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We show a general method allowing the solution calculation, in the form of a power series, for a very large class of nonlinear Ordinary Differential Equations (ODEs), namely the real analytic $\sigma\pi$-ODEs (and, more in general, the real…
We describe a solving semi-decision method based on examination of the rational structures of the generalized integrating factors of first-order ODEs. We propose a conjecture that for some family of equations of the type…
Let $F_1,\dotsc,F_R$ be quadratic forms with integer coefficients in $n$ variables. When $n\geq 9R$ and the variety $V(F_1,\dotsc,F_R)$ is a smooth complete intersection, we prove an asymptotic formula for the number of integer points in an…
In this paper we study linear projection methods for approximating the solution and simultaneously preserving first integrals of autonomous ordinary differential equations. We show that (linear) projection methods are a subset of discrete…
Quadratization of polynomial and nonpolynomial systems of ordinary differential equations is advantageous in a variety of disciplines, such as systems theory, fluid mechanics, chemical reaction modeling and mathematical analysis. A…
The purpose of this article is to propose ODE based approaches for the numerical evaluation of matrix functions $f(A)$, a question of major interest in the numerical linear algebra. To this end, we model $f(A)$ as the solution at a finite…
Given an approximation to a multiple isolated solution of a polynomial system of equations, we have provided a symbolic-numeric deflation algorithm to restore the quadratic convergence of Newton's method. Using first-order derivatives of…
The phase retrieval problem is a fundamental problem in many fields, which is appealing for investigation. It is to recover the signal vector $\tilde{x}\in\mathbb{C}^d$ from a set of $N$ measurements $b_n=|f^*_n\tilde{x}|^2,\ n=1,\cdots,…
In the article we discuss the notion of the generalized invariant manifold introduced in our previous study. In the literature the method of the differential constraints is well known as a tool for constructing particular solutions for the…
It is known, due to Mordukhai-Boltovski, Ritt, Prelle, Singer, Christopher and others, that if a given rational ODE has a Liouvillian first integral then the corresponding integrating factor of the ODE must be of a very special form of a…
New iterative methods for solving linear equations are presented that are easy to use, generalize good existing methods, and appear to be faster. The new algorithms mix two kinds of linear recurrence formulas. Older methods have either high…
The present work proposes a well-balanced finite volume-type numerical method for the solution of non-conservative hyperbolic partial differential equations (PDEs) with source terms. The method is characterized, first, by the use of a…
Partial differential equations (PDEs) with spatially-varying coefficients arise throughout science and engineering, modeling rich heterogeneous material behavior. Yet conventional PDE solvers struggle with the immense complexity found in…
An update of the ODEtools Maple package, for the analytical solving of 1st and 2nd order ODEs using Lie group symmetry methods, is presented. The set of routines includes an ODE-solver and user-level commands realizing most of the relevant…
We develop a general framework for numerically solving differential equations while preserving invariants. As in standard projection methods, we project an arbitrary base integrator onto an invariant-preserving manifold, however, our method…
The multiscale complexity of modern problems in computational science and engineering can prohibit the use of traditional numerical methods in multi-dimensional simulations. Therefore, novel algorithms are required in these situations to…
This paper presents a novel semi-analytical collocation method to solve multi-term variable-order time fractional partial differential equations (VOTFPDEs). In the proposed method it employs the Fourier series expansion for spatial…
In multi-phase fluid flow, fluid-structure interaction, and other applications, partial differential equations (PDEs) often arise with discontinuous coefficients and singular sources (e.g., Dirac delta functions). These complexities arise…
A trigonometric interpolation algorithm for non-periodic functions has been recently proposed and applied to study general ordinary differential equation (ODE). This paper enhances the algorithm to approximate functions in $2$-dim space.…
A transient magneto-quasistatic vector potential formulation involving nonlinear material is spatially discretized using the finite element method of first and second polynomial order. By applying a generalized Schur complement the…