Laplace transform solutions

Suggestion Use superposition of known solutions, or split the problem into two parts and use continuity to match Laplace-transformed solutions. 

The quasireversible LSV case was treated by Matsuda and Ayabe [389], who used a series sum as an approximation to the integral equation obtained from the Laplace-transform solution of the problem. The result depends on the heterogeneous rate constant, both the peak current and the peak potential varying with this parameter. Basha et al. [82] tried to improve on the results but it seems that those of Nicholson and Shain [417] were better. These also provided results for the totally irreversible case, first described by Delahay [199]. For this, the y(ai)-function has a constant maximum, given to four figures in [73,74], 0.4958, from the tables in [417]. Peak potential varies with rate constant, as with the quasireversible case. 

Here we see an exponential size distribution is predicted by the population balance. (B(Lq) is also the birth term due to nucleation of particles of size Lq. It could also be used in the population balance substituting for B directly, but this approach requires a LaPlace transform solution, which also results in equation (3.14).) Many inorganic precipitations operate in this way with small supersaturation that is, nearly all the mass is precipitated in one pass through the precipitation. This equation for a well-mixed constant volume aystallizer will be discussed in further detail in Chapter 6. 

This pair of coupled linear equations may be solved using Laplace transforms. Solution of Equations 20-23 yields... 

If f(t) = 0 for t < 0, then substitution of s = i into the above equation yields the definition of the Laplace transform. Thus, the Fourier transform is obtained from the Laplace transform solution (Equation 25) by the substitution s = i . The Fourier... 

The derivation of analytical expressions for the moments of a band in chromatography is tedious. It involves successive differentiations of the Laplace transform solution of the chromatography model used. Several more expedient methods have been proposed to simplify these derivations for axial chromatography [43,44]. A simple and generalized method was described by Lee et al. [45] for the moments in chromatographic elution peaks with any geometric configuration (axial or radial) and any kinetic models. 

A time periodic solution of the set of equations 15.17a to 15.17d with a period 2nr is easily derived by applying the residue theorem to the general inversion integral of the Laplace transform solution of these equations [15]. It can be written as ... 

A differential equation can be transformed into an algebraic equation by a Laplace transformation. Solution of this equation followed by inverse transformation provides a solution to the differential equation. 

Equations D-13 and D-14 are first order differential equations. Again, using the Laplace Transform solution method ... 

Such boundary conditions can be handled most easily using a Laplace transform solution method, as we demonstrate later in this chapter. 

The constitutive equations (115), (116), (118) are then used to developed governing equations for the closure of a long cylindrical tunnel in poroviscoelastic massif. Laplace transform solutions have been develofjed and discussed in detail in Dufour et al. (2009) to which interested readers may refer. 

This above solution could also have been obtained by the method of successive approximations discussed in Section 4.2 (see Exercise 4.2.3). Seinfeld and co-workers have made extension of such Laplace transform solutions to multidimensional problems in the coagulation of aerosols. ... 

Eq. (2.13) follows immediately from the Laplace transform solution of (2.1) ... 

Next we apply the Laplace transform solution to a higher-order differential equation. 

The high-order denominator polynomial in a Laplace transform solution arises from the differential equation terms (its characteristic polynomial) plus terms contributed by the inputs. The factors of the characteristic polynomial correspond to the roots of the characteristic polynomial set equal to zero. The input factors may be quite simple. Once the factors are obtained, the Laplace transform is then expanded into partial fractions, As an example, consider... 

It is clear from this example that the Laplace transform solution for complex or repeated roots can be quite cumbersome for transforms of ODEs higher than second order. In this case, using numerical simulation techniques may be more efficient to obtain a solution, as discussed in Chapters 5 and 6. 

The initial value of 1 corresponds to the initial condition given in Eq. 3-26. The final value of 0.5 agrees with the time-domain solution in Eq. 3-37. Both theorems are useful for checking mathematical errors that may occur in obtaining Laplace transform solutions. 

Moench, A.F. 1989. Convergent radial dispersion A Laplace transform solution for aquifer tracer testing. Water Resources Research 25(3) 439 47. 

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