Related papers: Series solutions of linear ODEs by Newton-Raphson …
When ${\cal{D}}$ is a linear partial differential operator of any order, a direct problem is to look for an operator ${\cal{D}}_1$ generating the compatibility conditions (CC) ${\cal{D}}_1\eta=0$ of ${\cal{D}}\xi=\eta$. We may thus…
The essentials of a new method in solving very large classes of nonlinear systems of PDEs, possibly associated with initial and/or boundary value problems, are presented. The PDEs can be defined by continuous, not necessarily smooth…
We are interested in the numerical solution of coupled nonlinear partial differential equations (PDEs) in two and three dimensions. Under certain assumptions on the domain, we take advantage of the Kronecker structure arising in standard…
This paper presents high-order numerical methods for solving boundary value problems associated with the Lane-Emden equation, which frequently arises in astrophysics and various nonlinear models. A major challenge in studying this equation…
We study algorithms for the fast computation of modular inverses. Newton-Raphson iteration over $p$-adic numbers gives a recurrence relation computing modular inverse modulo $p^m$, that is logarithmic in $m$. We solve the recurrence to…
This paper is part of a series of papers in which the asymptotic theory and appropriate symbolic computer code are developed to compute the asymptotic expansion of the solution of an n-th order ordinary differential equation. The paper…
We introduce a novel integrability-preserving discretization for a broad class of differential equations with variable coefficients, encompassing both linear and nonlinear cases. The construction is achieved via a categorical approach that…
In this paper we construct high order numerical methods for solving third and fourth orders nonlinear functional differential equations (FDE). They are based on the discretization of iterative methods on continuous level with the use of the…
In this paper, we give an algorithm for finding general rational solutions of a given first-order ODE with parametric coefficients that occur rationally. We present an analysis, complete modulo Hilbert's irreducibility problem, of the…
We propose a unified derivative-free proximal Newton-type algorithm framework for solving composite optimization problems formulated as the sum of a black-box function and a known regularization term. We establish the iteration and oracle…
In this paper, we derive systems of ordinary differential equations (ODEs) satisfied by modular forms of level three, which are level three versions of Ramanujan's system of ODEs satisfied by the classical Eisenstein series.
Some properties and relations satisfied by the polynomial solutions of a bispectral problem are studied. Given a finite order differential operator, under certain restrictions, its polynomial eigenfunctions are explicitly obtained, as well…
We describe inexact proximal Newton-like methods for solving degenerate regularized optimization problems and for the broader problem of finding a zero of a generalized equation that is the sum of a continuous map and a maximal monotone…
For a composite $n$ and an odd $c$ with $c$ not dividing $n$, the number of solutions to the equation $n+a\equiv b\mod c$ with $a,b$ quadratic residues modulus $c$ is calculated. We establish a direct relation with those modular solutions…
Linear programming relaxations are central to {\sc map} inference in discrete Markov Random Fields. The ability to properly solve the Lagrangian dual is a critical component of such methods. In this paper, we study the benefit of using…
We propose a new method to solve eigenvalue problems for linear and semilinear second order differential operators in high dimensions based on deep neural networks. The eigenvalue problem is reformulated as a fixed point problem of the…
We consider certain quotient algebras of tensor algebras of bimodules $M$ over a finite-dimensional algebra $R$, and we investigate Frobenius type properties of such algebras. Our main interest is in the case where $M=R^*$, the linear dual…
The development of functions of real variables in Taylor and Frobenius series (whole series which are formed in nonorthogonal, nonperiodic bases), in sinusoidal Fourier series (bases of orthogonal, periodic functions), in series of special…
Solutions to most nonlinear ordinary differential equations (ODEs) rely on numerical solvers, but this gives little insight into the nature of the trajectories and is relatively expensive to compute. In this paper, we derive analytic…
We compare and discuss the respective efficiency of three methods (with two variants for each of them), based respectively on Taylor (Maclaurin) series, Pad\'{e} approximants and conformal mappings, for solving quasi-analytically a…