Differential equations, positive function

Differentiation of Eq. (8) with respect to the position of a molecule gives a hierarchy of integro-differential equations, each of which relates a distribution function to the next higher order distribution function. Specifically,... 

Schrodinger (1926) postulated that this differential equation is also valid when the potential energy is not constant, but is a function of position. In that case the partial differential equation becomes dW(x, i) d f x, t)... 

The product of a function and its complex conjugate is always real and is positive everywhere. Accordingly, the wave function itself may be a real or a complex function. At any point x or at any time t, the wave function may be positive or negative. In order that F(x, t)p represents a unique probability density for every point in space and at all times, the wave function must be continuous, single-valued, and finite. Since F(x, /) satisfies a differential equation that is second-order in x, its first derivative is also continuous. The wave function may be multiplied by a phase factor e , where a is real, without changing its physical significance since... 

Note that the equation includes rates of gain and loss rather than total amounts. As a result, the mathematical form will be a differential equation rather than an algebraic one. The differential form is preferred in almost all mass transfer problems, because variations in the rates with position and time can be incorporated accurately. Each term in the equation will take on a specific functional form depending on the parameters and mass transfer characteristics of the problem of interest. 

The effect of advection and dispersion on the distribution of a chemical component within flowing groundwater is described concisely by the advection-dispersion equation. This partial differential equation can be solved subject to boundary and initial conditions to give the component s concentration as a function of position and time. 

To evaluate the effect of pressure drop on performance, differential equations for the pressure drop (15.2-11), material balance (15.2-4), and energy balance (15.2-10) must be integrated simultaneously to solve for P,fA, and T as functions of axial position, x ... 

The differential equation for dispersion in a cylindrical bed of voidage e may be obtained by taking a material balance over an annular element of height SI, inner radius r, and outer radius r + Sr (as shown in Figure 4.5). On the basis of a dispersion model it is seen that if C is concentration of a reference material as a function of axial position /, radial position r, time t, and DL and DR are the axial and radial dispersion coefficients, then ... 

For present purposes, the functions of time, f(f), which will be encountered will be piecewise continuous, of less than exponential order and defined for all positive values of time this ensures that the transforms defined by eqn. (A.l) do actually exist. Table 9 presents functional and graphical forms of f(t) together with corresponding Laplace transforms. The simpler of these forms can be readily verified using eqn. (A.l), but as extensive tables of functions and their transforms are available, derivation is seldom necessary, (see, for instance, ref. 75). A simple introduction to the Laplace transform, to some of its properties and to its use in solving linear differential equations, is presented in Chaps. 2—4 of ref. 76, whilst a more complete coverage is available in ref. 77. 

Note that setting one of the terms on the left side of the equation equal to zero yields either the batch reactor equation or the steady-state PFTR equation. However, in general we must solve the partial differential equation because the concentration is a function of both position and time in the reactor. We will consider transients in tubular reactors in more detail in Chapter 8 in connection with the effects of axial dispersion in altering the perfect plug-flow approximation. 

Note that now Tj is a variable that is a function of position Zc in the cooling coif while T, the reactor temperature in the CSTR reactor, is a constant. We can solve this differential equation separately to obtain an average coolant temperature to insert in the reactor energy-balance equation. However, the heat load on the cooling coil can be comphcated to calculate because the heat transfer coefficient may not be constant. 

Dennis and Walker (D3) expanded and Z as a series of Legendre functions in the position coordinates. Equations (5-1) and (5-2) were reduced to a set of ordinary differential equations, solved numerically. This approach is inconvenient for high Re since the number of terms which must be included becomes prohibitive. Solutions to the steady equations were obtained for Re < 40 (D3) and for impulsively started motion for Re < 100 (D4). 

Equations (5.95), (5.96) and (5.97) are suitable for constant critical melting porosity. In a one dimensional steady state melting column as a result of decompression melting, the porosity may increase from the bottom to the top of the column. If melting porosities change as a function of the spatial position, the related differential equations need to be solved numerically. More details of various melt transport models by porous flow have been given by Spiegelman and Elliott (1993), Iwamori, (1994), and Lundstrom (2000). 

The first step in the solution procedure is discretization in the radial dimension, which involves writing the three-dimensional differential equations as an enlarged set of two-dimensional equations at the radial collocation points with the assumed profile identically satisfying the radial boundary conditions. An examination of experimental measurements (Valstar et al., 1975) and typical radial profiles in packed beds (Finlayson, 1971) indicates that radial temperature profiles can be represented adequately by a quadratic function of radial position. The quadratic representation is preferable to one of higher order since only one interior collocation point is then necessary,6 thus not increasing the dimensionality of the system. The assumed radial temperature profile for either the gas or solid is of the form... 

Since the resulting system after radial collocation is still too complex for direct mathematical solution, the next step in the solution process is discretization of the two-dimensional system by orthogonal collocation in the axial direction. Although elimination of the spatial derivatives by axial collocation increases the number of equations,8 they become ordinary differential equations and are easily solved using traditional techniques. Since the position and number of points are the only factors affecting the solution obtained by collocation, any set of linearly independent polynomials may be used as trial functions. The Lagrangian polynomials of degree N based on the collocation points... 

Equation (26) is a differential equation with a solution that describes the concentration of a system as a function of time and position. The solution depends on the boundary conditions of the problem as well as on the parameter D. This is the basis for the experimental determination of the diffusion coefficient. Equation (26) is solved for the boundary conditions that apply to a particular experimental arrangement. Then, the concentration of the diffusing substance is measured as a function of time and location in the apparatus. Fitting the experimental data to the theoretical concentration function permits the evaluation of the diffusion coefficient for the system under consideration. 

The next problem is to express the charge density as a function of the potential so the differential equation (26) can be solved for f/. The procedure is to describe the ion concentrations in terms of the potential by means of a Boltzmann factor in which the work required to bring an ion from infinity to a position at which the potential p is given by z,e p. The probability of finding an ion at this position is given by the Boltzmann factor, with this work appearing as the exponential of energy ... 

The local stability of a given stationary-state profile can be determined by the same sort of test applied to the solutions for a CSTR. Of course now, when we substitute in a = ass + Aa etc., we have the added complexity that the profile is a function of position, as may be the perturbation. Stability and instability again are distinguished by the decay or growth of these small perturbations, and except for special circumstances the governing reaction-diffusion equation for SAa/dr will be a linear second-order partial differential equation. Thus the time dependence of Aa will be governed by an infinite series of exponential terms ... 

The only function of interest in the given context is w(Ar). The stability question is then answered if the rate, w(A), has been found to be positive or negative at any value of k or wavelength A of the perturbation. The validity of this argument is due to the linearized differential equations, for which we know their solutions can be superposed. Negative w(A) means that 0- O for t- o°. Insertion of Eqns. (11.16) and (11.17) into the transport equation and the boundary condition yields an implicit equation for w(k). If we use the following transformations to express w and tin terms of the characteristic parameters Dv and v of the system, namely... 

This is a functional equation for the boundary position X and the unknown constant parameter n. Upon substituting Eq. (256) into Eq. (251) an ordinary differential equation is obtained for X(t, n), and a family of curves in the phase plane (X, X) can be obtained. For n sufficiently close to unity two functions in the phase plane can be determined which serve as upper and lower bounds for the trajectories. The choice is guided by reference to the exact solution for the limiting case of constant surface temperature. It is shown that the upper and lower bounds are quite close to the one-parameter phase plane solution, although no comparison is made with a direct numerical solution. The one-parameter solution also agrees well with experiments on the solidification of aluminum under conditions of low surface heat transfer coefficient (hi = 0.02 cm.-1). 

The trial-and-error method for differential equations used in Section 2.3 suggests that we should look for a function which, when differentiated twice, gives a positive multiple of itself. The exponential function has this property, and so we try... 

This time-dependent partial differential equation relates the change in concentration at any point as a function of time to the change in the concentration gradient with respect to position. This important equation is the starting point for a large amount of literature in mass transfer that deals primarily with solving this equation subject to various initial and boundary conditions. 

In Eq. (32), we have included the full spectrum of a second-order ordinary differential equation with negative bound state eigenvalues and the continuum being the positive real axis. The free-particle background is mbee = i /k. In this case, the full m-function becomes [in the equation below we have introduced a natural generalization of Js, the "jump" or imaginary part, see Eq. (36) for the general case]... 

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