We solve the elliptic equations associated with the Hamiltonian and momentum constraints, corresponding to a system composed of two black holes with arbitrary linear and angular momentum. These new solutions are based on a Kerr-Schild spacetime slicing which provides more physically realistic solutions than the initial data based on conformally flat metric/maximal slicing methods. The singulari...

We study the initial boundary value problem for Einstein’s vacuum field equation. We prescribe initial data on an orientable, compact, 3-dimensional manifold S with boundary 6 6= ∅ and boundary conditions on the manifold T = R0 × 6. We assume the boundaries 6 and {0} × 6 of S and T to be identified in the natural way. Furthermore, we prescribe certain gauge source functions which determine the ...

The problem solving environments (PSEs) Maple [8] and Matlab [7] are in very wide use. Although they have much in common, they are clearly distinguished by the emphasis in Maple on algebraic computation and in Matlab on numerical computation. We discuss here a program, IVPsolve, for solving numerically initial value problems (IVPs) for systems of first order ordinary differential equations (ODE...

The form of the initial value constraints in Ashtekar’s hamiltonian formulation of general relativity is recalled, and the problem of solving them is compared with that in the traditional metric variables. It is shown how the general solution of the four diffeomorphism constraints can be obtained algebraically provided the curvature is nondegenerate, and the form of the remaining (Gauss law) co...

The secular variation of the geomagnetic field as observed at the Earth's surface results from the complex magnetohydrodynamics taking place in the fluid core of the Earth. One way to analyze this system is to use the data in concert with an underlying dynamical model of the system through the technique of variational data assimilation, in much the same way as is employed in meteorology and oce...

Interval methods for solving problems in differential equations and their implementation in floating-point interval arithmetic are interesting due to interval-solutions obtained which contain not only the errors of methods, but also all possible numerical errors. Such methods have been analyzed in a number of papers and monographs (see e.g. [1], [5], [9], and [10]). In our previous papers expli...

We present a new class of 3D black hole initial data sets for numerical relativity. These data sets go beyond the axisymmetric, " gravity wave plus rotating black hole " single black hole data sets by creating a dynamic, distorted hole with adjustable distortion parameters in 3D. These data sets extend our existing test beds for 3D numerical relativity, representing the late stages of binary bl...

For a system of functional differential equations of an arbitrary order the conditions are established for the initial value problem to be solvable on an infinite interval, and the structure of the set of solutions to this problem is studied. Introduction For the p-th order functional differential system x(p)(t) = f(t, xt, . . . , x (p−1) t ), b ≤ t < +∞, (1) we consider the initial value probl...

Our aim was to study what kind of bases can be provided to understand the basic terms of differential equation through teaching mathematical material in secondary school and to what extent this basis has to be expanded so that we can help the demonstration of differential equation. So if we give up the usual expansion of mathematical device we have to find another method which is easy to algori...