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The aim of this article is to study a nonlinear system modeling a Non-Newtonian fluid of polymer aqueous solutions. We are interested here in the existence of weak solutions for the stationary problem in a bounded plane domain or in…
We describe a method to model nonlinear dynamical systems using periodic solutions of delay-differential equations. We show that any finite-time trajectory of a nonlinear dynamical system can be loaded approximately into the initial…
A new non-perturbative method of solution of the nonlinear Heisenberg equations in the finite-dimensional subspace is illustrated. The method, being a counterpart of the traditional Schrodinger picture method, is based on a finite operator…
Various versions of the Dynamical Systems Method (DSM) are proposed for solving linear ill-posed problems with bounded and unbounded operators. Convergence of the proposed methods is proved. Some new results concerning discrepancy principle…
The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an…
We consider a pair of coupled non-linear partial differential equations describing a biochemical model system. The Weiss-algorithm for the Painle\'{e} test, that has been succesfully used in mathematical physics for the KdV-equation,…
Spectral parameter power series (SPPS) method is a recently introduced technique for solving linear differential equations and related spectral problems. In the present work we develop an approach based on the SPPS for analysis of…
A very simple and accurate numerical method which is applicable to systems of differentio-integral equations with quite general boundary conditions has been devised. Although the basic idea of this method stems from the Keller Box method,…
The aim of this paper is to solve numerically, using the meshless method via radial basis functions, time-space-fractional partial differential equations of type Black-Scholes. The time-fractional partial differential equation appears in…
The differential equations of chemical kinetics are systems of nonlinear (polynomial) differential equations, therefore their solutions cannot usually be found in symbolic form. Here we offer a method to solve classes of kinetic…
This paper proposes an efficient method for computing partial eigenvalues of large sparse matrices what can be called the inexact inverse power method (IIPM). It is similar to the inexact Rayleigh quotient method and inexact Jacobi-Davidson…
Symmetry group methods are applied to obtain all explicit group-invariant radial solutions to a class of semilinear Schrodinger equations in dimensions $n\neq 1$. Both focusing and defocusing cases of a power nonlinearity are considered,…
In this paper, we investigate the existence and nonexistence of entire solutions to a general class of Cauchy problems in the positive half line. Our results provide a unified approach to proving sharp local and entire solvability of…
Factorial series played a major role in Stirling's classic book "Methodus Differentialis" (1730), but now only a few specialists still use them. This article wants to show that this neglect is unjustified, and that factorial series are…
Partial ordinary Bell polynomials are used to formulate and prove a version of the Fa\`{a} di Bruno's formula which is convenient for handling nonlinear terms in the differential transformation. Applicability of the result is shown in two…
Carleman linearization is a technique that embeds systems of ordinary differential equations with polynomial nonlinearities into infinite dimensional linear systems in a procedural way. In this paper we generalize the method for systems of…
A version of the Dynamical Systems Method (DSM) for solving ill-posed nonlinear equations with monotone operators in a Hilbert space is studied in this paper. An a posteriori stopping rule, based on a discrepancy-type principle is proposed…
The theory of matrix splitting is a useful tool for finding solution of rectangular linear system of equations, iteratively. The purpose of this paper is two-fold. Firstly, we revisit theory of weak regular splittings for rectangular…
In this paper, a nonlinear system of fractional ordinary differential equations with multiple scales in time is investigated. We are interested in the effective long-term computation of the solution. The main challenge is how to obtain the…
We describe a set of Gaussian Process based approaches that can be used to solve non-linear Ordinary Differential Equations. We suggest an explicit probabilistic solver and two implicit methods, one analogous to Picard iteration and the…