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We consider discrete nonlinear hyperbolic equations on quad-graphs, in particular on the square lattice. The fields are associated to the vertices and an equation Q(x_1,x_2,x_3,x_4)=0 relates four fields at one quad. Integrability of…
Fundamentals on Lie group methods and applications to differential equations are surveyed. Many examples are included to elucidate their extensive applicability for analytically solving both ordinary and partial differential equations.
A classification of ordinary differential equations and finite-difference equations in one variable having polynomial solutions (the generalized Bochner problem) is given. The method used is based on the spectral problem for a polynomial…
We discuss the last version as well as applications of a method for obtaining exact solutions of nonlinear partial differential equations. As this version is based on more than one simple equation we call it Simple Equations Method (SEsM).…
Properties of partial integrals such as real and complex-valued polynomial, multiple polynomial, exponential, and conditional for ordinary differential systems are studied. The possibilities of constructing first integrals and last…
A fast algorithm (linear in the degrees of freedom) for the solution of linear variable-coefficient rational-order fractional integral and differential equations is described. The approach is related to the ultraspherical method for…
We determine a considerable class of nonlinear partial differential equation systems which have global regular solutions. Uniqueness is not a direct general consequence of this method. The scheme can be applied to the incompressible Navier…
This paper examines some solutions for confluent and double-confluent Heun equations. In the first place, we review two Leaver's solutions in series of regular and irregular confluent hypergeometric functions for the confluent equation and…
We consider a matrix nonlinear partial differential equation that generalizes Heisenberg ferromagnet equation. This generalized Heisenberg ferromagnet equation is completely integrable with a linear bundle Lax pair related to the…
This \textquoteleft research-survey' is meant for beginners in the studies of integrable systems. Here we outline some analytical methods for dealing with a class of nonlinear partial differential equations. We pay special attention to…
Nonlinear matrix equations arise in many practical contexts related to control theory, dynamical programming and finite element methods for solving some partial differential equations. In most of these applications, it is needed to compute…
A new approach for obtaining the transformations of solutions of nonlinear ordinary differential equations representable as the compatibility condition of the overdetermined linear systems is proposed. The corresponding transformations of…
In this paper we consider some new classes of integral equations that arise from Lie symmetry analysis. Specifically, we consider the task of obtaining solutions of a Cauchy problem for some classes of second order hyperbolic partial…
In recent years a large literature on deep learning based methods for the numerical solution partial differential equations has emerged; results for integro-differential equations on the other hand are scarce. In this paper we study deep…
In this paper, we propose an efficient exponential integrator finite element method for solving a class of semilinear parabolic equations in rectangular domains. The proposed method first performs the spatial discretization of the model…
We consider the problem of obtaining higher order in regularization parameter $\epsilon$ analytical results for master integrals with elliptics. The two commonly employed methods are provided by the use of differential equations and direct…
Duality methods are used to generate explicit solutions to nonlinear Hodge systems, demonstrate the well-posedness of boundary value problems, and reveal, via the Hodge-B\"acklund transformation, underlying symmetries among superficially…
Here the polynomial interpolation approach is used to introduce the main results on multivariate normal algebraic systems. Next we bring a construction which shows that any standard algebraic system, with finite set of solutions, can be…
We propose a new approach that allows one to reduce nonlinear equations on Lie groups to equations with a fewer number of independent variables for finding particular solutions of the nonlinear equations. The main idea is to apply the…
Neural Networks have been widely used to solve Partial Differential Equations. These methods require to approximate definite integrals using quadrature rules. Here, we illustrate via 1D numerical examples the quadrature problems that may…