Related papers: Non-point symmetry reduction method of partial dif…
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 shall deal with the periodic problem for nonlinear perturbations of abstract hyperbolic evolution equations generating an evolution system of contractions. We prove an averaging principle for the translation along trajectories operator…
It is well recognized that new types of exact travelling wave solutions to nonlinear partial differential equations can be obtained by modifications of the methods which are in hand. In this study, we extend the class of auxiliary equations…
We present a general method for studying long time asymptotics of nonlinear parabolic partial differential equations. The method does not rely on a priori estimates such as the maximum principle. It applies to systems of coupled equations,…
A powerful method for solving non-linear first-order ordinary differential equations, which is based on geometrical understanding of the corresponding dynamics of the so called Lie systems, is developed. This method allows us not only to…
For any $n\in\mathbb{N}$ a nonlinear ordinary differential equation with Lie algebra of point symmetries isomorphic to $\frak{sl}(2,\mathbb{R})$ is given.
In this work, we propose and analyze a residual-minimization strategy for the numerical solution of nonlinear PDEs posed in Banach spaces. Given a finite-dimensional trial space and a suitably enriched discrete test space (of higher…
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…
We investigate slowly converging solutions for non-linear evolution equations of elliptic or parabolic type. These equations arise from the study of isolated singularities in geometric variational problems. Slowly converging solutions have…
We obtain the complete Lie point symmetry algebras of two sequences of odd-order evolution equations. This includes equations that are fully-nonlinear, i.e. nonlinear in the highest derivative. Two of the equations in the sequences have…
We give two theorems which show that the Lie point and the Noether symmetries of a second-order ordinary differential equation of the form (D/(Ds))(((Dx^{i}(s))/(Ds)))=F(x^{i}(s),x^{j}(s)) are subalgebras of the special projective and the…
We present a new view onto the successive approximations' approach in study of the two-point nonlinear fractional boundary value problems. In order to reduce the original problem and further construct its approximate solution we use the…
A semi-classical approach to the study of the evolution of anyonic excitations--elementary particles with fractional statistics, complementing bosons and fermions--is through the Boltzmann equation for anyons. This work reviews a…
We present an algorithm for determining the Lie point symmetries of differential equations on fixed non transforming lattices, i.e. equations involving both continuous and discrete independent variables. The symmetries of a specific…
For a class of partial differential algebraic equations (PDAEs) of quasi-linear type which include nonlinear terms of convection type a possibility to determine a time and spatial index is considered. As a typical example we investigate an…
We consider an initial and Dirichlet boundary value problem for a semilinear, two dimensional heat equation over a rectangular domain. The problem is discretized in time by a version of the Relaxation Scheme proposed by C. Besse (C. R.…
We give a complete point-symmetry classification of all third-order evolution equations of the form $u_t=F(t,x,u,u_x, u_{xx})u_{xxx}+G(t,x,u,u_x, u_{xx})$ which admit semi-simple symmetry algebras and extensions of these semi-simple Lie…
We propose certain approach of solving two-dimensional non-stationary and stationary advection-diffusion-reaction boundary value problems through their reduction to the set of corresponding one-dimensional problems. This method leverages…
Using a modified version of Schauder's fixed point theorem, measures of non-compactness and classical techniques, we provide new general results on the asymptotic behavior and the non-oscillation of second order scalar nonlinear…
Multiple solutions are common in various non-convex problems arising from industrial and scientific computing. Nonetheless, understanding the nontrivial solutions' qualitative properties seems limited, partially due to the lack of efficient…