Infinite power series method

Adomian s Decomposition Method is used to solve the model equations that are in the form of nonlinear differential equation(s) with boundary conditions.2,3 Approximate analytical solutions of the models are obtained. The approximate solutions are in the forms of algebraic expressions of infinite power series. In terms of the nonlinearities of the models, the first three to seven terms of the series are generally sufficient to meet the accuracy required in engineering applications. 

A method similar to the iterative, is the partial closure method [37], It was formulated originally as an approximated extrapolation of the iterative method at infinite number of iterations. A subsequent more general formulation has shown that it is equivalent to use a truncated Taylor expansion with respect to the nondiagonal part of T instead of T-1 in the inversion method. An interpolation of two sets of charges obtained at two consecutive levels of truncations (e.g. to the third and fourth order) accelerates the convergence rate of the power series [38], This method is no longer in use, because it has shown serious numerical problems with CPCM and IEFPCM. 

The second general approach to correlation theory, also based on perturbation theory, is the coupled-cluster method, which can be thought of as an infinite-order perturbation method. The coupled-cluster wave function T cc is expressed as a power series,... 

By using the general power series expansion for U all the infinitely many parametrisations of a unitary transformation are treated on equal footing. However, the question about the equivalence of these parametrisations for application in the Douglas-Kroll method, which represents a crucial point, is more subtle and will be analysed in the next section. It is especially not clear a priori, if the antihermitean matrix W can always be chosen in the appropriate way the mandatory properties of W, i.e., its oddness, antihermiticity and behaviour as a correct power in the external potential, have to be checked for every single transformation Ui of Eq. (73). 

Taylor series expansions, as described above, provide a very general method for representing a large class of mathematical functions. For the special case of periodic functions, a powerful alternative method is expansion in an infinite sum of sines and cosines, known as a trigonometric series or Fourier series. A periodic function is one that repeats in value when its argument is increased by multiples of a constant L, called the period or wavelength. For example. 

We have introduced new designations JI2 = Fi Fe Fi = FiIFc-The factors of hydrodynamic resistance hy depend on the relative distance s between the drops. Paper [43] solves the problem of slow central motion of two drops in a liquid where the two drops and the liquid all have different viscosities. The factors hy are found in the infinite series form. The appoximate solution of a similar problem is obtained by method of reflection in [46], and hy are found as power series in ratios Ri/r and R2/T which may be considered as asymptotic expressions for factors hy at s 2. An asymptotic expression for h is obtained in [39] for small values of the gap between the drops at s 2 ... 

The integral Gsb may be developed in a power series of Cs the coefficients of which involve distribution functions of sets of n molecules of the solute at infinite dilution in the solvent (McMillan and Mayee [1945]. This approach is especially useful in the case in whidi the dis-s3rmmetry between solvent and solute is such that the molecular structure of the solvent may be completely neglected. This is so when the dimertsinn.s of the molecules of the solvent are sufficiently small with reflect to those of the solute (cf. Guggenheim [1953]) as may be realised in a high poljmaer solution. In this case the problem of the expansion Gbb in powers of Cb becomes identical to that of the expansion of J(g — 1) dr in powers of the density for an imperfect gas. However as this approach is at present limited to dilute solutions we shall not develop this method here. 

In this chapter we consider several methods of expanding functions in infinite series. Two, which are particularly useful in physical chemistry, are the power series known as the Maclaurin series and the Taylor series. Let us consider the Maclaurin series first. Suppose that a function y(x) can be expanded in a power series... 

Bigeleisen and Ishida (BI) (see reading list) have explored the use of expansion methods to evaluate RPFR. The Bernoulli expansion is an infinite series in even powers of frequencies and is expressed... 

Fourier demonstrated that any periodic function, or wave, in any dimension, could always be reconstructed from an infinite series of simple sine waves consisting of integral multiples of the wave s own frequency, its spectrum. The trick is to know, or be able to find, the amplitude and phase of each of the sine wave components. Conversely, he showed that any periodic function could be decomposed into a spectrum of sine waves, each having a specific amplitude and phase. The former process has come to be known as a Fourier synthesis, and the latter as a Fourier analysis. The methods he proposed for doing this proved so powerful that he was rewarded by his mathematical colleagues with accusations of witchcraft. This reflects attitudes which once prevailed in academia, and often still do. 

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