Related papers: Polygons for finding exact solutions of nonlinear …
Linear differential equations of arbitrary order with polynomial coefficients are considered. Specifically, necessary and sufficient conditions for the existence of polynomial solutions of a given degree are obtained for these equations. An…
The paper develops the method for construction of the families of particular solutions to the nonlinear Partial Differential Equations (PDE) without relation to the complete integrability. Method is based on the specific link between…
In order to find closed form solutions of nonintegrable nonlinear ordinary differential equations, numerous tricks have been proposed. The goal of this short review is to recall classical, 19th-century results, completed in 2006 by…
A novel approach is introduced for deriving exact solutions to nonlinear systems of ordinary differential equations. This method consists of four parts. In the initial part, the examined nonlinear differential equation system is transformed…
Exact solutions are derived for an n-dimensional radial wave equation with a general power nonlinearity. The method, which is applicable more generally to other nonlinear PDEs, involves an ansatz technique to solve a first-order PDE system…
Complete discrimination system for polynomial and direct integral method were discussed systematically. In particularly, we pointed out some mistaken viewpoints. Combining with complete discrimination system for polynomial, direct integral…
We discuss the application of a variant of the method of simplest equation for obtaining exact traveling wave solutions of a class of nonlinear partial differential equations containing polynomial nonlinearities. As simplest equation we use…
Our goal is to find closed form analytic expressions for the solitary waves of nonlinear nonintegrable partial differential equations. The suitable methods, which can only be nonperturbative, are classified in two classes. In the first…
A novel method, connecting the space of solutions of a linear differential equation, of arbitrary order, to the space of monomials, is used for exploring the algebraic structure of the solution space. Apart from yielding new expressions for…
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…
This chapter is mainly a tutorial introduction to methods recently developed in order to find all (as opposed to some) meromorphic particular solutions of given nonintegrable, autonomous, algebraic ordinary differential equations of any…
We seek multi-order exact solutions of a generalized shallow water wave equation along with those corresponding to a class of nonlinear systems described by the KdV, modified KdV, Boussinesq, Klein-Gordon and modified Benjamin-Bona-Mahony…
The method of simplest equation is applied for obtaining exact solitary traveling-wave solutions of nonlinear partial differential equations that contain monomials of odd and even grade with respect to participating derivatives. The used…
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…
An explicit analytic solution to the nonlinear differential equation d^k y (--) ^n = y^l dx^kk is obtained for arbitrary integer values of k, l and n.
In this article, we follow an idea that is opposite to the idea of Hopf and Cole: we use transformations in order to transform simpler linear or nonlinear differential equations (with known solutions) to more complicated nonlinear…
We outline a new geometric method of constructing exact solutions of gravitational field equations parametrized by generic off-diagonal metrics, anholonomic frames and possessing, in general, nontrivial torsion and nonmetricity. The…
A modified method of functional constraints is used to construct the exact solutions of nonlinear equations of reaction-diffusion type with delay and which are associated with variable coefficients. This study considers a most generalized…
Recently, finding exact solutions of nonlinear fractional differential equations has attracted great interest. In this paper, the space time-fractional Klein-Gordon equation with cubic nonlinearities is examined. Firstly, suitable exact…
A method to find exact solutions to nonlinear Schr\"odinger equation, defined on a line and on a plane, is found by connecting it with second order linear ordinary differential equation. The connection is essentially made using Riccati…