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The purpose of this paper is twofold. An immediate practical use of the presented algorithm is its applicability to the parametric solution of underdetermined linear ordinary differential equations (ODEs) with coefficients that are…
We present a quantum algorithm for systems of (possibly inhomogeneous) linear ordinary differential equations with constant coefficients. The algorithm produces a quantum state that is proportional to the solution at a desired final time.…
This article revisits the approximation problem of systems of nonlinear delay differential equations (DDEs) by a set of ordinary differential equations (ODEs). We work in Hilbert spaces endowed with a natural inner product including a point…
We propose a novel quantum algorithm for solving linear autonomous ordinary differential equations (ODEs) using the Pad\'e approximation. For linear autonomous ODEs, the discretized solution can be represented by a product of matrix…
We extend the Deep Galerkin Method (DGM) introduced in Sirignano and Spiliopoulos (2018)} to solve a number of partial differential equations (PDEs) that arise in the context of optimal stochastic control and mean field games. First, we…
In recent years, a significant amount of attention has been paid to solve partial differential equations (PDEs) by deep learning. For example, deep Galerkin method (DGM) uses the PDE residual in the least-squares sense as the loss function…
We apply the discontinuous Galerkin finite element method with a degree $p$ polynomial basis to the linear advection equation and derive a PDE which the numerical solution solves exactly. We use a Fourier approach to derive polynomial…
A fast algorithm (linear in the degrees of freedom) for the solution of linear variable-coefficient rational-order fractional integral and differential equations is described. The approach is related to the ultraspherical method for…
This paper exhibits a very simple formula for a particular solution of a linear ordinary differential equation with constant real coefficients, P(d/dt)x = f, f a function given by a linear combination of polynomials, trigonometrical and…
Linear differential equations are ubiquitous in science and engineering. Quantum computers can simulate quantum systems, which are described by a restricted type of linear differential equations. Here we extend quantum simulation algorithms…
This paper is concerned with developing accurate and efficient numerical methods for one-dimensional fully nonlinear second order elliptic and parabolic partial differential equations (PDEs). In the paper we present a general framework for…
We investigate and derive second solutions to linear homogeneous second-order difference equations using a variety of methods, in each case going beyond the purely formal solution and giving explicit expressions for the second solution. We…
This paper presents a quantum algorithm for the solution of prototypical second-order linear elliptic partial differential equations discretized by $d$-linear finite elements on Cartesian grids of a bounded $d$-dimensional domain. An…
This article presents a new high-order accurate algorithm for finding a particular solution to a linear, constant-coefficient partial differential equation (PDE) by means of a convolution of the volumetric source function with the Green's…
In this work we apply the Deep Galerkin Method (DGM) described in Sirignano and Spiliopoulos (2018) to solve a number of partial differential equations that arise in quantitative finance applications including option pricing, optimal…
There are many methods for finding a particular solution to a nonhomogeneous linear ordinary differential equation (ODE) with constant coefficients. The method of undetermined coefficients, Laplace transform method and differential operator…
We construct quasi-Monte Carlo methods to approximate the expected values of linear functionals of Galerkin discretizations of parametric operator equations which depend on a possibly infinite sequence of parameters. Such problems arise in…
We study the problem of identifying unknown processes embedded in time-dependent partial differential equation (PDE) using observational data, with an application to advection-diffusion type PDE. We first conduct theoretical analysis and…
While quantum computing provides an exponential advantage in solving linear differential equations, there are relatively few quantum algorithms for solving nonlinear differential equations. In our work, based on the homotopy perturbation…
We propose a numerical method to approximate the solution of second order elliptic problems in nonvariational form. The method is of Galerkin type using conforming finite elements and applied directly to the nonvariational (nondivergence)…