Higher-Order ODEs

Higher order ODEs are reduced to a set of first order equations for solution by these softwares. Thus the third order equation,... 

The higher order ODEs are reduced to systems of first-order equations and solved by the Runge-Kutta method. The missing condition at the initial point is estimated until the condition at the other end is satisfied. After two trials, linear interpolation is applied after three or more, Lagrange interpolation is applied. 

Higher order ODEs (of order n) were converted to a system of n coupled linear first order ODEs in section 2.1.4. This system was then solved using the exponential matrix developed earlier. This approach yields analytical solutions for linear ODEs of any order. In section 2.1.5, the given system of coupled linear ODEs was converted to Laplace domain. The resulting linear system of algebraic equations was then solved for the solution in the Laplace domain. The solution obtained in the Laplace domain was then converted to the time domain. 

Similarly higher order ODEs can be solved using Maple s dsolve command as shown in the next example. 

Frequently, the equations of motion occur in such a form that the effort for solving the corrector equation (4.1.29) can be reduced significantly. This concerns the solution of the linear systems (4.1.30) and the computation the Jacobian by finite differences. This can be done for all ODEs in first order form originating from higher order ODEs. 

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 ... 

The results obtained in the last two sections for simple first- and second-order systems can now be generalized to higher-order systems. Consider the iVth-order ODE... 

Before moving on to real Rosenbrock methods, consider again (4.66). The left-hand side contains a term in fy if we are dealing with a system of odes, this is called the Jacobian of the system. It is often constant, evaluable in advance. It will be seen in Chap. 9 that unless the diffusion problem has nonlinear concentration terms (for example from higher-order homogeneous reactions), the Jacobian is constant. If not, it must be evaluated at every step. 

Higher order linear ODEs can also be solved by changing them into a system of first order ODEs and using the exponential matrix approach discussed earlier. The most general form of a linear ODE of n order is[l]... 

Again, the solution is obtained by finding the exponential matrix and the non-homogeneous part (see equation 2.12). The procedure used to solve higher order linear ODEs can be summarized as follows ... 

Higher order linear ODEs can also be solved using the dsolve command. It should be noted that Maple solves equations in symbolic form. Therefore, even if the constants are numerical, the output is in symbolic form. Sometimes, this output can be messy. It should be noted that when more than two equations are solved the dsolve command may not be able to give an elegant solution. For illustration, the heat transfer problem solved in example 2.3 is solved below using Maple s dsolve command. 

The integration of a DAE system can be performed by transformation in an ODE system. It is worthy to note that this operation might be confronted with the index problem. Index is the minimum number order of differentiation needed to transform a DAE system into a set of first-order ODEs. Problems of index one can be solved by means of standard differentiation methods. When the index is higher than one then the DAE system needs a special treatment. Modem codes have capabilities for automatic detection of index higher-than-one, diagnose the problem and suggest modifications. 

We begin this chapter by studying first order ODE, classifying techniques based on certain properties. We next extend these techniques to second and higher order equations. Finally, we present detailed treatment and very general methods for classes of linear equations, about which much is known. 

Although the R-K approach (and other companion methods) have traditionally been employed to solve first-order ODEs, it can also treat higher ODEs. The procedure requires reducing an nth order ODE to n first-order ODEs. For example, if the equation is of the form ... 

Systems of coupled ODEs might arise from reformulating a higher-order differential equation to a system of first-order differential equations, or as a description of a system that consists of coupled variables. Systems of differential equations can be solved as an extention of the methodology for a single differential equation. The principle is shown in Example 6.4, which considers a stirred tank reactor. 

Finally, even within the bifurcation approach, the ODE model presented is not complete. For example, including more variables in the model can affect its dynamics, as can higher-order terms in the functions /, g, and h in Equations (10). (These effects have not yet been considered because, as a first step, I wanted to consider only the simplest case.) Moreover, no direct correspondence has been established between parameters of the ODE modql and the parameters of any excitable media, though a comparison of the phase diagrams for the ODE model and the reaction-diffusion equations suggests that this might be accomplished. Nevertheless, there is every reason to believe that soon it will be possible to capture completely the dynamics of spiral waves in excitable media with a low-dimensional model similar to the ODE model considered in this chapter, in spirit if not in form. 

What if we have ODEs of higher order than one ... 

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. 

We have found that dynamics can be more conveniently handled in the Russian transfer-function language than in the English ODE language. However, the manipulation of the algebraic equations becomes more and more difficult as the system becomes more complex and higher in order, if the system is th-order, an Afth-order polynomial in s must be factored into its N roots. For N greater than 2, we usually abandon analytical methods and turn to numerical... 

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