Ordinary differential equations higher order

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. 

Higher Order Linear Ordinary Differential Equations... 

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 chapter, a few methods were presented for obtaining a solution to the linear second-order (or higher) ordinary differential equations. To the inexperienced practitioners, these many options could present a dilemma that is, given a problem, which method should one use ... 

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. 

Hindmarsh AC, Petzold LR (1995) Algorithms and software for ordinary differential equations and differential-algebraic equations. Part II Higher-order methods and software packages. Comput Phys 9 148-155... 

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... 

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