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The goal of this work is to discuss how should we impose initial values in fractional problems to ensure that they have exactly one smooth unique solution, where smooth simply means that the solution lies in a certain suitable space of…

General Mathematics · Mathematics 2019-10-09 Daniel Cao Labora

In this work, we introduce a novel numerical method for solving initial value problems associated with a given differential. Our approach utilizes a spline approximation of the theoretical solution alongside the integral formulation of the…

Numerical Analysis · Mathematics 2024-10-01 Gustavo H. O. Salgado , João P. R. Romanelli

This paper is devoted to the analysis of linear second order discrete-time descriptor systems (or singular difference equations (SiDEs) with control). Following the algebraic approach proposed by Kunkel and Mehrmann for pencils of matrix…

Numerical Analysis · Mathematics 2020-05-13 Vu Hoang Linh , Ha Phi

This paper introduces a second-order hyperplane search, a novel optimization step that generalizes a second-order line search from a line to a $k$-dimensional hyperplane. This, combined with the forward-mode stochastic gradient method,…

Machine Learning · Computer Science 2024-08-21 Adam D. Cobb , Atılım Güneş Baydin , Barak A. Pearlmutter , Susmit Jha

In this paper, a general integral identity for twice differentiable functions is derived. By using of this identity, the author establish some new estimates on Hermite-Hadamard type and Simpson type inequalities for s-convex via Riemann…

Classical Analysis and ODEs · Mathematics 2013-04-16 Imdat Iscan

We obtain integral representations of the $n$-th derivatives of the Bessel functions with respect to the order. The numerical evaluation of these expressions is very efficient using a double exponential integration strategy. Also, from the…

Classical Analysis and ODEs · Mathematics 2018-08-17 J. L. González-Santander

Recent computational strategies based on screening tests have been proposed to accelerate algorithms addressing penalized sparse regression problems such as the Lasso. Such approaches build upon the idea that it is worth dedicating some…

Machine Learning · Statistics 2015-10-28 Antoine Bonnefoy , Valentin Emiya , Liva Ralaivola , Rémi Gribonval

The paper proposes a linesearch for a primal-dual method. Each iteration of the linesearch requires to update only the dual (or primal) variable. For many problems, in particular for regularized least squares, the linesearch does not…

Optimization and Control · Mathematics 2018-03-26 Yura Malitsky , Thomas Pock

We present an algorithm that transforms, if possible, a given ODE or PDE with radical function coefficients into one with rational coefficients by means of a rational change of variables. It also applies to systems of linear ODEs. It is…

Classical Analysis and ODEs · Mathematics 2020-03-16 Jorge Caravantes , J. Rafael Sendra , David Sevilla , Carlos Villarino

Conventionally, data driven identification and control problems for higher order dynamical systems are solved by augmenting the system state by the derivatives of the output to formulate first order dynamical systems in higher dimensions.…

Optimization and Control · Mathematics 2021-06-02 Joel A. Rosenfeld , Benjamin P. Russo , Rushikesh Kamalapurkar

This paper studies first-order algorithms for solving fully composite optimization problems over convex and compact sets. We leverage the structure of the objective by handling its differentiable and non-differentiable components…

Optimization and Control · Mathematics 2023-07-13 Maria-Luiza Vladarean , Nikita Doikov , Martin Jaggi , Nicolas Flammarion

We describe a method for the rapid numerical evaluation of the Bessel functions of the first and second kinds of nonnegative real orders and positive arguments. Our algorithm makes use of the well-known observation that although the Bessel…

Numerical Analysis · Mathematics 2017-05-23 James Bremer

This paper introduces an efficient algorithm for computing the general oscillatory matrix functions. These computations are crucial for solving second-order semi-linear initial value problems. The method is exploited using the scaling and…

Numerical Analysis · Mathematics 2024-06-11 Dongping Li , Xue Wang , Xiuying Zhang

For a large class of systems of o.d.e.'s which have first integrals, the method of arrays yields the following results: i) The first integrals $I$ can be found by solving systems of linear equations. ii) How the first integral $I$ responds…

Mathematical Physics · Physics 2007-05-23 Lawrence Goldman

We consider second-order functional differential operators with a constant delay. Properties of their spectral characteristics are obtained and a nonlinear inverse problem is studied, which consists in recovering the operators from their…

Spectral Theory · Mathematics 2018-02-08 Natalia Bondarenko , Vjacheslav Yurko

In this paper, we develop a novel class of linear energy-preserving integrating factor methods for the 2D nonlinear Schr\"odinger equation with wave operator (NLSW), combining the scalar auxiliary variable approach and the integrating…

Numerical Analysis · Mathematics 2022-09-27 Xuelong Gu , Wenjun Cai , Chaolong Jiang , Yushun Wang

A set of Maple V R.3/4 computer algebra routines for the analytical solving of 1st. order ODEs, using Lie group symmetry methods, is presented. The set of commands includes a 1st. order ODE-solver and routines for, among other things: the…

General Relativity and Quantum Cosmology · Physics 2009-10-28 E. S. Cheb-Terrab , L. G. S. Duarte , L. A. C. P. da Mota

The main objective of this paper is to introduce an algorithm for solving fractional and classical differential equations based on a new generalized fractional power series. The algorithm relies on expanding the solution of an FDE or an ODE…

General Mathematics · Mathematics 2024-06-26 Youness Assebbane , Mohamed Echchehira , Mohamed Bouaouid , Mustapha Atraoui

This article provides an effective computational algorithm based on Legendre wavelet (LW) and standard tau approach to approximate the solution of multi-dimensional distributed order time-space fractional weakly singular integro-partial…

Numerical Analysis · Mathematics 2022-06-01 Yashveer Kuma , Somveer Singh , Reshma Singh , Vineet Kumar Singh

These notes focus on the minimization of convex functionals using first-order optimization methods, which are fundamental in many areas of applied mathematics and engineering. The primary goal of this document is to introduce and analyze…

Optimization and Control · Mathematics 2024-10-28 Charles Dossal , Samuel Hurault , Nicolas Papadakis