Laplace transform techniques

The Laplace transform technique also allows the reduction of the partial differential equation in two variables to one of a single variable In the present case. 

Solving equation (1-8) (using Laplace transform techniques) yields the time evolution of the current of a spherical electrode ... 

This chapter has presented time-domain solutions of unsteady material and energy balances. The more usual undergraduate treatment of dynamic systems is given in a course on control and relies heavily on Laplace transform techniques. One suitable reference is... 

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

These can be solved by classical methods (i.e., eliminate Sout to obtain a second-order ODE in Cout), by Laplace transformation techniques, or by numerical integration. The initial conditions for the washout experiment are that the entire system is full of tracer at unit concentration, Cout = Sout = L Figure 15.7 shows the result of a numerical simulation. The difference between the model curve and that for a normal CSTR is subtle, and would not normally be detected by a washout experiment. The semilog plot in Figure 15.8 clearly shows the two time constants for the system, but the second one emerges at such low values of W t) that it would be missed using experiments of ordinary accuracy. 

Pearson, E. M. Halicioglu, T. Tiller, W. A., Laplace-transform technique for deriving thermodynamic equations from the classical microcanonical ensemble, Phys. Rev. A 1988, 32, 3030-3039... 

The initial material charged to the vessel contains only A at a concentration Cao Use Laplace transform techniques to solve for the changes in Ca and Cb with time during the batch cycle for ... 

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

The statement cA = c0/ (1 + K) in Eqs. (157a and b) above is tantamount to saying that cA + cB = Co, where c0 is the total concentration of both species of the dissolved solute. If the diffusivities SDA8 and DBs are assumed to be equal, then cB can be eliminated from Eqs. (155) and (156) and a fourth-order, linear partial-differential equation is obtained. The solution of this equation consistent with the conditions in Eq. (157) is obtainable by Laplace transform techniques (S9). Sherwood and Pigford discuss the results in terms of the behavior of the liquid-film mass transfer coefficient. 

Use of Computer Simulation to Solve Differential Equations Pertaining to Diffusion Problems. As shown earlier (Section 4.2.11), differential equations used in the solutions of Fick s second law can often be solved analytically by the use of Laplace transform techniques. However, there are some cases in which the equations can be solved more quickly by using an approximate technique known as the finite-difference method (Feldberg, 1968). 

Equation (122) together with the conditions (127)—(131) can be solved by the Laplace transform technique, details of which are given in Chap. 2, Sect. 3 and Appendix A. Collins and Kimball [4] showed that time-dependent recombination probability q(t r0, t0) was... 

These were solved thus using generating function techniques, though Laplace transform technique could equally well have been used. The most important quantity of interest for comparison with other analysis is the average number of reactants present at any time, t, (AD = 2 NPn. This was shown to be... 

This equation can be solved by Laplace transform techniques and Mt expressed as modified spherical Bessel functions [28]. However, because the boundary conditions on M are radically symmetric, only the / = 0 (i.e. S-wave) component is of interest. 

This same pair of reactions was also studied by Ishida,10 who used a Laplace transform technique. Although superficially quite different results were obtained, they are in fact identical to those presented above. 

This is a problem which may be solved by the Laplace-transform technique. The solution is given in Ref. I as... 

Jung et al. [81] reported a variant of this approach that used the multiplicative factor of zero for the SS component and 1.3 for the OS component. This calculation, SOS-MP2 (scaled opposite-spin MP2), can be performed with only an 0(n4) operation cost when combined with Almlof s Laplace transform technique [82], The SOS approximation can be applied to CIS(D) [69], A similar simplification was often adopted in the GW method under the name COHSEX approximation [32] also partly from an operation cost consideration. 

Equations (3-55) to (3-59) were rewritten in dimensionless form and solved analytically by the Laplace transformation technique. The final solution, the theoretical impulse response, can be obtained as follows 69... 

This poses a problem By applying to the system of Eq. (2.27) the Laplace transformation technique illustrated in Chapter I for evaluating the correlation function... 

ODEs can be solved by applying Laplace transform technique. Consider a set of linear ODEs ... 

The I VP solved in example 2.1 is solved below using the Laplace transform technique following the procedure described above. 

This IVP can be solved easily using the Laplace transform technique. The procedure presented above for the Laplace transformation technique can be used for solving this example. When solving a second order differential equation, the... 

In the previous sections we solved linear ODEs using exponential matrix (section 2.1.2 - 2.1.4) and the Laplace transform technique (section 2.1.5). Alternatively, Maple s dsolve command can be used to solve linear ODEs. However, the analytical solution obtained from the dsolve command may not be in a simplified form. 

Maple s dsolve command was used to solve linear ODEs in section 2.1.6. In our opinion, exponential matrix method is the best method to arrive at an elegant analytical solution. The Laplace transform technique illustrated in section 2.1.5 could be used for integro-differential equations. Maple s dsolve command has to be used if the exponential matrix method fails. 

The matrix exponential can also be obtained by using the Laplace transform technique. Taking the Laplace transform of the governing equations now written in lower case X (see Ogata page 725).[1]... 

Transient heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a parabolic partial differential equation. For steady state heat or mass transfer in solids, potential distribution in electrochemical cells is usually represented by elliptic partial differential equations. In this chapter, we describe how one can arrive at the analytical solutions for linear parabolic partial differential equations and elliptic partial differential equations in semi-infinite domains using the Laplace transform technique, a similarity solution technique and Maple. In addition, we describe how numerical similarity solutions can be obtained for nonlinear partial differential equations in semi-infinite domains. 

Parabolic partial differential equations are solved using the Laplace transform technique in this section. Diffusion like partial differential equations are first order... 

Laplace Transform Technique for Parabolic PDEs The dimensionless temperature profile is given by > u =convert(u,erfc) ... 

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