Higher-order equations

Here we have neglected derivatives of the local velocity of third and higher orders. Equation (A3.1.23) has the fonn of the phenomenological Newton s law of friction... 

The higher-order equation can be written as a set of first-order equations. 

It was shown, in Eqs. (1-73), (1-74), (1-75), that a = 1, afy r> => 0, a = 0. As the zeroth approximation we shall assume that A mid /a are zero (their effects are negligibly small) if Eqs. (1-86) and (1-87) are multiplied by /a and A, respectively, we obtain the condition that og0 and -oSi are zero higher order equations would show that all the coefficients are zero. Thus, the coefficients are proportional to some power of /a (or A). The zero-order approximation to the distribution function is just the local maxwellian distribution... 

This exercise is identical to how we handle higher order equations in numerical analysis and... 

We need to study the numerical integration of only first-order ODEs. Any higher-order equations, say with Mth-order derivatives, can be reduced to N first-order ODEs. For example, suppose we have a third-order ODE ... 

Hirata, S. Higher-order equation-of-motion coupled-cluster methods. J. Chem. Phys. 2004, 121, 51-9. 

These equations were developed further using similar techniques to those already discussed. The more detailed analysis of liquid structures required to describe the recombination process, than the homogeneous reaction, requires higher-order equations for the liquid structure to be used. This necessarily means that approximations have to be made [286]. 

Surds, of the form v2, y/2, which are obtained from the solution of a quadratic or higher order equation. 

Equations solving quadratic and higher order equations finding the factors and roots of simple polynomial equations using either algebraic or graphical procedures. 

For a higher order equation, the general form is given by... 

In general there is no exact solution to any sequence of consecutive higher-order equations. The reason for this is that the differential equations are no longer linear equations (as they were in the case of first-order reactions), and nonlinear equations do not have exact solutions except in very particular cases. However, two exact methods are available for studying some aspects of these systems, and there is one more commonly used... 

Although the linear model is the model most commonly encountered in analytical science, not all relationships between a pair of variables can be adequately described by linear regression. A calibration curve does not have to approximate a straight line to be of practical value. The use of higher-order equations to model the association between dependent and independent variables may be more appropriate. The most popular function to model non-linear data and include curvature in the graph is to fit a power-series polynomial of the form... 

Successive approximations A procedure for solving higher order equations through the use of intermediate estimates of the... 

We often encounter quadratic or higher-order equations in equilibrium calculations. With modern programmable calculators, solving such problems by iterative methods is often feasible. But frequently a problem can be made much simpler by using some mathematical common sense. 

It we cross-differentiate (12 122a) and (12 122b) and subtract the resulting two equations, we can eliminate the pressure and obtain a single higher-order equation ... 

The governing equations (12-307) for a 2D disturbance (ay = v = 0) can be combined to obtain a single higher-order equation that can be used to study the stability conditions. First, we note that the continuity equation (12-307d) can be satisfied by introducing a function [Pg.875]

Many authors propose alternative mathematical treatments for kinetics equations. Some examples are a general approach based on a matrix formulation of the differential kinetic equations (Berberan-Santos Martinho, 1990) spreadsheets in which rate equations are integrated using the simple Euler approximation (Blickensderfer, 1990) a method for the accurate determination of the first-order rate constant (Borderie, Lavabre, Levy Micheau, 1990) a simplification of half-life methods that provides a fast way of determining reaction orders and rate constants (Eberhart Levin, 1991) a general approach to reversible processes, the special cases of which are shown to be equivalent to basic kinetic equations (Simonyi Mayer, 1985) an equation from which zero-, first- and higher order equations can be derived (Tan, Lindenbaum Meltzer, 1994). 

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