We apply the RPSM to find out series solution for this equation subject to given initial conditions by replacing its power series expansion
with its truncated residual function.
Then the equation becomes time-harmonic and the solution can be approximated with a truncated Fourier series expansion
. Due to the orthogonality of the trigonometric functions, for linear problems the computation of the Fourier coefficients separate and one can compute the solution for each period fully in parallel.
Neumann series expansion
was proposed in  to replace the matrix inversion in MMSE detection, the performance and computational complexity of which scaled with the number of selected terms of Neumann series.
First, a series expansion
of the ODF into spherical harmonics will be performed (Section 2); second, after a short introduction to symmetric irreducible tensors (Section 3), the correspondence of symmetric irreducible tensors with spherical harmonics will be shown (Section 4).
where [c.sub.i] is the ith series expansion
coefficient, which can be determined by standard deviation and central moments of X.
We obtain the energies and wave functions of reduced two-dimensional Schrodinger equation by using the double Fourier-Bessel series expansion
A review of the literature reveals that the power series expansion
was exploited by several researchers [10-12, 20-24] to develop powerful numerical methods for solving nonlinear differential equations.
If b(t) [member of] [C.sub.[gamma]] and [absolute value of ([f.sup.0.sub.1]([x.sub.i]))] [less than or equal to] [xi] exp(-[beta]([p.sup.2.sub.i]/2)), then the Boltzmann-Grad limit of mean value functional (12) exists under the condition that : [mathematical expression not reproducible], and it is determined by the following series expansion
We may try to find a series expansion
in powers of t of
The half-normal key function without series expansion
provided the best fit to the distance data (Kolmogorov--Smirnov test: [D.sub.n] = 0.17, p = 0.47; Cramer-von Mises family tests: [W.sup.2] = 0.15, p > 0.40, [C.sup.2] = 0.09, p > 0.40; AIC = 3949.26).
Using the Taylor series expansion
, the PDF can be written as
Kloeden and Platen  described a method based on the stochastic Taylor series expansion
but the major difficulty with this approach is that the double stochastic integrals cannot be easily expressed in terms of simpler stochastic integrals when the Wiener process is multidimensional.