2,060 research outputs found
A comparison between numerical solutions to fractional differential equations: Adams-type predictor-corrector and multi-step generalized differential transform method
In this note, two numerical methods of solving fractional differential
equations (FDEs) are briefly described, namely predictor-corrector approach of
Adams-Bashforth-Moulton type and multi-step generalized differential transform
method (MSGDTM), and then a demonstrating example is given to compare the
results of the methods. It is shown that the MSGDTM, which is an enhancement of
the generalized differential transform method, neglects the effect of non-local
structure of fractional differentiation operators and fails to accurately solve
the FDEs over large domains.Comment: 12 pages, 2 figure
Analytical solution of linear ordinary differential equations by differential transfer matrix method
We report a new analytical method for exact solution of homogeneous linear
ordinary differential equations with arbitrary order and variable coefficients.
The method is based on the definition of jump transfer matrices and their
extension into limiting differential form. The approach reduces the th-order
differential equation to a system of linear differential equations with
unity order. The full analytical solution is then found by the perturbation
technique. The important feature of the presented method is that it deals with
the evolution of independent solutions, rather than its derivatives. We prove
the validity of method by direct substitution of the solution in the original
differential equation. We discuss the general properties of differential
transfer matrices and present several analytical examples, showing the
applicability of the method. We show that the Abel-Liouville-Ostogradski
theorem can be easily recovered through this approach
Reconstruction of Planar Domains from Partial Integral Measurements
We consider the problem of reconstruction of planar domains from their
moments. Specifically, we consider domains with boundary which can be
represented by a union of a finite number of pieces whose graphs are solutions
of a linear differential equation with polynomial coefficients. This includes
domains with piecewise-algebraic and, in particular, piecewise-polynomial
boundaries. Our approach is based on one-dimensional reconstruction method of
[Bat]* and a kind of "separation of variables" which reduces the planar problem
to two one-dimensional problems, one of them parametric. Several explicit
examples of reconstruction are given.
Another main topic of the paper concerns "invisible sets" for various types
of incomplete moment measurements. We suggest a certain point of view which
stresses remarkable similarity between several apparently unrelated problems.
In particular, we discuss zero quadrature domains (invisible for harmonic
polynomials), invisibility for powers of a given polynomial, and invisibility
for complex moments (Wermer's theorem and further developments). The common
property we would like to stress is a "rigidity" and symmetry of the invisible
objects.
* D.Batenkov, Moment inversion of piecewise D-finite functions, Inverse
Problems 25 (2009) 105001Comment: Proceedings of Complex Analysis and Dynamical Systems V, 201
Approximate computations with modular curves
This article gives an introduction for mathematicians interested in numerical
computations in algebraic geometry and number theory to some recent progress in
algorithmic number theory, emphasising the key role of approximate computations
with modular curves and their Jacobians. These approximations are done in
polynomial time in the dimension and the required number of significant digits.
We explain the main ideas of how the approximations are done, illustrating them
with examples, and we sketch some applications in number theory
Hyperelliptic Theta-Functions and Spectral Methods
A code for the numerical evaluation of hyperelliptic theta-functions is
presented. Characteristic quantities of the underlying Riemann surface such as
its periods are determined with the help of spectral methods. The code is
optimized for solutions of the Ernst equation where the branch points of the
Riemann surface are parameterized by the physical coordinates. An exploration
of the whole parameter space of the solution is thus only possible with an
efficient code. The use of spectral approximations allows for an efficient
calculation of all quantities in the solution with high precision. The case of
almost degenerate Riemann surfaces is addressed. Tests of the numerics using
identities for periods on the Riemann surface and integral identities for the
Ernst potential and its derivatives are performed. It is shown that an accuracy
of the order of machine precision can be achieved. These accurate solutions are
used to provide boundary conditions for a code which solves the axisymmetric
stationary Einstein equations. The resulting solution agrees with the
theta-functional solution to very high precision.Comment: 25 pages, 12 figure
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