Related papers: Maurice Janet's algorithms on systems of linear pa…
The paper develops the method for construction of families of particular solutions to some classes of nonlinear Partial Differential Equations (PDE). Method is based on the specific link between algebraic matrix equations and PDE.…
The paper represents the method for construction of the families of particular solutions to some new classes of $(n+1)$ dimensional nonlinear Partial Differential Equations (PDE). Method is based on the specific link between algebraic…
In this article, we study systems of $n \geq 1$, not necessarily linear, discrete differential equations (DDEs) of order $k \geq 1$ with one catalytic variable. We provide a constructive and elementary proof of algebraicity of the solutions…
The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an…
Discrete Differential Equations (DDEs) are functional equations that relate polynomially a power series $F(t,u)$ in $t$ with polynomial coefficients in a "catalytic" variable $u$ and the specializations, say at $u=1$, of $F(t,u)$ and of…
The objective of this paper is to investigate the existence and the forms of the pair of finite order entire and meromorphic solutions of some certain systems of Fermat-type partial differential-difference equations of several complex…
In this paper the relation between Pommaret and Janet bases of polynomial ideals is studied. It is proved that if an ideal has a finite Pommaret basis then the latter is a minimal Janet basis. An improved version of the related algorithm…
Physics-informed neural networks (PINNs) have recently emerged as a popular approach for solving forward and inverse problems involving partial differential equations (PDEs). Compared to fully connected neural networks, PINNs based on…
We introduce DDE-Solver, a Maple package designed for solving Discrete Differential Equations (DDEs). These equations are functional equations relating algebraically a formal power series F(t, u) with polynomial coefficients in a…
Ordinary and partial differential equations (ODEs/PDEs) play a paramount role in analyzing and simulating complex dynamic processes across all corners of science and engineering. In recent years machine learning tools are aspiring to…
We consider a linear algebra approach to establishing a discrete comparison principle for a nonmonotone class of quasilinear elliptic partial differential equations. In the absence of a lower order term, we require local conditions on the…
In present paper we propose seemingly new method for finding solutions of some types of nonlinear PDEs in closed form. The method is based on decomposition of nonlinear operators on sequence of operators of lower orders. It is shown that…
In theory, boundary and initial conditions are important for the wellposedness of partial differential equations (PDEs). Numerically, these conditions can be enforced exactly in classical numerical methods, such as finite difference method…
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…
The Painlev\'{e} and weak Painlev\'{e} conjectures have been used widely to identify new integrable nonlinear dynamical systems. For a system which passes the Painlev\'{e} test, the calculation of the integrals relies on a variety of…
There exist several methods for computing exact solutions of algebraic differential equations. Most of the methods, however, do not ensure existence and uniqueness of the solutions and might fail after several steps, or are restricted to…
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…
A large family of linear, usually overdetermined, systems of partial differential equations that admit a multiplication of solutions, i.e, a bi-linear and commutative mapping on the solution space, is studied. This family of PDE's contains…
We propose and implement an algorithm for solving an overdetermined system of partial differential equations in one unknown. Our approach relies on Bour-Mayer method to determine compatibility conditions via Jacobi-Mayer brackets. We solve…
This article demonstrates how variation of parameters can be successfully implemented in combination with other classical techniques, such as the method of characteristics, to derive novel classes of solutions to nonlinear partial…