We apply Feynman’s principle, “The same equations have the same solutions”, to Kepler’s problem and show that Newton’s dynamics in a properly curved 3-D space is identical with that described by Einstein’s theory in the 3-D optical geometry of Schwarzschild’s spacetime. For this reason, rather unexpectedly, Newton’s formulae for Kepler’s problem, in the case of nearly circular motion in a stati...

This paper discusses application of the piecewise sinusoidal (PWS) basis function (BF) over a chain of several wire segments, i.e. as a multiple domain basis functions. The usage of PWS BF is compared to results based on the piecewise linear (PWL) basis functions. An example of meander monopole demonstrates the advantages associated with the PWS BF.

In this paper, we consider the problem of estimating the eigenvalues and eigenfunctions of the covariance kernel (i.e., the functional principal components) from sparse and irregularly observed longitudinal data. We approach this problem through a maximum likelihood method assuming that the covariance kernel is smooth and finite dimensional. We exploit the smoothness of the eigenfunctions to re...

‎Newton method is one of the most famous numerical methods among the line search‎ ‎methods to minimize functions. ‎It is well known that the search direction and step length play important roles ‎in this class of methods to solve optimization problems. ‎In this investigation‎, ‎a new modification of the Newton method to solve ‎unconstrained optimization problems is presented‎. ‎The significant ...

In this paper, An effective and simple numerical method is proposed for solving systems of integral equations using radial basis functions (RBFs). We present an algorithm based on interpolation by radial basis functions including multiquadratics (MQs), using Legendre-Gauss-Lobatto nodes and weights. Also a theorem is proved for convergence of the algorithm. Some numerical examples are presented...

The purpose of this work is to construct a robust numerical scheme for class nonlinear free boundary identification problems. First, shape optimization problem constructed based on least square functional. Schauder’s fixed point theorem manipulated show the existence solution state solution. an optimal proved. proposed Radial Basis Functions method as discretization approach, minimization proce...

We derive a posteriori error estimates for nonconforming discretizations of Poisson's and Stokes' equations. The estimates are residual based and make use of weight factors obtained by a duality argument. Crouzeix-Raviart elements on triangles and rotated bilinear elements are considered. The quadrilateral case involves the introduction of additional local trial functions. We show that their in...

This work concerns the solution of generalized Riemann problems. To this end, we consider the ADER scheme of Titarev & Toro (2002), which relies on a generalization of the classical Godunov scheme. Another solution method is the power series expansion of LeFloch & Raviart (1988). We analyze the two resulting approximation schemes, where we show that for scalar 1d problems the Toro-Titarev solve...

In this paper, we give a new characterization of the approximate solution given by hybridized mixed methods for second-order, self-adjoint elliptic problems. We apply this characterization to obtain an explicit formula for the entries of the matrix equation for the Lagrange multiplier unknowns resulting from hybridization. We also obtain necessary and sufficient conditions under which the multi...