Ordinary Differential Equations of Higher Order

The higher-order differential equations, especially those of order 2, are of great importance because of physical situations describable by them. 

Equation fn) = (xf Such a differential equation can be solved by n integrations. The solution will contain n arbitrary constants. 

Linear Differential Equations with Constant Coefficients and Right-Hand Member Zero (Homogeneous) The solution of y + ay + by = 0 depends upon the nature of the roots of the characteristic equation mr + am + b = 0 obtained by substituting the trial solution y = emx in the equation. 

Example The differential equation My + Ay + ky = 0 represents the vibration of a linear system of mass M, spring constant k, and damping constant A. If A 2 VkM, the roots of the characteristic equation 

This solution is oscillatory, representing undercritical damping. 

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Bagmut, G. (1969) Difference schemes of higher-order accuracy for an ordinary differential equations with singularity. Zh. Vychisl. Mat. i Mat. Fiz., 9, 221-226 (in Russian) English transl. in USSR Comput. Mathem. and Mathem. Physics. 

This equation must be solved for yn +l. The Newton-Raphson method can be used, and if convergence is not achieved within a few iterations, the time step can be reduced and the step repeated. In actuality, the higher-order backward-difference Gear methods are used in DASSL [Ascher, U. M., and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia (1998) and Brenan, K. E., S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North Holland Elsevier (1989)]. 

The solution of these six ordinary differential equations is readily obtained by any ODE software. Note that the rate equation could be any function of C without complicating the solution procedure unduly. When higher axial derivatives are present, each radial equation must be reduced to a set of first order ones. 

A differential equation that involves only ordinary derivatives is called an ordinary dlffecenKal equation, and a differential equation that involves partial derivatives is called a partial differential equation. Then it follows that pfi blems that involve a single independent variable result in ordinary differencial equations, and problems that involve two or more independent variable result in partial differeittial equations. A differential equation may involve several derivatives of various orders of an unknown fiinction. The order of the highest derivative in a differential equation is the order of the eqit tion, yFor example, the order of y" + (y") is 3 since it contains no fourth or higher order derivatives. 

Note that all the conditions are known at one time, t = 0. Thus it is possible to calculate the function on the right-hand side at f = 0 to obtain the derivative there. This makes the set of equations initial value problems. The equations are ordinary differential equations because there is only one independent variable. Any higher-order ordinary differential equation can be turned into a set of first-order ordinary differential equations they are initial value problems if all the conditions are known at the same value of the independent variable [Finlayson, 1980, 1997 (p. 3-54), 1990 (Vol. BI, p. 1-55)]. The methods for initial value problems are explained here for a single equation extension to multiple equations is straightforward. These methods are used when solving plug-flow reactors (Chapter 8) as well as time-dependent transport problems (Chapters 9-11). 

An arbitrary function.of. the. variables must now be added to the integral of a partial differential equation instead of the constant hitherto, employed for ordinary differential equations. -If the number of arbitrary constants to, Jbe eliminated is equal to the number of independent variables, the resulting differential equation is of the first order. The higher orders occur when the number of constants to be eliminated, exceeds that of the independent yariables. 

In this section, a few applications of the theory and methods that were previously outlined will be illustrated. However, it should be noted that a substantial percentage of the application of second (and higher) order ordinary differential equations is in association with solving partial differential equations, a topic discussed in Chapter 6. 

Solution of a Higher-Order Ordinary Differential Equation... 

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... 

One method to solve this boundary value problem of the ordinary differential equations is elimination it attempts to obtain the uncoupled equations with higher order differentiations by eliminating the coupled terms (Humi and Miller 1988). In this method, the boundary... 

In all the above examples, the systems were chosen so that the models resulted in sets of simultaneous first-order ordinary differential equations. These are the most commonly encountered types of problems in the analysis of multicomponent and/or multistage operations. Closed-form solutions for such sets of equations are not usually obtainable. However, numerical methods have been thoroughly developed for the solution of sets of simultaneous differential equations. In this chapter, we discuss the most useful techniques for the solution of such problems. We first show that higher-order differential equations can be reduced to first order by a series of substitutions. 

Higher-order Runge-Kutta formulas are derived in an analogous manner. Several of these are listed in Table 5.2. The fourth-order Runge-Kutta, which has an error of O(h ), is probably the most widely used numerical integration method for ordinary differential equations. 

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