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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…
We present the package SADE (Symmetry Analysis of Differential Equations) for the determination of symmetries and related properties of systems of differential equations. The main methods implemented are: Lie, nonclassical, Lie-B\"acklund…
This work unifies pseudo-time and inexact regularization techniques for nonmonotone classes of partial differential equations, into a regularized pseudo-time framework. Convergence of the residual at the predicted rate is investigated…
The ultimate goal of any numerical scheme for partial differential equations (PDEs) is to compute an approximation of user-prescribed accuracy at quasi-minimal computational time. To this end, algorithmically, the standard adaptive finite…
Differential equation models are crucial to scientific processes. The values of model parameters are important for analyzing the behaviour of solutions. A parameter is called globally identifiable if its value can be uniquely determined…
We represent an algorithm reducing a big class of systems of ($M+1$)-dimensional nonlinear partial differential equations (PDEs) to the systems of $M$-dimensional first order PDEs. Thus, we integrate the original system with respect to only…
In this work, we present an adjoint-based method for discovering the underlying governing partial differential equations (PDEs) given data. The idea is to consider a parameterized PDE in a general form and formulate a PDE-constrained…
We study systems of $n \geq 1$ discrete differential equations of order $k\geq1$ in one catalytic variable and provide a constructive and elementary proof of algebraicity of their solutions. This yields effective bounds and a systematic…
Parallel-in-time (PinT) techniques have been proposed to solve systems of time-dependent differential equations by parallelizing the temporal domain. Among them, Parareal computes the solution sequentially using an inaccurate (fast) solver,…
Many important systems across biology, engineering, physics, and economics are characterized by polynomial ordinary differential equations (ODEs), yet analytical solutions are rare. We develop a framework for identifying and solving a broad…
The analysis of many physical phenomena can be reduced to the study of solutions of differential equations with polynomial coefficients. In the present work, we establish the necessary and sufficient conditions for the existence of…
The approximation of solutions of partial differential equations (PDEs) with numerical algorithms is a central topic in applied mathematics. For many decades, various types of methods for this purpose have been developed and extensively…
First-order systems of hyperbolic partial differential equations (PDEs) occur ubiquitously throughout computational physics, commonly used in simulations of fluid turbulence, shock waves, electromagnetic interactions, and even general…
In this paper we present an algorithmic approach to the generation of fully conservative difference schemes for linear partial differential equations. The approach is based on enlargement of the equations in their integral conservation law…
Contrary to widespread perception, there is ever since 1994 a unified, general type independent theory for the existence of solutions for very large classes of nonlinear systems of PDEs. This solution method is based on the Dedekind order…
Many physical and engineering systems require solving direct problems to predict behavior and inverse problems to determine unknown parameters from measurement. In this work, we study both aspects for systems governed by differential…
This paper presents a novel approach for numerical solution of a class of fourth order time fractional partial differential equations (PDE's). The finite difference formulation has been used for temporal discretization, whereas, the space…
Learning underlying dynamics from data is important and challenging in many real-world scenarios. Incorporating differential equations (DEs) to design continuous networks has drawn much attention recently, however, most prior works make…
The Painleve test is very useful to construct not only the Laurent-series solutions but also the elliptic and trigonometric ones. Such single-valued functions are solutions of some polynomial first order differential equations. To find the…
Nonlinear systems of partial differential equations (PDEs) may permit several distinct solutions. The typical current approach to finding distinct solutions is to start Newton's method with many different initial guesses, hoping to find…