相关论文: Symbolic computation of exact solutions expressibl…
A new algorithm is presented to find exact traveling wave solutions of differential-difference equations in terms of tanh functions. For systems with parameters, the algorithm determines the conditions on the parameters so that the…
This work presents a brief discussion and a plan towards the analytical solving of Partial Differential Equations (PDEs) using symbolic computing, as well as an implementation of part of this plan as the PDEtools software-package of…
This paper discusses the algorithms and implementations of three Mathematica packages for the study of integrability and the computation of closed-form solutions of nonlinear polynomial PDEs. The first package, PainleveTest.m, symbolically…
We describe a neural-based method for generating exact or approximate solutions to differential equations in the form of mathematical expressions. Unlike other neural methods, our system returns symbolic expressions that can be interpreted…
We present direct methods and symbolic software for the computation of conservation laws of nonlinear partial differential equations (PDEs) and differential-difference equations (DDEs).The methods are applied to nonlinear PDEs in (1+1)…
There are many numerical methods for solving partial different equations (PDEs) on manifolds such as classical implicit, finite difference, finite element, and isogeometric analysis methods which aim at improving the interoperability…
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 work develops a class of probabilistic algorithms for the numerical solution of nonlinear, time-dependent partial differential equations (PDEs). Current state-of-the-art PDE solvers treat the space- and time-dimensions separately,…
The authors proposed a general way to find particular solutions for overdetermined systems of PDEs previously, where the number of equations is greater than the number of unknown functions. In this paper, we propose an algorithm for finding…
The solution of systems of non-autonomous linear ordinary differential equations is crucial in a variety of applications, such us nuclear magnetic resonance spectroscopy. A new method with spectral accuracy has been recently introduced in…
The implementation of discontinuous Galerkin finite element methods (DGFEMs) represents a very challenging computational task, particularly for systems of coupled nonlinear PDEs, including multiphysics problems, whose parameters may consist…
Algorithms for the symbolic computation of polynomial conservation laws, generalized symmetries, and recursion operators for systems of nonlinear differential-difference equations (DDEs) are presented. The algorithms can be used to test the…
Differential-elimination algorithms apply a finite number of differentiations and eliminations to systems of partial differential equations. For systems that are polynomially nonlinear with rational number coefficients, they guarantee the…
Whether integrable, partially integrable or nonintegrable, nonlinear partial differential equations (PDEs) can be handled from scratch with essentially the same toolbox, when one looks for analytic solutions in closed form. The basic tool…
The compact explicit expressions for formal exact operator solutions to Cauchy problem for sufficiently general systems of nonlinear differential equations (ODEs and PDEs) in the form of chronological operator exponents are given. The…
High-dimensional partial-differential equations (PDEs) arise in a number of fields of science and engineering, where they are used to describe the evolution of joint probability functions. Their examples include the Boltzmann and…
We are concerned with the arithmetic of solutions to ordinary or partial nonlinear differential equations which are algebraic in the indeterminates and their derivatives. We call these solutions D-algebraic functions, and their equations…
A symbolic computational algorithm which detects " linear "` solutions of nonlinear polynomial differential equations of single functions, is developed throughout this paper.
We present a new algorithm for computing hyperexponential solutions of ordinary linear differential equations with polynomial coefficients. The algorithm relies on interpreting formal series solutions at the singular points as analytic…
Recent work has introduced a simple numerical method for solving partial differential equations (PDEs) with deep neural networks (DNNs). This paper reviews and extends the method while applying it to analyze one of the most fundamental…