Related papers: Solving ordinary differential equations on the Inf…
There exist many applications where it is necessary to approximate numerically derivatives of a function which is given by a computer procedure. In particular, all the fields of optimization have a special interest in such a kind of…
Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not…
An efficient approximate version of implicit Taylor methods for initial-value problems of systems of ordinary differential equations (ODEs) is introduced. The approach, based on an approximate formulation of Taylor methods, produces a…
In this paper, we present a new numerical method to solve fractional differential equations. Given a fractional derivative of arbitrary real order, we present an approximation formula for the fractional operator that involves integer-order…
We consider Continuous Ordinary Differential Equations (CODE) y'=f(y), where f is a continuous function. They are known to always have solutions for a given initial condition y(0)=y0, these solutions being possibly non unique. We restrict…
Probabilistic solvers for ordinary differential equations (ODEs) have emerged as an efficient framework for uncertainty quantification and inference on dynamical systems. In this work, we explain the mathematical assumptions and detailed…
This paper studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs), a.k.a. (Ordinary) Difference Equations. It presents a new framework using these equations as a central tool for computation and…
We introduce basic aspects of new operator method, which is very suitable for practical solving differential equations of various types. The main advantage of the method is revealed in opportunity to find compact exact operator solutions of…
When a computer algebra system fails to solve an Ordinary Differential Equation, is this a limitation of its implementation, or a genuine computational barrier? Three traditions bear on the question. Modern computer algebra algorithms can…
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…
In this paper we investigate the computational complexity of solving ordinary differential equations (ODEs) $y^{\prime}=p(y)$ over \emph{unbounded time domains}, where $p$ is a vector of polynomials. Contrarily to the bounded (compact) time…
In this work, a novel quantum Fourier ordinary differential equation (ODE) solver is proposed to solve both linear and nonlinear partial differential equations (PDEs). Traditional quantum ODE solvers transform a PDE into an ODE system via…
The simplicity and the efficiency of a quasi-analytical method for solving nonlinear ordinary differential equations (ODE), is illustrated on the study of anharmonic oscillators (AO) with a potential $V(x) =\beta x^{2}+x^{2m}$ ($m>0$). The…
A method of representation of a solution as segments of the series in powers of the step of the independent variable is expanded for solving complex systems of ordinary differential equations (ODE): the Lorenz system and other systems. A…
When mathematical/computational problems reach infinity, extending analysis and/or numerical computation beyond it becomes a notorious challenge. We suggest that, upon suitable singular transformations (that can in principle be…
On computers, discrete problems are solved instead of continuous ones. One must be sure that the solutions of the former problems, obtained in real time (i.e., when the stepsize h is not infinitesimal) are good approximations of the…
In this paper we construct high order numerical methods for solving third and fourth orders nonlinear functional differential equations (FDE). They are based on the discretization of iterative methods on continuous level with the use of the…
A new computational methodology for executing calculations with infinite and infinitesimal quantities is described in this paper. It is based on the principle `The part is less than the whole' introduced by Ancient Greeks and applied to all…
Computers calculate transcendental functions by approximating them through the composition of a few limited-precision instructions. For example, an exponential can be calculated with a Taylor series. These approximation methods were…
Algorithms for the computation of the real zeros of hypergeometric functions which are solutions of second order ODEs are described. The algorithms are based on global fixed point iterations which apply to families of functions satisfying…