47,004 research outputs found
A One Step Method for the Solution of General Second Order Ordinary Differential Equations
In this paper, an implicit one step method for the numerical solution of second order initial value problems of ordinary differential equations has been developed by collocation and interpolation technique. The introduction of an o step point guaranteed the zero stability and consistency of the method. The implicit method developed was implemented as a block which gave simultaneous solutions, as well as their rst derivatives, at both o step and the step point. A comparison of our method to the predictor-corrector method after solving some sample problems reveals that our method performs better
Solving General Second Order Ordinary Differential Equations by a One-Step Hybrid Collocation Method
A one-step hybrid method is developed for the numerical approximation of second order initial value problems of ordinary differential equations by interpolation and collocation at nonstop and step points respectively. The method is zero stable and consistent with very small error term. Numerical experiment of the method on sample problem shows that the method is more efficient and accurate than the results obtained from our earlier methods
Modified Block Method for the Direct Solution of Second Order Ordinary Differential Equations
The direct solution of general second order ordinary differential equations is considered
in this paper. The method is based on the collocation and interpolation of the power series approximate solution to generate a continuous linear multistep method. We modified
the existing block method in order to accommodate the general nth order ordinary differential equation. The method was found to be efficient when tested on second order ordinary
differential equation
Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model
We present a computer-assisted proof of heteroclinic connections in the
one-dimensional Ohta-Kawasaki model of diblock copolymers. The model is a
fourth-order parabolic partial differential equation subject to homogeneous
Neumann boundary conditions, which contains as a special case the celebrated
Cahn-Hilliard equation. While the attractor structure of the latter model is
completely understood for one-dimensional domains, the diblock copolymer
extension exhibits considerably richer long-term dynamical behavior, which
includes a high level of multistability. In this paper, we establish the
existence of certain heteroclinic connections between the homogeneous
equilibrium state, which represents a perfect copolymer mixture, and all local
and global energy minimizers. In this way, we show that not every solution
originating near the homogeneous state will converge to the global energy
minimizer, but rather is trapped by a stable state with higher energy. This
phenomenon can not be observed in the one-dimensional Cahn-Hillard equation,
where generic solutions are attracted by a global minimizer
Sparse spectral-tau method for the three-dimensional helically reduced wave equation on two-center domains
We describe a multidomain spectral-tau method for solving the
three-dimensional helically reduced wave equation on the type of two-center
domain that arises when modeling compact binary objects in astrophysical
applications. A global two-center domain may arise as the union of Cartesian
blocks, cylindrical shells, and inner and outer spherical shells. For each such
subdomain, our key objective is to realize certain (differential and
multiplication) physical-space operators as matrices acting on the
corresponding set of modal coefficients. We achieve sparse banded realizations
through the integration "preconditioning" of Coutsias, Hagstrom, Hesthaven, and
Torres. Since ours is the first three-dimensional multidomain implementation of
the technique, we focus on the issue of convergence for the global solver, here
the alternating Schwarz method accelerated by GMRES. Our methods may prove
relevant for numerical solution of other mixed-type or elliptic problems, and
in particular for the generation of initial data in general relativity.Comment: 37 pages, 3 figures, 12 table
An intelligent processing environment for real-time simulation
The development of a highly efficient and thus truly intelligent processing environment for real-time general purpose simulation of continuous systems is described. Such an environment can be created by mapping the simulation process directly onto the University of Alamba's OPERA architecture. To facilitate this effort, the field of continuous simulation is explored, highlighting areas in which efficiency can be improved. Areas in which parallel processing can be applied are also identified, and several general OPERA type hardware configurations that support improved simulation are investigated. Three direct execution parallel processing environments are introduced, each of which greatly improves efficiency by exploiting distinct areas of the simulation process. These suggested environments are candidate architectures around which a highly intelligent real-time simulation configuration can be developed
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