Related papers: \QTR{group}{Exact solutions of nonlinear equations…
In this paper, we propose quasilinearization methods that convert nonlocal fully-nonlinear parabolic systems into the nonlocal quasilinear parabolic systems. The nonlocal parabolic systems serve as important mathematical tools for modelling…
We develop a numerical method for solving a system of nonlinear integral equations involving two integral terms: at the current time t, one integral is taken from 0 to t, and a different integral is taken from t to infinity. We prove the…
We give a direct analytic proof of the classical Boundary Harnack inequality for solutions to linear uniformly elliptic equations in either divergence or non-divergence form.
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.…
In this article we determine several theorems and methods for solving linear congruences and systems of linear congruences, and we find the number of distinct solutions. Many examples of solving congruences are given.
Discretization of non-linear Poisson-Boltzmann Equation equations results in a system of non-linear equations with symmetric Jacobian. The Newton algorithm is the most useful tool for solving non-linear equations. It consists of solving a…
We discuss a version the methodology for obtaining exact solutions of nonlinear partial differential equations based on the possibility for use of: (i) more than one simplest equation; (ii) relationship that contains as particular cases the…
We consider the problem of robust matrix completion, which aims to recover a low rank matrix $L_*$ and a sparse matrix $S_*$ from incomplete observations of their sum $M=L_*+S_*\in\mathbb{R}^{m\times n}$. Algorithmically, the robust matrix…
Let $C$ be a nonempty closed and convex subset of a uniformly smooth and uniformly convex real Banach space $E$ with dual space $E^*$. We present a novel hybrid method for finding a common solution of a family of equilibrium problems, a…
This article demonstrates how variation of parameters can be successfully implemented in combination with other classical techniques, such as the method of characteristics, to derive novel classes of solutions to nonlinear partial…
A nonlinear inequality is formulated in the paper. An estimate of the rate of growth/decay of solutions to this inequality is obtained. This inequality is of interest in a study of dynamical systems and nonlinear evolution equations. It can…
A novel symmetry method for finding exact solutions to nonlinear PDEs is illustrated by applying it to a semilinear reaction-diffusion equation in multi-dimensions. The method uses a separation ansatz to solve an equivalent first-order…
The paper describes a number of simple but quite effective methods for constructing exact solutions of PDEs, that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions…
The Monotonicity inequality is an important tool in the understanding of existence and uniqueness of strong solutions for Stochastic PDEs. In this article, we discuss three approaches to establish this deterministic inequality explicitly.
A full Lie point symmetry analysis of rational difference equations is performed. Non-trivial symmetries are derived and exact solutions using these symmetries are obtained.
Nonlinear equations are challenging to solve due to their inherently nonlinear nature. As analytical solutions typically do not exist, numerical methods have been developed to tackle their solutions. In this article, we give a quantum…
An effective method to obtain exact analytical solutions of equations describing the coherent dynamics of multilevel systems is presented. The method is based on the usage of orthogonal polynomials, integral transforms and their discrete…
There is no general existence theorem for solutions for nonlinear difference equations, so we must prove the existence of solutions in accordance with models one by one. In our work, we found theorems for the existence of analytic solutions…
A Hamiltonian formulation of generic many-body systems with balanced loss and gain is presented. It is shown that a Hamiltonian formulation is possible only if the balancing of loss and gain terms occur in a pairwise fashion. It is also…
We study the existence and numerical computation of traveling wave solutions for a family of nonlinear higher-order Boussinesq evolution systems with a Hamiltonian structure. This general Boussinesq evolution system includes a broad class…