Related papers: Similarity reductions for a nonlinear diffusion eq…
A method is presented for finding the Lie point symmetry transformations acting simultaneously on difference equations and lattices, while leaving the solution set of the corresponding difference scheme invariant. The method is applied to…
Lie symmetry analysis is one of the powerful tools to analyze nonlinear ordinary differential equations. We review the effectiveness of this method in terms of various symmetries. We present the method of deriving Lie point symmetries,…
Q-conditional symmetries (nonclassical symmetries) for a general class of two-component reaction-diffusion systems with non-constant diffusivities are studied. The work is a natural continuation of our paper (Cherniha and Davydovych, 2012)…
Symmetry preserving difference schemes approximating second and third order ordinary differential equations are presented. They have the same three or four-dimensional symmetry groups as the original differential equations. The new…
We consider four different models of nonlinear diffusion equations involving fractional Laplacians and study the existence and properties of classes of self-similar solutions. Such solutions are an important tool in developing the general…
Reduction operators (called often nonclassical symmetries) of variable coefficient semilinear reaction-diffusion equations with power nonlinearity $f(x)u_t=(g(x)u_x)_x+h(x)u^m$ ($m\neq0,1,2$) are investigated using the algorithm suggested…
A full symmetry classification is given for models of energy transport in radiant plasma when the mass density is spatially variable and the diffusivity is nonlinear. A systematic search for conservation laws also leads to some potential…
We give a classification into conjugacy classes of subalgebras of the symmetry algebra generated by the Zabolotskaya-Khokhlov equation, and obtain all similarity reductions of this equation into $(1+1)$-dimensional equations. We thus show…
Our purpose is to obtain gradient estimates for certain nonlinear partial differential equations by coupling methods. First we derive uniform gradient estimates for a certain semi-linear PDEs based on the coupling method introduced in Wang…
Mathematical modeling of many physical processes such as diffusion, viscosity of fluids and combustion involves differential equations with small coefficients of higher derivatives. These may be small diffusion coefficients for modeling the…
This paper deals with the case of using nonlinear diffusion filters to obtain piecewise constant images as a previous process for segmentation techniques. We first show an intrinsic formulation for the nonlinear diffusion equation to…
One of old methods for finding exact solutions of nonlinear differential equations is considered. Modifications of the method are discussed. Application of the method is illustrated for finding exact solutions of the Fisher equation and…
Singular perturbation theory plays a central role in the approximate solution of nonlinear differential equations. However, applying these methods is a subtle art owing to the lack of globally applicable algorithms. Inspired by the fact…
A method for finding exact solutions of nonlinear differential equations is presented. Our method is based on the application of the Newton polygons corresponding to nonlinear differential equations. It allows one to express exact solutions…
Symmetries play an critical role in finding analytic solutions to nonlinear differential equations. A symmetry is a mapping of the solutions of the differential equation into the solutions and have been studied extensively for over a…
A mathematical model for the discrete nonlinear fragmentation (collision-induced breakage) equation with diffusion is studied. The existence of global weak solutions is established in arbitrary spatial dimensions without assuming a strictly…
We establish sharp energy decay rates for a large class of nonlinearly first-order damped systems, and we design discretization schemes that inherit of the same energy decay rates, uniformly with respect to the space and/or time…
We apply the theory of Lie symmetries in order to study a fourth-order $1+2$ evolutionary partial differential equation which has been proposed for the image processing noise reduction. In particular we determine the Lie point symmetries…
The paper takles a procedure which allow to extend some linear, wave type equations to the study of nonlinear models. More concretely, we present a practical way to generate the largest class of a given form of second order differential…
Exact self-consistent particle-like solutions with spherical and/or cylindrical symmetry to the equations governing the interacting system of scalar, electromagnetic and gravitational fields have been obtained. As a particular case it is…