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To approximate solutions of a linear differential equation, we project, via trigonometric interpolation, its solution space onto a finite-dimensional space of trigonometric polynomials and construct a matrix representation of the…
The existence of sufficiently many finite order meromorphic solutions of a differential equation, or difference equation, or differential-difference equation, appears to be a good indicator of integrability. In this paper, we investigate…
The differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the determining…
The existence and analyticity of solutions to linear systems of moment differential equations with analytic coefficients is studied. The relation of solutions of such systems with respect to linear moment differential equations is…
In this work, we propose a new approach called ``stationary reduction method based on nonisospectral deformation of orthogonal polynomials" for deriving discrete Painlev\'{e}-type (d-P-type) equations. We apply this approach to…
A large family of linear, usually overdetermined, systems of partial differential equations that admit a multiplication of solutions, i.e, a bi-linear and commutative mapping on the solution space, is studied. This family of PDE's contains…
We study the problem of solvability of linear differential systems with small coefficients in the Liouvillian sense (or, by generalized quadratures). For a general system, this problem is equivalent to that of solvability of the Lie algebra…
Discrete Differential Equations (DDEs) are functional equations that relate polynomially a power series $F(t,u)$ in $t$ with polynomial coefficients in a "catalytic" variable $u$ and the specializations, say at $u=1$, of $F(t,u)$ and of…
In this paper, we propose some algorithms for analytical solution construction to nonlinear polynomial partial differential equations with constant function coefficients. These schemes are based on one-(single), two- (double) or three-…
We prove a necessary and sufficient condition for the existence of a $T$-periodic solution for the time-periodic second order differential equation $\ddot{x}+f(t,x)+p(t,x,\dot x)=0$, where $f$ grows superlinearly in $x$ uniformly in time,…
We present several second-order linear differential equations that are associated to a particular Riccati equation with only one constant parameter in its coefficients through the technique of supersymmetric factorizations and through a…
Recently, a new fractional derivative called the conformable fractional derivative is given which is based on the basic limit definition of the derivative in [1]. Then, the fractional versions of chain rules, exponential functions,…
This study shows how to obtain least-squares solutions to initial and boundary value problems to nonhomogeneous linear differential equations with nonconstant coefficients of any order. However, without loss of generality, the approach has…
A numerical explicit method to evaluates transient solutions of linear partial differential inhomogeneous equation with constant coefficients is proposed. A general form of the scheme for a specific linear inhomogeneous equation is shown.…
In this paper we give a necessary and suffcient conditions for the existence and uniqueness of periodic solutions of functional differential equations with n delay d dt x(t) = Ax(t) + n j=1 Bx(t -- r j) + f (t). The conditions are obtained…
Selection of 25 examples from extensive nontrivial families for different types of nonlinear PDEs and their formal general solutions are given. The main goal here is to show on examples the types of solvable PDEs and what their general…
In this paper we show that an arbitrary solution of one ordinary difference equation is also a solution for a hierarchy of integrable difference equations. We also provide an example of such a solution that is related to sequence generated…
We show a general method allowing the solution calculation, in the form of a power series, for a very large class of nonlinear Ordinary Differential Equations (ODEs), namely the real analytic $\sigma\pi$-ODEs (and, more in general, the real…
It is well-known that one-dimensional time fractional diffusion-wave equations with variable coefficients can be reduced to ordinary fractional differential equations and systems of linear fractional differential equations via scaling…
This article is devoted to the study of solutions of non-homogenous linear differential equations having entire coefficients. We get all non-trivial solutions of infinite order of equation $f^{(n)}+a_{n-1}(z)f^{(n-1)}+\ldots…