1,150 research outputs found
Methods in Mathematica for Solving Ordinary Differential Equations
An overview of the solution methods for ordinary differential equations in
the Mathematica function DSolve is presented.Comment: 13 page
The calculation of expectations for classes of diffusion processes by Lie symmetry methods
This paper uses Lie symmetry methods to calculate certain expectations for a
large class of It\^{o} diffusions. We show that if the problem has sufficient
symmetry, then the problem of computing functionals of the form
can be reduced to evaluating a single
integral of known functions. Given a drift we determine the functions
for which the corresponding functional can be calculated by symmetry.
Conversely, given , we can determine precisely those drifts for which
the transition density and the functional may be computed by symmetry. Many
examples are presented to illustrate the method.Comment: Published in at http://dx.doi.org/10.1214/08-AAP534 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Difference schemes with point symmetries and their numerical tests
Symmetry preserving difference schemes approximating second and third order
ordinary differential equations are presented. They have the same three or
four-dimensional symmetry groups as the original differential equations. The
new difference schemes are tested as numerical methods. The obtained numerical
solutions are shown to be much more accurate than those obtained by standard
methods without an increase in cost. For an example involving a solution with a
singularity in the integration region the symmetry preserving scheme, contrary
to standard ones, provides solutions valid beyond the singular point.Comment: 26 pages 7 figure
Exact Wave Functions for Generalized Harmonic Oscillators
We transform the time-dependent Schroedinger equation for the most general
variable quadratic Hamiltonians into a standard autonomous form. As a result,
the time-evolution of exact wave functions of generalized harmonic oscillators
is determined in terms of solutions of certain Ermakov and Riccati-type
systems. In addition, we show that the classical Arnold transformation is
naturally connected with Ehrenfest's theorem for generalized harmonic
oscillators.Comment: 10 pages, no figure
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