569 research outputs found
Numerical integration of asymptotic solutions of ordinary differential equations
Classical asymptotic analysis of ordinary differential equations derives approximate solutions that are numerically stable. However, the analysis also leads to tedious expansions in powers of the relevant parameter for a particular problem. The expansions are replaced with integrals that can be evaluated by numerical integration. The resulting numerical solutions retain the linear independence that is the main advantage of asymptotic solutions. Examples, including the Falkner-Skan equation from laminar boundary layer theory, illustrate the method of asymptotic analysis with numerical integration
Improved convergence of scattering calculations in the oscillator representation
The Schr\"odinger equation for two and tree-body problems is solved for
scattering states in a hybrid representation where solutions are expanded in
the eigenstates of the harmonic oscillator in the interaction region and on a
finite difference grid in the near-- and far--field. The two representations
are coupled through a high--order asymptotic formula that takes into account
the function values and the third derivative in the classical turning points.
For various examples the convergence is analyzed for various physics problems
that use an expansion in a large number of oscillator states. The results show
significant improvement over the JM-ECS method [Bidasyuk et al, Phys. Rev. C
82, 064603 (2010)]
Multi-channel analog of the effective-range expansion
Similarly to the standard effective range expansion that is done near the
threshold energy, we obtain a generalized power-series expansion of the
multi-channel Jost-matrix that can be done near an arbitrary point on the
Riemann surface of the energy within the domain of its analyticity. In order to
do this, we analytically factorize its momentum dependencies at all the
branching points on the Riemann surface. The remaining single-valued matrix
functions of the energy are then expanded in the power-series near an arbitrary
point in the domain of the complex energy plane where it is analytic. A
systematic and accurate procedure has been developed for calculating the
expansion coefficients. This means that near an arbitrary point in the domain
of physically interesting complex energies it is possible to obtain a
semi-analytic expression for the Jost-matrix (and therefore for the S-matrix)
and use it, for example, to locate the spectral points (bound and resonant
states) as the S-matrix poles.Comment: 33 pages, 10 figure
COMPARISON OF VARIOUS FRACTIONAL BASIS FUNCTIONS FOR SOLVING FRACTIONAL-ORDER LOGISTIC POPULATION MODEL
Three types of orthogonal polynomials (Chebyshev, Chelyshkov, and Legendre) are employed as basis functions in a collocation scheme to solve a nonlinear cubic initial value problem arising in population growth models. The method reduces the given problem to a set of algebraic equations consist of polynomial coefficients. Our main goal is to present a comparative study of these polynomials and to asses their performances and accuracies applied to the logistic population equation. Numerical applications are given to demonstrate the validity and applicability of the method. Comparisons are also made between the present method based on different basis functions and other existing approximation algorithms
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