Related papers: Efficient Explicit Taylor ODE Integrators with Sym…
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…
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…
A Taylor method for solving an ordinary differential equation initial-value problem $\dot x = f(t,x)$, $x(t_0) = x_0$, computes the Taylor series (TS) of the solution at the current point, truncated to some order, and then advances to the…
This paper demonstrates new methods and implementations of nonlinear solvers with higher-order of convergence, which is achieved by efficiently computing higher-order derivatives. Instead of computing full derivatives, which could be…
In this study, we propose high-order implicit and semi-implicit schemes for solving ordinary differential equations (ODEs) based on Taylor series expansion. These methods are designed to handle stiff and non-stiff components within a…
In this brief, we discuss the implementation of a third order semi-implicit differentiator as a complement of the recent work by the author that proposes an interconnected semi-implicit Euler double differentiators algorithm through Taylor…
In this work, an approximate family of implicit multiderivative Runge-Kutta (MDRK) time integrators for stiff initial value problems is presented. The approximation procedure is based on the recent Approximate Implicit Taylor method (Baeza…
The numerical solution of implicit and stiff differential equations by implicit numerical integrators has been largely investigated and there exist many excellent efficient codes available in the scientific community, as Radau5 (based on a…
Numerically solving ordinary differential equations (ODEs) is a naturally serial process and as a result the vast majority of ODE solver software are serial. In this manuscript we developed a set of parallelized ODE solvers using…
In research problems that involve the use of numerical methods for solving systems of ordinary differential equations (ODEs), it is often required to select the most efficient method for a particular problem. To solve a Cauchy problem for a…
Stiff ordinary differential equations (ODEs) are common in many science and engineering fields, but standard neural ODE approaches struggle to accurately learn these stiff systems, posing a significant barrier to widespread adoption of…
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…
Algorithms are presented for the tanh- and sech-methods, which lead to closed-form solutions of nonlinear ordinary and partial differential equations (ODEs and PDEs). New algorithms are given to find exact polynomial solutions of ODEs and…
Stiff systems of ordinary differential equations (ODEs) arise in a wide range of scientific and engineering disciplines and are traditionally solved using implicit integration methods due to their stability and efficiency. However, these…
Traditional solvers for delay differential equations (DDEs) are designed around only a single method and do not effectively use the infrastructure of their more-developed ordinary differential equation (ODE) counterparts. In this work we…
High-quality ordinary differential equation (ODE) solver libraries have a long history, going back to the 1970s. Over the past several years we have implemented, on top of the PETSc linear and nonlinear solver package, a new…
A new approach for the construction of high order A-stable explicit integrators for ordinary differential equations (ODEs) is theoretically studied. Basically, the integrators are obtained by splitting, at each time step, the solution of…
High-order Lie derivatives are essential in nonlinear systems analysis. If done symbolically, their evaluation becomes increasingly expensive as the order increases. We present a compact and efficient numerical approach for computing Lie…
Neural ordinary differential equations (NODEs) -- parametrizations of differential equations using neural networks -- have shown tremendous promise in learning models of unknown continuous-time dynamical systems from data. However, every…
Symbolic computation for systems of differential equations is often computationally expensive. Many practical differential models have a form of polynomial or rational ODE system with specified outputs. A basic symbolic approach to analyze…