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Finite-difference solutions of tenth-order boundary-value problems
This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.In this thesis finite difference methods are used to obtain numerical solutions for a class of high-order ordinary differential equations with applications to eigenvalue problems. Two families of numerical methods are developed for tenth-order boundary-value
problems and global extrapolations on two and three grids are considered for the special problem. Special nonlinear tenth-order boundary-value problems are solved using a family of direct finite difference methods which are adapted to solve a general linear and nonlinear boundary-value problem. These methods convert the ordinary differential equation into a set of algebraic equations. If the original
ordinary differential equations are linear, the finite difference equations will give linear algebraic equations. If the ordinary differential equation are nonlinear, the resulting finite difference equations will be nonlinear algebraic equations. These nonlinear equations are first linearized by Newton's method. The methods developed are of orders two, four, six, eight, ten and twelve. The error analyses are discussed. A generalized form is given to solve a class of high-order boundary-value problems by converting the differential equation to
a system of first-order equations. The method based on using a Pade rational
approximant to the exponential function for general boundary-value problems is applied to a tenth-order eigenvalue problem associated with instability in a Benard layer and numerical results are compared with asymtotic estimates appearing in the literature. This method may be implernented on a parallel computer. The method is extended to a twelfth-order eigenvalue problern in an appendix. The algorithms developed are tested on a variety of problems from the literature. The REDUCE package is used to obtain the parameters in the numerical methods and all computations are carried out on a Sun Workstation at Brunel University using Fortran 77 with double precision arithmetic.This study is funded by the Ministry of Education of Pakistan, Islamabad
Algebraic Solutions of the Lam\'e Equation, Revisited
A minor error in the necessary conditions for the algebraic form of the
Lam\'e equation to have a finite projective monodromy group, and hence for it
to have only algebraic solutions, is pointed out. [See F. Baldassarri, "On
algebraic solutions of Lam\'e's differential equation", J. Differential
Equations 41 (1981), 44-58.] It is shown that if the group is the octahedral
group S_4, then the degree parameter of the equation may differ by +1/6 or -1/6
from an integer; this possibility was missed. The omission affects a recent
result on the monodromy of the Weierstrass form of the Lam\'e equation. [See R.
C. Churchill, "Two-generator subgroups of SL(2,C) and the hypergeometric,
Riemann, and Lam\'e equations", J. Symbolic Computation 28 (1999), 521-545.]
The Weierstrass form, which is a differential equation on an elliptic curve,
may have, after all, an octahedral projective monodromy group.Comment: 20 pages, elsart document class, no figure
Explicit formula for the generating series of diagonal 3D rook paths
Let denote the number of ways in which a chess rook can move from a
corner cell to the opposite corner cell of an
three-dimensional chessboard, assuming that the piece moves closer to the goal
cell at each step. We describe the computer-driven \emph{discovery and proof}
of the fact that the generating series admits
the following explicit expression in terms of a Gaussian hypergeometric
function: G(x) = 1 + 6 \cdot \int_0^x \frac{\,\pFq21{1/3}{2/3}{2} {\frac{27
w(2-3w)}{(1-4w)^3}}}{(1-4w)(1-64w)} \, dw.Comment: To appear in "S\'eminaire Lotharingien de Combinatoire
New Fundamental Symmetries of Integrable Systems and Partial Bethe Ansatz
We introduce a new concept of quasi-Yang-Baxter algebras. The quantum
quasi-Yang-Baxter algebras being simple but non-trivial deformations of
ordinary algebras of monodromy matrices realize a new type of quantum dynamical
symmetries and find an unexpected and remarkable applications in quantum
inverse scattering method (QISM). We show that applying to quasi-Yang-Baxter
algebras the standard procedure of QISM one obtains new wide classes of quantum
models which, being integrable (i.e. having enough number of commuting
integrals of motion) are only quasi-exactly solvable (i.e. admit an algebraic
Bethe ansatz solution for arbitrarily large but limited parts of the spectrum).
These quasi-exactly solvable models naturally arise as deformations of known
exactly solvable ones. A general theory of such deformations is proposed. The
correspondence ``Yangian --- quasi-Yangian'' and `` spin models ---
quasi- spin models'' is discussed in detail. We also construct the
classical conterparts of quasi-Yang-Baxter algebras and show that they
naturally lead to new classes of classical integrable models. We conjecture
that these models are quasi-exactly solvable in the sense of classical inverse
scattering method, i.e. admit only partial construction of action-angle
variables.Comment: 49 pages, LaTe
A Characterization of Reduced Forms of Linear Differential Systems
A differential system , with
is said to be in reduced form if where
is the Lie algebra of the differential Galois group of
. In this article, we give a constructive criterion for a system to be in
reduced form. When is reductive and unimodular, the system is in
reduced form if and only if all of its invariants (rational solutions of
appropriate symmetric powers) have constant coefficients (instead of rational
functions). When is non-reductive, we give a similar characterization via
the semi-invariants of . In the reductive case, we propose a decision
procedure for putting the system into reduced form which, in turn, gives a
constructive proof of the classical Kolchin-Kovacic reduction theorem.Comment: To appear in : Journal of Pure and Applied Algebr
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