相关论文: Semi-analytical Solutions for Breakage and Aggrega…
We deal with the numerical solution of linear partial differential equations (PDEs) with focus on the goal-oriented error estimates including algebraic errors arising by an inaccurate solution of the corresponding algebraic systems. The…
The $\Sigma$-method for structural analysis of a differential-algebraic equation (DAE) system produces offset vectors from which the sparsity pattern of a system Jacobian is derived. This pattern implies a block-triangular form (BTF) of the…
Semi-Lagrangian methods are numerical methods designed to find approximate solutions to particular time-dependent partial differential equations (PDEs) that describe the advection process. We propose semi-Lagrangian one-step methods for…
Partial differential equations (PDEs) are used to describe a variety of physical phenomena. Often these equations do not have analytical solutions and numerical approximations are used instead. One of the common methods to solve PDEs is the…
Solving high-dimensional parabolic partial differential equations (PDEs) with deep learning methods is often computationally and memory intensive, primarily due to the need for automatic differentiation (AD) to compute large Hessian…
An exact arithmetic, memory efficient direct solution method for finite element method (FEM) computations is outlined. Unlike conventional black-box or low-rank direct solvers that are opaque to the underlying physical problem, the proposed…
In this article, a new deep learning architecture, named JDNN, has been proposed to approximate a numerical solution to Partial Differential Equations (PDEs). The JDNN is capable of solving high-dimensional equations. Here, Jacobi Deep…
In this paper, approximate analytical solutions of nonlinear Emden-Fowler type equations are obtained by the differential transform method (DTM). The DTM is a numerical as well as analytical method for solving integral equations, ordinary…
We address a new numerical scheme based on a class of machine learning methods, the so-called Extreme Learning Machines with both sigmoidal and radial-basis functions, for the computation of steady-state solutions and the construction of…
In this work we develop a novel domain splitting strategy for the solution of partial differential equations. Focusing on a uniform discretization of the $d$-dimensional advection-diffusion equation, our proposal is a two-level algorithm…
Abstract differential-algebraic equations (ADAEs) of a semilinear type are studied. Theorems on the existence and uniqueness of solutions and the maximal interval of existence, on the global solvability of the ADAEs, the boundedness of…
A fractional Adomian decomposition method for fractional nonlinear differential equations is proposed. The iteration procedure is based on Jumarie's fractional derivative. An example is given to elucidate the solution procedure, and the…
Neural network-based solvers for partial differential equations (PDEs) have attracted considerable attention, yet they often face challenges in accuracy and computational efficiency. In this work, we focus on time-dependent PDEs and observe…
This article aims to demonstrate and discuss the applications of automatic differentiation (AD) for finding derivatives in PDE-constrained optimization problems and Jacobians in non-linear finite element analysis. The main idea is to…
As fractional diffusion equations can describe the early breakthrough and the heavy-tail decay features observed in anomalous transport of contaminants in groundwater and porous soil, they have been commonly employed in the related…
Analytical and numerical techniques have been developed for solving fractional partial differential equations (FPDEs) and their systems with initial conditions. However, it is much more challenging to develop analytical or numerical…
In this article, we explore the effectiveness of two polynomial methods in solving non-linear time and space fractional partial differential equations. We first outline the general methodology and then apply it to five distinct experiments.…
The non-linear collision induced fragmentation plays a crucial role in modeling several engineering and physical problems. In contrast to linear breakage, it has not been thoroughly investigated in the existing literature. This study…
In this paper we present a Fourier feature based deep domain decomposition method (F-D3M) for partial differential equations (PDEs). Currently, deep neural network based methods are actively developed for solving PDEs, but their efficiency…
The Fractional Diffusion Equation (FDE) is a mathematical model that describes anomalous transport phenomena characterized by non-local and long-range dependencies which deviate from the traditional behavior of diffusion. Solving this…