106,325 research outputs found
Numerical Solution of Partial Differential Equations Using Polynomial Particular Solutions
Polynomial particular solutions have been obtained for certain types of partial differential operators without convection terms. In this dissertation, a closed-form particular solution for more general partial differential operators with constant coefficients has been derived for polynomial basis functions. The newly derived particular solutions are further coupled with the method of particular solutions (MPS) for numerically solving a large class of elliptic partial differential equations. In contrast to the use of Chebyshev polynomial basis functions, the proposed approach is more flexible in selecting the collocation points inside the domain. Polynomial basis functions are well-known for yielding ill-conditioned systems when their order becomes large. The multiple scale technique is applied to circumvent the difficulty of ill-conditioning.
The derived polynomial particular solutions are also applied in the localized method of particular solutions to solve large-scale problems. Many numerical experiments have been performed to show the effectiveness of the particular solutions on this algorithm.
As another part of the dissertation, a modified method of particular solutions (MPS) has been used for solving nonlinear Poisson-type problems defined on different geometries. Polyharmonic splines are used as the basis functions so that no shape parameter is needed in the solution process. The MPS is also applied to compute the sizes of critical domains of different shapes for a quenching problem. These sizes are compared with the sizes of critical domains obtained from some other numerical methods. Numerical examples are presented to show the efficiency and accuracy of the method
On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem
We prove that the number of limit cycles generated by a small
non-conservative perturbation of a Hamiltonian polynomial vector field on the
plane, is bounded by a double exponential of the degree of the fields. This
solves the long-standing tangential Hilbert 16th problem. The proof uses only
the fact that Abelian integrals of a given degree are horizontal sections of a
regular flat meromorphic connection (Gauss-Manin connection) with a
quasiunipotent monodromy group.Comment: Final revisio
Moment and SDP relaxation techniques for smooth approximations of problems involving nonlinear differential equations
Combining recent moment and sparse semidefinite programming (SDP) relaxation
techniques, we propose an approach to find smooth approximations for solutions
of problems involving nonlinear differential equations. Given a system of
nonlinear differential equations, we apply a technique based on finite
differences and sparse SDP relaxations for polynomial optimization problems
(POP) to obtain a discrete approximation of its solution. In a second step we
apply maximum entropy estimation (using moments of a Borel measure associated
with the discrete solution) to obtain a smooth closed-form approximation. The
approach is illustrated on a variety of linear and nonlinear ordinary
differential equations (ODE), partial differential equations (PDE) and optimal
control problems (OCP), and preliminary numerical results are reported
Generalised Heine-Stieltjes and Van Vleck polynomials associated with degenerate, integrable BCS models
We study the Bethe Ansatz/Ordinary Differential Equation (BA/ODE)
correspondence for Bethe Ansatz equations that belong to a certain class of
coupled, nonlinear, algebraic equations. Through this approach we numerically
obtain the generalised Heine-Stieltjes and Van Vleck polynomials in the
degenerate, two-level limit for four cases of exactly solvable
Bardeen-Cooper-Schrieffer (BCS) pairing models. These are the s-wave pairing
model, the p+ip-wave pairing model, the p+ip pairing model coupled to a bosonic
molecular pair degree of freedom, and a newly introduced extended d+id-wave
pairing model with additional interactions. The zeros of the generalised
Heine-Stieltjes polynomials provide solutions of the corresponding Bethe Ansatz
equations. We compare the roots of the ground states with curves obtained from
the solution of a singular integral equation approximation, which allows for a
characterisation of ground-state phases in these systems. Our techniques also
permit for the computation of the roots of the excited states. These results
illustrate how the BA/ODE correspondence can be used to provide new numerical
methods to study a variety of integrable systems.Comment: 24 pages, 9 figures, 3 table
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