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Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks. We introduce a generalization for these methods that manifests as a…
This article firstly develops a proximal explicit approach for the generalized method of lines. In such a method, the domain of the PDE in question is discretized in lines and the equation solution is written on these lines as functions of…
We develop a high accuracy power series method for solving partial differential equations with emphasis on the nonlinear Schr\"odinger equations. The accuracy and computing speed can be systematically and arbitrarily increased to orders of…
In this paper we are interested in the numerical solution of stochastic differential equations with non negative solutions. Our goal is to construct explicit numerical schemes that preserve positivity, even for super linear stochastic…
In this work, a new technique has been presented to find approximate solution of linear integro-differential equations. The method is based on modified orthonormal Bernoulli polynomials and an operational matrix thereof. The method converts…
Nonlinear elliptic problems arise in many fields, including plasma physics, astrophysics, and optimal transport. In this article, we propose a novel operator-splitting/finite element method for solving such problems. We begin by introducing…
The use of nonlinear PDEs has led to significant advancements in various fields, such as physics, biology, ecology, and quantum mechanics. However, finding multiple solutions for nonlinear PDEs can be a challenging task, especially when…
In this paper, a symmetrized two-scale finite element method is proposed for a class of partial differential equations with symmetric solutions. With this method, the finite element approximation on a fine tensor product grid is reduced to…
This paper concerns the inclusion of Newton's method into an adaptive finite element method (FEM) for the solution of nonlinear partial differential equations (PDEs). It features an adaptive choice of the damping parameter in the Newton…
A nonlinear partial differential equation is a nonlinear relationship between an unknown function and how it changes due to two or more input variables. A numerical method reduces such an equation to arithmetic for quick visualization, but…
In this paper, based on the combination of finite element mesh and neural network, a novel type of neural network element space and corresponding machine learning method are designed for solving partial differential equations. The…
Designing efficient and accurate numerical solvers for high-dimensional partial differential equations (PDEs) remains a challenging and important topic in computational science and engineering, mainly due to the "curse of dimensionality" in…
To obtain new types of exact travelling wave solutions to nonlinear partial differential equations, a number of approximate methods are known in the literature. In this study, we extend the class of auxiliary equations of Fibonnacci&Lucas…
A general formalism to solve nonlinear differential equations is given. Solutions are found and reduced to those of second order nonlinear differential equations in one variable. The approach is uniformized in the geometry and solves…
This short communication develops a new numerical procedure suitable for a large class of ordinary differential equation systems found in models in physics and engineering. The main numerical procedure is analogous to those concerning the…
Solutions to fractional models inherently exhibit non-smooth behavior, which significantly deteriorates the accuracy and therefore efficiency of existing numerical methods. We develop a two-stage data-infused computational framework for…
Our purpose is to obtain gradient estimates for certain nonlinear partial differential equations by coupling methods. First we derive uniform gradient estimates for a certain semi-linear PDEs based on the coupling method introduced in Wang…
We consider the Serre system of equations which is a nonlinear dispersive system that models two-way propagation of long waves of not necessarily small amplitude on the surface of an ideal fluid in a channel. We discretize in space the…
By using truncation technique, minimization method and Morse theory, we obtain three nontrivial solutions for a class of semilinear elliptic equations.
In this paper, it is shown how a combination of approximate symmetries of a nonlinear wave equation with small dissipations and singularity analysis provides exact analytic solutions. We perform the analysis using the Lie symmetry algebra…