First-order ordinary differential separable

Equations T.ll.iii and V.ll.iv are two coupled first-order ordinary differential eqnations in dcldz and d I dz. These can be separated and integrated to yield... 

Once the effective rate forms at the particle/bubble level are established, and flow patterns as assumed in Figure 6.2 are available, one simply uses these effective rate expressions to write down the corresponding steady-state material balances for the reactor for the assumed flow patterns. Under steady-state conditions, this involves either first-order ordinary differential equations for the phases in which plug flow is assumed or simple difference equations in species concentration in phases in which completely mixed flow is assumed. The treatment in all these cases is very similar to what will be in a single-phase reactor (see, e.g.. Ref [48]), except that one has a separate differential equation balancing for each species concentration and they are coupled through the effective reaction rate term. 

There are several ways to solve a third-order ordinary-differential-equation boundary-value problem. One is shooting, which is discussed in Section 6.3.4.1. Here, we choose to separate the equation into a system of two equations—one second-oider and one first-order equation. The two-equation system is formed in the usual way by defining a new variable g = /, which itself serves as one of the equations,... 

To solve the 2nd order ordinary differential equation in Rqs. 4-39, wc define a new variable w as w = dT/dj). This reduces Eq. 4-39u into a first order differential equation than can be solved by separating variables. 

However, such an equality is impossible, since by changing one of arguments, for example, R, the first term varies while the second one remains the same, and correspondingly the sum of these terms cannot be equal to zero for arbitrary values of R and 0. Therefore, we have to conclude that neither term depends on the coordinates and each is constant. This fact constitutes the key point of the method of separation of variables, allowing us to describe the function C/ as a product of two functions, each of them depending on one coordinate only. For convenience, let us represent this constant in the form +m, where m is called a constant of separation. Thus, instead of Laplace s equation we have two ordinary differential equations of second order ... 

Decaying exponents were chosen because in this representation the memory kernels (6.2.22) separate and the integro-difFerential equation (6.2.21) can be reduced to a set of ordinary coupled first order differential equations. This is achieved by introducing the functions ... 

Each of the first three terms in Eq (3.40) depends on one variable only, independent of the other two. This is possible only if each term separately equals a constant, say, —a, and — respectively. These constants must be negative in order that > 0. Eq (3.40) is thereby transformed into three ordinary differential equations ... 

The moment model approach provides a set of ordinary differential equations (ODEs). Prom the definition of i-th moment in Equation 10.12, we can convert the population balance in Equation 10.10 to moment equations by multiplying both sides by P, and integrating over aU particle sizes. The moments of order four and higher do not affect those of order three and lower, implying that only the first four moments and concentration can adequately represent the crystallization dynamics[100j. Separate moment equations are used for the seed and nuclei classes of crystals, and are defined as follows... 

This section deals with the construction of optimal higher order FDTD schemes with adjustable dispersion error. Rather than implementing the ordinary approaches, based on Taylor series expansion, the modified finite-difference operators are designed via alternative procedures that enhance the wideband capabilities of the resulting numerical techniques. First, an algorithm founded on the separate optimization of spatial and temporal derivatives is developed. Additionally, a second method is derived that reliably reflects artificial lattice inaccuracies via the necessary algebraic expressions. Utilizing the same kind of differential operators as the typical fourth-order scheme, both approaches retain their reasonable computational complexity and memory requirements. Furthermore, analysis substantiates that important error compensation... 

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