Scalar difference equations

Dissolution of a species We consider the dynamics of dissolution of a solid mounted on a revolving assembly to show that the mathematical description of its dissolution gives rise to a simple scalar difference equation. 

Let Xq = initial concentration of the dissolved solid in the solution Xj = solubility of the solid species in a solvent x = concentration of dissolved species at time f = (At)n, where n stands for the number of discrete time steps, (At). For dissolution to occur, we must of course assume that Xq Xj. The rate law for the concentration of dissolved solid in the solution follows the equation 

Thus Equation (2.2) represents the dissolution process in terms of a scalar difference equation that has a straightforward solution. 

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The discrete-time solution of the state equation may be considered to be the vector equivalent of the scalar difference equation method developed from a z-transform approach in Chapter 7. 

The above continuous-time model obviously has a discrete-time counterpart, which for the simpler single-input single-output case may be approximated by a scalar difference equation of the following form (Andersen 1997) ... 

The last term on the right-hand side of (1.28) is the chemical source term. As will be seen in Chapter 5, the chemical source term is often a complex, non-linear function of the scalar fields , and thus solutions to (1.28) are very different than those for the z nm-scalar transport equation wherein S is null. 

If the frequency of the exponential fitting is different from the frequency of the scalar test equation the method is not P-stable i.e. there are areas in the Fig. 28 that are white and in which the conditions of P-stability are not satisfied). 

Applying the above method to the scalar test equation (54) we obtain the difference equation (55) with k = 2 and the associated characteristic equation (56) with also k = 2... 

A common feature for all the different formulations of the VOF model is that the location and orientation of the interface are defined through a volume fraction function. The evolution of this volume fraction function in time and space is determined by a scalar advection equation defined by ... 

In this section, particular pressure-based methods designed to solve the momentum equation are outlined. The numerical methods for solving the momentum component equations differ considerably from those designed to solve the generic scalar transport equation, because the velocity is a vector quantity. The special treatment of the terms in the momentum equations that are different from those in the generic transport equation is summarized. 

For the sum and difference correlation functions, the SSOZ equation decouples into two scalar equations. The equation involving the sum correlation functions and Cj, and their closures, is identical to the scalar SSOZ equation for the nonpolar symmetric diatomic molecule, so its solution is known. The scalar equation for the difference correlation functions is given by... 

We may classify the difference equations encountered in chemical engineering applications into two types, namely, scalar and vector difference equations. The unknowns in difference equations are functions of discrete independent variables. 

In DNS of single-phase flows, a complete set of compressible Navier-Stokes, energy, and scalar transport equations are calculated together with the equation of state and some constitutive relations [3]. In DNS of particle-laden flows, in addition to carrier-gas equations, the Lagrangian form of particle (droplet) equations are solved via standard difference schemes [5]. 

Phase-lag Analysis of General Symmetric 2A-step, keN Methods. - Following the same analysis as above, when a symmetric 2 method is applied to the scalar test equation (130) a difference equation of the form... 

Since elwx is the analytic solution of the scalar test equation (130), the difference equation must satisfied by it. So, we have the following relations... 

The inference from all this is that the vacuum is a universal wave field of long wavelength. In an open universe the wavelength becomes infinitely long, resulting in a quiet flat space-time. In a universe of closed topology there is endless wave motion, which we identify here with the quantum potential and which Milo Wolff (1990)describes as the solution of a universal scalar wave equation. Our interpretation of cosmic physics differs from Wolff s model only in his proposal (p. 171)... 

In case of molecular dipole moments (Section II.B.2) of ketene (43) and methylene ketene (48) which are related to each other by a homology (chain lengthening of the skeleton) retaining the overall symmetry it has been shown that the physical properties only differ by a scalar factor (Equation 20). The magnitude of this factor is determined by the overall length of the cumulenic skeleton. 

In [162] the authors given the necessary relation in order to have the maximum tenth algebraic order and two free parameters. The application of the above method to the scalar test equation (4) gives the following difference equation ... 

The scalar product R+i — flflexcitonl " ) gives the following difference equation for the exciton wavefimction, (r, i ),... 

Based on Lambert and Watson, when a symmetric multistep method is applied to the scalar test equation y" = —q y a difference equation (63) is obtained. The characteristic equation associated with (63) is given by ... 

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