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We present a new approach to solving polynomial ordinary differential equations by transforming them to linear functional equations and then solving the linear functional equations. We will focus most of our attention upon the first-order…
This paper reveals that a common and central role, played in many error bound (EB) conditions and a variety of gradient-type methods, is a residual measure operator. On one hand, by linking this operator with other optimality measures, we…
Sextic oscillator in D dimensions is considered as a typical quasi-exactly solvable (QES) model. Usually, its QES N-plets of bound states have to be computed using the coupled Magyari's nonlinear algebraic equations. We propose and describe…
The aim of this study is to present a good modernistic strategy for solving some well-known classes of Lane-Emden type singular differential equations. The proposed approach is based on the reproducing kernel Hilbert space (RKHS) and…
We study the Lane-Emden system $$\begin{cases} -\Delta u=v^p,\quad u>0,\quad\text{in}~\Omega, -\Delta v=u^q,\quad v>0,\quad\text{in}~\Omega, u=v=0,\quad\text{on}~\partial\Omega, \end{cases}$$ where $\Omega\subset\mathbb{R}^2$ is a smooth…
In this study, we firstly propose an auxiliary equation neural networks method (AENNM), an innovative analytical method that integrates neural networks (NNs) models with the auxiliary equation method to obtain exact solutions of nonlinear…
Differentiable systems in this paper means systems of equations that are described by differentiable real functions in real matrix variables. This paper proposes algorithms for finding minimal rank solutions to such systems over (arbitrary…
We prove the quasi-optimal convergence of a standard adaptive finite element method (AFEM) for nonlinear elliptic second-order equations of monotone type. The adaptive algorithm is based on residual-type a posteriori error estimators and…
Approximate solutions to elliptic partial differential equations with known kernel can be obtained via the boundary element method (BEM) by discretizing the corresponding boundary integral operators and solving the resulting linear system…
We present in this paper a rigorous theoretical framework to show stability, convergence and accuracy of improved edge-based and face-based smoothed finite element methods (bESFEM and bFS-FEM) for nearly-incompressible elasticity problems.…
The Poisson-Boltzmann equation is widely used to model molecular electrostatics; however, it is usually solved in linearised form because the sinh nonlinearity is challenging, limiting its applicability in highly charged systems such as…
We study existence, nonexistence, and uniqueness of positive radial solutions for a class of nonlinear systems driven by Pucci extremal operators under a Lane-Emden coupling configuration. Our results are based on the analysis of the…
We present two approximation methods for computing eigenfrequencies and eigenmodes of large-scale nonlinear eigenvalue problems resulting from boundary element method (BEM) solutions of some types of acoustic eigenvalue problems in…
We address the problem of the best uniform approximation by linear combinations of a finite system of functions. If the system is Chebyshev and the problem is unconstrained, then the classical Remez algorithm provides a fast and precise…
Consider \[ \begin{cases} F(D^2 u,Du,u,x) = u^{-p}v^{-q},~\text{in}~\Omega\\ F(D^2 v,Dv,v,x)=u^{-r}v^{-s},~~\text{in}~~\Omega\\ u,v>0~~\text{in}~~\Omega\\ u=v=0~\quad~\text{on}~~\partial\Omega, \end{cases} \] where $\Omega$ is an open…
Regression with a spherical response is challenging due to the absence of linear structure, making standard regression models inadequate. Existing methods, mainly parametric, lack the flexibility to capture the complex relationship induced…
Extreme learning machine (ELM) is a network model that arbitrarily initializes the first hidden layer and can be computed speedily. In order to improve the classification performance of ELM, a $\ell_2$ and $\ell_{0.5}$ regularization ELM…
Nonlinear differential equations exhibit rich phenomena in many fields but are notoriously challenging to solve. Recently, Liu et al. [1] demonstrated the first efficient quantum algorithm for dissipative quadratic differential equations…
We present a new numerical method for the isometric embedding of 2-geometries specified by their 2-metrics in three dimensional Euclidean space. Our approach is to directly solve the fundamental embedding equation supplemented by six…
We derive exact analytic solutions for density and velocity fields to all orders in Eulerian standard perturbation theory for $\Lambda$CDM cosmology. In particular, we show that density and velocity field kernels can be written in a…