Heaviside operator

In manipulating the Heaviside operator D, the laws of algebraic operation must be followed. These basic laws are as follows. 

This method parallels the Heaviside operator method used in Chapter 2. We first define the incrementing operator as... 

There is some similarity to the Heaviside operator, in the sense that first, second, and so forth derivatives yield s, operators in a manner similar to D, in Heaviside operators however, Laplace transforms also require additional knowledge of initial conditions. The transform of integrals is also similar in form to the Heaviside operator, as we see next. 

We see that s appears in the denominator, just as in the Heaviside operation. 

The nature of the Laplace transform has now been sufficiently studied so that direct applications to solution to physicochemical problems are possible. Because the Laplace transform is a linear operator, it is not suitable for nonlinear problems. Moreover, it is a suitable technique only for initial-value problems. We have seen (Table 9.1) that certain classes of variable coefficient ODE can also be treated by Laplace transforms, so we are not constrained by the constant coefficient restriction required using Heaviside operators. 

Classical algebra < Laplace transformation < Heaviside operational calculus < Formal Graph... 

Here 0 is the Heaviside function. The projection operator formalism must be carried out in matrix from and in this connection it is useful to define the orthogonal set of variables, k,uk,5k > where the entropy density is sk = ek — CvTrik with Cv the specific heat. In terms of these variables the linearized hydrodynamic equations take the form... 

We consider in the following the proof that the projection operators in Eq. (F.27) are equivalent. We introduce the unit operator exp(—iHot/h)exp(iHot/h) on both sides of the Heaviside step functions. We find... 

This is done by writing down the coordinate representations of the operators using the methodology presented in this appendix. We introduce unit operators on both sides of the Heaviside step function using the momentum eigenstates and find... 

On the other hand, for a step input pressure p t) = pQH t), where H t) is the Heaviside step function, the Laplace transform of Eq, (16.66a), in combination with Eq. (16.68), gives the displacement in terms of the operational... 

Goldstein, S. The application of Heaviside s operational method to the solution of a problem in heat conduction. Z. angew. Math. Mech. 12 (1932) 234-243, and On the calculation of the surface temperature of geometrically simple bodies. Z. angew. Math. Mech. 14 (1934) 158-162... 

In order to avoid mathematical difficulties when applying the averaging operators to the generic equation we introduce the phase indicator function Xfc(r,t p) in any realization. This Heaviside function picks the phase k while ignoring the other phases and the interface. The characteristic function is defined by ... 

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