Related papers: Numerical integration rules based on B-spline base…
A recent type of B-spline functions, namely trigonometric cubic B-splines, are adapted to the collocation method for the numerical solutions of the Kuramoto-Sivashinsky equation. Having only first and second order derivatives of the…
Computing accurate splines of degree greater than three is still a challenging task in today's applications. In this type of interpolation, high-order derivatives are needed on the given mesh. As these derivatives are rarely known and are…
The Numerical Recipes series of books are a useful resource, but all the algorithms they contain cannot be used within open-source projects. In this paper we develop drop-in alternatives to the two algorithms they present for cubic spline…
A new differential-recurrence relation for the B-spline functions of the same degree is proved. From this relation, a recursive method of computing the coefficients of B-spline functions of degree $m$ in the Bernstein-B\'{e}zier form is…
Particle tracing through numerical integration is a well-known approach to generating pathlines for visualization. However, for particle simulations, the computation of pathlines is expensive, since the interpolation method is complicated…
We propose a collocation method based on multivariate polynomial splines over triangulation or tetrahedralization for the numerical solution of partial differential equations. We start with a detailed explanation of the method for the…
An accurate method to compute enclosures of Abelian integrals is developed. This allows for an accurate description of the phase portraits of planar polynomial systems that are perturbations of Hamiltonian systems. As an example, it is…
We present techniques for successfully performing numerical relativity simulations of binary black holes with fourth-order accuracy. Our simulations are based on a new coding framework which currently supports higher order finite…
This work is an extension of previous work by Alazah et al. [M. Alazah, S. N. Chandler-Wilde, and S. La Porte, Numerische Mathematik, 128(4):635-661, 2014]. We split the computation of the Fresnel Integrals into 3 cases: a truncated Taylor…
Two-step hybrid methods specially adapted to the numerical integration of perturbed oscillators are obtained. The formulation of the methods is based on a refinement of classical Taylor expansions due to Scheifele [{\em Z. Angew. Math.…
We used a collocation method in refinable spline space to solve a linear dynamical system having fractional derivative in time. The method takes advantage of an explicit derivation rule for the B-spline basis that allows us to efficiently…
We extend the technique of using the Trapezoidal Rule for efficient evaluation of the Special Functions of Mathematical Physics given by integral representations. This technique was recently used for Bessel functions, and here we treat…
We present methods for the numerical evaluation of the master integrals that appear in the calculation of scattering amplitudes at higher order in perturbative quantum field theory. We follow the general strategy of solving first-order…
Using Suzuki-Trotter decompositions of exponential operators we describe new algorithms for the numerical integration of the equations of motion for classical spin systems. These techniques conserve spin length exactly and, in special…
We develop a numerical method for solving a system of nonlinear integral equations involving two integral terms: at the current time t, one integral is taken from 0 to t, and a different integral is taken from t to infinity. We prove the…
This paper presents a new approach and methodology to solve the second order one dimensional hyperbolic telegraph equation with Dirichlet and Neumann boundary conditions using the cubic trigonometric B-spline collocation method. The usual…
We develop a method for the rigorous estimation of Hausdorff dimensions of limit sets produced by continued fraction iterated function systems. Our method is based on the approximation of a Perron-Frobenius operator using the finite element…
Extensions to the trapezoidal rule using derivative information are studied for periodic integrands and integrals along the entire real line. Integrands which are analytic within a half plane or within a strip containing the path of…
The Operator axioms have produced new real numbers with new operators. New operators naturally produce new equations and thus extend the traditional mathematical models which are selected to describe various scientific rules. So new…
We study numerical integration of smooth functions defined over the $s$-dimensional unit cube. A recent work by Dick et al. (2019) has introduced so-called extrapolated polynomial lattice rules, which achieve the almost optimal rate of…