Laplace operator method

In a surfactant mixture the initial conditions, for each component, are equivalent to those for a single surfactant system. When solving the given initial and boundary condition problem the result is Eq. (4.1). The derivation of the solution was performed using Green s functions (Ward Tordai 1946, Petrov Miller 1977) or by the Laplace operator method (Hansen 1961). Appendix 4E demonstrates the application of the operator method for solving such types of transport problems. 

The solution can be found by applying the Laplace operator method (Hansen 1961), leading to two equivalent relationships. 

Integral-Transform Method A number of integral transforms are used in the solution of differential equations. Only one, the Laplace transform, will be discussed here [for others, see Integral Transforms (Operational Methods) ]. The one-sided Laplace transform indicated by L[f t)] is defined by the equation L[f t)] = /(O dt. It has... 

Equations of Convolution Type The equation u x) = f x) + X K(x — t)u(t) dt is a special case of the linear integral equation of the second land of Volterra type. The integral part is the convolution integral discussed under Integral Transforms (Operational Methods) so the solution can be accomplished by Laplace transforms L[u x)] = E[f x)] + XL[u x)]LIK x)] or... 

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. 

Remark Quite often, the Dirichlet problem is approximated by the method based on the difference approximation at the near-boundary nodes of the Laplace operator on an irregular pattern, with the use of formulae (14) instead of (16) at the nodes x G However, in some cases the difference operator so constructed does not possess several important properties intrinsic to the initial differential equation, namely, the self-adjointness and the property of having fixed sign, For this reason iterative methods are of little use in studying grid equations and will be excluded from further consideration. 

This problem can be solved by the method of separation of variables. The eigenvalue problem for the difference Laplace operator i. y = - -... 

This problem can be solved by the method of separation of variables. The eigenvalue problem for the difference Laplace operator Ay = ySlXl + Vx2x2 supplied by the first kind boundary conditions may be set up in a quite similar manner as follows it is required to find the values of the parameter A (eigenvalues) associated with nontrivial solutions of the homogeneous equation subject to the homogeneous boundary conditions... 

The Laplace operator V2 takes the form given in equations (2.11), (2.12) and (2.13) for the different coordinate systems. The differential equations (2.15) and (2.16) for steady-state temperature fields, are linear and elliptical, as long as W is independent of, or changes linearly with d. This leads to different methods of solution than those used in transient conduction where the differential equations are parabolic. 

We use the same symbol as in Section IX.D, but this should not lead to confusion.) Now, however, since F(12)=F(12, t=0) is arbitrary rather than an equilibrium distribution function, it is no longer true that F(12) = 0. Thus using projection operator methods on the Laplace transform of (11.1), we find... 

Here is the generalized Laplace operator, defined by equation (37), while R is the hyperradius (equation (5)). In a later chapter of this book. Professor Fano will discuss the application of the hyperspherical method to nonseparable dynamical problems. Here we shall only note that if mass-weighted coordinates axe used, the Schrodinger equation for any system interacting through Coulomb forces can be written in the form ... 

Several different integration methods are available in the literature. Many of them approximate the effect of the Laplace operator by finite differences [32,39]. Among the more elaborate schemes of this kind we mention the Crank-Nicolson scheme [32] and the Dufort-Frankel scheme [39]. 

Let us consider an example of using the operator method for solving a differential equation describing a concentration change of an intermediate during a first-order consecutive reaction (Fig. 2.8). The original function here is the intermediate current concentration CB(t). A use of the Mathcad operator Laplace leads to the Laplace transform of this function in the form Laplace (CbIL), t,s). Pay attention to the result of the transform of the derivative Cb (t). Besides the... 

The same sequence of the operations can be performed by the Maple suites. But in contrast to Mathcad, where a user has to find a Laplace transform and recover an original function himself, the Maple s operator method for solving an ODE is almost completely automated. If it is necessary to find a solution by means of mathematical apparatus of operational calculus, it is enough to specify an additional option in the body of dsolve in the form of the expression method = laplace. Let us illustrate this for seeking the general solution of the linear second-order differential equation... 

For this type of reactions, ideology of the operator method in respect to solving systems of linear differential equations remains completely the same but in this case, when applying the Laplace transform, we obtain not a separate algebraic... 

Two examples will illustrate the operations which have been performed when it is stated the equation has been solved by the Laplace transform method . First we solve the rate equation for a case when the function equals zero at / = 0. We re-investigate the behav-... 

The two ODEs are solved simultaneously by an extension of the D-operator method outlined in Appendix A1 or by Laplace transformation. The results can be arranged into the following dimensionless form ... 

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