Analysis of boundary value problems using Hermite interpolation
Recent Papers
R E Grundy. A novel approach to the analysis of
nonlocal initial-boundary value problems using two-point interpolating
polynomials. (2010).
Abstract
In this paper we analyse
initial-boundary value problems for partial differential equations
involving
nonlocal nonlinearities using two-point Hermite interpolation. The
technique
enables us to construct polynomial approximations which provide
extensive
qualitative and quantitative information regarding the structure of
solutions.
We proceed via a series of examples from the literature chosen for
their
extensive numerical and analytic provision where we can compute steady
states
and their stability properties, blow-up thresholds and blow-up times.
The method
is particularly apposite for studying feedback problems an example of
which is
given in the paper.
R E Grundy.
The analysis of boundary value problems using multipoint Hermite
expansions. (2010).
Abstract
In this paper we examine the
general utility of multipoint Hermite type expansions in the analysis
of
two-point ordinary boundary value problems. We show that when the
solutions are
entire functions or any singularities in the complex plane are
sufficiently
distant from the real interval over which a solution is sought, it is
possible
to construct convergent Hermite expansions simply by fitting an equal
number of
derivatives using the end points as nodes. When singularities do
influence
convergence we show it is possible to change the convergence domains in
the
complex plane either by introducing additional nodes or by fitting
different numbers of derivatives at each node. In
this way convergence can often be restored and an appropriate Hermite
expansion
constructed. We illustrate the application of these ideas to the
solution of
ordinary boundary value problems via a series of linear and nonlinear
examples
for which a symbolic computational facility is an indispensable tool in
the
analysis.
R E Grundy. A novel method for obtaining periodic
solutions to ordinary differential equations. (2010).
Abstract
This paper
introduces what we believe is a novel method for obtaining periodic
solutions
of ordinary differential equations using Hermite trigonometric
polynomials. By
seeking solutions in terms of such representations we show how we can
derive
equations relating the frequency and any parameters which are present
enabling
periodic solutions to be constructed. We illustrate the idea using an
example
which models the nonlinear pendulum. If small parameters are present we
show as
a useful adjunct, how we can reproduce the results of classical
asymptotic
analysis without recourse to the elimination of secular terms.
Throughout the
paper MAPLE is used as an essential aid in all the algebraic
manipulations and
computations that we need to do.
R E Grundy.
A novel method for obtaining periodic solutions to ordinary
differential
equations: the spiral spring pendulum. (2011).
Abstract
This paper
introduces what we believe is a novel method for obtaining periodic
solutions
of ordinary differential equations using Hermite trigonometric
polynomials. By
seeking solutions in terms of such representations we show how we can
derive
periodicity conditions relating the frequency and any parameters which
are
present thereby enabling the solution to be constructed. We illustrate
the idea
via an example which models an inverted nonlinear pendulum with a
linear
restoring force. If small parameters are present we show as a useful
adjunct,
how we can reproduce the results of classical asymptotic analysis
without
recourse to the elimination of secular terms. Throughout the paper
MAPLE is
used as an essential aid in all the algebraic manipulations and
computations
that we need to do.
R E Grundy. The analysis of forced oscillations using
Hermite trigonometric interpolation. (2011).
Abstract
This paper
extends the use of Hermite trigonometric interpolation to the
computation of
periodic solutions of ordinary differential equations with periodic
forcing.
Without loss of generality we may consider 2π-periodic solutions and
construct trigonometric
polynomials
which fit function values and any number of derivatives at the end
points
of the interval [0, π] and which are
thereby rendered automatically of period 2π. By seeking solutions in
terms of these
trigonometric
polynomials we show how we can derive periodicity conditions relating
the
frequency, amplitude and phase of the periodic solutions to the
parameters
appearing in the given equation. We proceed via the example of the
forced
Duffing equation wherein a major aim is to demonstrate numerical
convergence as
the number of derivatives fitted increases. We compare our results with
exact
solutions when these are available or more generally by numerical
confirmation.
If small parameters are present we show as a useful adjunct, how we can
reproduce the results of classical asymptotic analysis without recourse
to the
elimination of secular terms. Throughout the paper MAPLE is used as an
essential aid in all the algebraic manipulations and numerical
calculations
that we need to do.
