Higher-Order Differentiation

The method of Ishida et al [84] includes a minimization in the direction in which the path curves, i.e. along (g/ g -g / gj), where g and g are the gradient at the begiiming and the end of an Euler step. This teclmique, called the stabilized Euler method, perfomis much better than the simple Euler method but may become numerically unstable for very small steps. Several other methods, based on higher-order integrators for differential equations, have been proposed [85, 86]. 

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

Again the specific remarks for y" + ay + hy =f x) apply to differential equations of similar type but higher order. We shall discuss two general methods. 

Method of Variation of Parameters This method is apphcable to any linear equation. The technique is developed for a second-order equation but immediately extends to higher order. Let the equation be y" + a x)y + h x)y = R x) and let the solution of the homogeneous equation, found by some method, he y = c f x) + Cofoix). It is now assumed that a particular integral of the differential equation is of the form P x) = uf + vfo where u, v are functions of x to be determined by two equations. One equation results from the requirement that uf + vfo satisfy the differential equation, and the other is a degree of freedom open to the analyst. The best choice proves to be... 

Partial Differential Equations of Second and Higher Order... 

Equation (8-14) shows that starts from 0 and builds up exponentially to a final concentration of Kcj. Note that to get Eq. (8-14), it was only necessaiy to solve the algebraic Eq. (8-12) and then find the inverse of C (s) in Table 8-1. The original differential equation was not solved directly. In general, techniques such as partial fraction expansion must be used to solve higher order differential equations with Laplace transforms. 

Differentiation of Eq. (8) with respect to the position of a molecule gives a hierarchy of integro-differential equations, each of which relates a distribution function to the next higher order distribution function. Specifically,... 

In practice the finite-field calculation is not so simple because the higher-order terms in the induced dipole and the interaction energy are not negligible. Normally we use a number of applied fields along each axis, typically multiples of 10 " a.u., and use the standard techniques of numerical analysis to extract the required data. Such calculations are not particularly accurate, because they use numerical methods to find differentials. 

Differentials of higher orders are of little significance unless dx is a constant, in which case the first, second, third, etc. differentials approximate the first, second, third, etc. differences and may be used in constructing difference tables (see Algebra ). 

A simple repetition of the iteration procedure (2.20)-(2.22) results in divergence of higher order solutions. However, a perturbation theory series may be summed up so that all unbound diagrams are taken into account, just as is usually done for derivation of the Dyson equation [120]. As a result P satisfies the integral-differential equation... 

Our results indicate that dispersion coefficients obtained from fits of pointwise given frequency-dependent hyperpolarizabilities to low order polynomials can be strongly affected by the inclusion of high-order terms. A and B coefficients derived from a least square fit of experimental frequency-dependent hyperpolarizibility data to a quadratic function in ijf are therefore not strictly comparable to dispersion coefficients calculated by analytical differentiation or from fits to higher-order polynomials. Ab initio calculated dispersion curves should therefore be compared with the original frequency-dependent experimental data. 

Alekseevskii, M. (1984) Difference schemes of higher-order accuracy for some singular-perturbed boundary-value problems. Differential Equations, 17, 1177-1183 (in Russian). 

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. 

As shown in this chapter, in the simulation of systems described by partial differential equations, the differential terms involving variations with respect to length are replaeed by their finite-differenced equivalents. These finite-differenced forms of the model equations are shown to evolve as a natural eonsequence of the balance equations, according to the manner of Franks (1967). The approximation of the gradients involved may be improved, if necessary, by using higher order approximations. Forward and end sections can... 

We now move onto a few so-called higher order or complex processes. We should remind ourselves that all linearized higher order systems can be broken down into simple first and second order units. Other so-called "complex" processes like two interacting tanks are just another math problem in coupled differential equations these problems are still linear. The following sections serve to underscore these points. 

Higher Differentials The first derivative offfx) with respect to x is denoted by/ or df/dx. The derivative of the first derivative is called the second derivative offfx) with respect to x and is denoted by or d2f/dx2 and similarly for the higher-order derivatives. 

Partial Differential Equations of Second and Higher Order Many of the applications to scientific problems fall naturally into partial differential equations of second order, although there are important exceptions in elasticity, vibration theory, and elsewhere. 

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 most important quantitative measure for the degree of chaotic-ity is provided by the Lyapunov exponents (LE) (Eckmann and Ru-elle, 1985 Wolf et. al., 1985). The LE defines the rate of exponential divergence of initially nearby trajectories, i.e. the sensitivity of the system to small changes in initial conditions. A practical way for calculating the LE is given by Meyer (Meyer, 1986). This method is based on the Taylor-expansion method for solving differential equations. This method is applicable for systems whose equations of motion are very simple and higher-order derivatives can be determined analytically (Schweizer et.al., 1988). 

Numerical solution of higher order differential equations is accomplished most conveniently by first converting them into an equivalent set of first order equations. Thus the second order equation... 

Assuming Taylor series expansion using only zero- and first-order terms (dropping second and higher order terms), we arrive at the linear or linearized system described by... 

Special attention should be paid to the selection of additives, as they drive the higher-order quality factors that differentiate one product from another. Table 7 summarizes commonly used additives in detergent products along with their... 

---