Related papers: Using a template engine as a computer algebra tool
From the discovery that the template system of C++ forms a Turing complete language in 1994, a programming technique called Template Metaprogramming has emerged that allows for the creation of faster, more generic and better code. Here, we…
Runge-Kutta formulas are some of the workhorses of numerical solving of differential equations. However, they are extremely difficult to generate; the algebra involved can be very complicated indeed. It is now standard, following the work…
Taylor series methods show a newfound promise for the solution of non-stiff ordinary differential equations (ODEs) given the rise of new compiler-enhanced techniques for calculating high order derivatives. In this paper we detail a new…
Classical and new numerical schemes are generated using evolutionary computing. Differential Evolution is used to find the coefficients of finite difference approximations of function derivatives, and of single and multi-step integration…
A general purpose, modular program package for the integration of large number of independent ordinary differential equation systems capable of using professional graphics cards is presented. The available numerical schemes are the explicit…
Background. It is assumed that the introduction of stochastic in mathematical model makes it more adequate. But there is virtually no methods of coordinated (depended on structure of the system) stochastic introduction into deterministic…
There exist many Runge-Kutta methods (explicit or implicit), more or less adapted to specific problems. Some of them have interesting properties, such as stability for stiff problems or symplectic capability for problems with energy…
We present an algorithm for the numerical solution of ordinary differential equations by random enumeration of the Butcher trees used in the implementation of the Runge-Kutta method. Our Monte Carlo scheme allows for the direct numerical…
Many problems in science and engineering require an efficient numerical approximation of integrals or solutions to differential equations. For systems with rapidly changing dynamics, an equidistant discretization is often inadvisable as it…
A new method for the numerical solution of ODEs is presented. This approach is based on an approximate formulation of the Taylor methods that has a much easier implementation than the original Taylor methods, since only the functions in the…
We consider the efficient numerical solution of coupled dynamical systems, consisting of a small nonlinear part and a large linear time invariant part, possibly stemming from spatial discretization of an underlying partial differential…
Runge-Kutta methods are the classic family of solvers for ordinary differential equations (ODEs), and the basis for the state of the art. Like most numerical methods, they return point estimates. We construct a family of probabilistic…
Runge-Kutta methods have an irreplaceable position among numerical methods designed to solve ordinary differential equations. Especially, implicit ones are suitable for approximating solutions of stiff initial value problems. We propose a…
Deriving analytical solutions of ordinary differential equations is usually restricted to a small subset of problems and numerical techniques are considered. Inevitably, a numerical simulation of a differential equation will then always be…
We study Runge-Kutta methods for rough differential equations which can be used to calculate solutions to stochastic differential equations driven by processes that are rougher than a Brownian motion. We use a Taylor series representation…
This paper illuminates the derivation, the applicability, and the efficiency of the Multiplicative Runge-Kutta Method, derived in the frame- work of geometric multiplicative calculus. The removal of the restrictions of geometric…
Differential equations arise in mathematics, physics,medicine, pharmacology, communications, image processing and animation, etc. An Ordinary Differential Equation (ODE) is a differential equation if it involves derivatives with respect to…
Statistical regression models whose mean functions are represented by ordinary differential equations (ODEs) can be used to describe phenomenons dynamical in nature, which are abundant in areas such as biology, climatology and genetics. The…
Direct shooting is an efficient method to solve numerical optimal control. It utilizes the Runge-Kutta scheme to discretize a continuous-time optimal control problem making the problem solvable by nonlinear programming solvers. However,…
This work presents a new evolutionary optimization algorithm in theoretical mathematics with important applications in scientific computing. The use of the evolutionary algorithm is justified by the difficulty of the study of the…