Cb(L,s) in the Laplace domain. Making use of the formula (14.2-16), we obtain the following first normalised moment and the second central moment. 

This equation was first derived by Ward and Tordai (1946) and later by Hansen (1961), who used the method of Laplace transforms. 

Kipp (1985) transformed the above equations to the dimensionless form by dimensionless factors and variable parameters and solved the equations by the method of Laplace transformation, and finally got the solution through Laplace inverse transformation principle. 

In the sections that follow, solution techniques for linear boundary value problems are developed. Specifically, the method of separation of variables in Section 6.2 is illustrated. In Section 6.3, the method of eigenfunction expansion is outlined. In Section 6.4, the method of Laplace transform is illustrated. The method of combination of variables is outlined in Section 6.5. In Section... 

In this section, the method of Laplace transform will be used. The properties of Laplace transforms, and especially Theorem 3.7 (Section 3.6.1), will be applicable. The Laplace transform was introduced earlier for use in solving ordinary differential equations. Now we will emphasize its use in solving PDEs. 

Use the method of Laplace transform to solve the following problem ... 

A fluid of constant density, p, and viscosity, fi, is contained in a very long horizontal pipe of length L and radius R. Initially, the fluid is at rest. At t = 0, a pressure gradient (po - Pi)IL is impressed on the system. Can the method of Laplace transform be used to determine how the velocity profiles change with time ... 

The method of separation of variables was used to solve the dimensionless system of Equation 1.37a through Equation 1.39a however one could have used the methods of Laplace transform. 

The formula for the induction time of the overall system can be calculated by the method of Laplace transformation, see (Carslaw and Jaeger 1959) and (Barrie et al. 1963). The induction time t of the composite liner is composed of induction times h and t2 and permeabilities P and P2 of the components as follows ... 

Baumeister and Hamill [32] solved the hyperbolic heat conduction equation in a semi-infinite medium subjected to a step change in temperature at one of its ends using the method of Laplace transform. The space-integrated expression for the temperature in the Laplace domain had the inversion readily available within the tables. This expression was differentiated using Leibniz s rule, and the resulting temperature distribution was given for x > X as... 

Allow for particle growth with X(x) = kx in the population balance Eq. (4.3.4) and solve by the method of Laplace transforms for the monodisperse initial condition. 

Solve by the method of Laplace transforms the population balance equation given in Section 3.3.6.2 under equilibrium conditions. And show that the solution is an exponential distribution. 

Similarly, solutions for the product frequency can also be obtained by the method of Laplace transforms (Scott, 1968). For an integrated treatment of the constant, sum and product aggregation frequencies, the reader is referred to Hidy and Brock. 

Establish directly by solving Eq. (5.2.16) via the method of Laplace transforms for the case of constant aggregation frequency, given by a x, x ) = the self-similar solution il/ rj) = e. (Hint Recognize the convolution on the right-hand side of (5.2.16). Letting = ij/ where ij/ is the derivative of the Laplace transform ij o ij/ respect to the transform variable s, obtain and solve a (separable) differential equation for the derivative of 0 with respect to ij/). 

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