Related papers: Generalized damped Milne-Pinney equation and Chiel…
We discuss alternative iteration methods for differential equations. We provide a convergence proof for exactly solvable examples and show more convenient formulas for nontrivial problems.
We use the methods of group theory to reduce the equations of motion of the $CP^{1}$ model in (2+1) dimensions to sets of two coupled ordinary differential equations. We decouple and solve many of these equations in terms of elementary…
We study the Cauchy problem of the semilinear damped wave equation with polynomial nonlinearity, and establish the local and global existence of the solution for slowly decaying initial data not belonging to $L^2(\mathbb{R}^n)$ in general.…
We present numerical solutions for differential equations by expanding the unknown function in terms of Chebyshev polynomials and solving a system of linear equations directly for the values of the function at the extrema (or zeros) of the…
The initial boundary-value problem (IBVP) and the Cauchy problem for the Kuramoto--Sivashinsky equation and other related $2m$th-order semilinear parabolic partial differential equations in one and N dimensions are considered. Global…
In this article, we introduce and study three numerical methods for the Dirichlet Monge Amp\`ere equation in two dimensions. The approaches consist in considering new equivalent problems. The latter are discretized by a wide stencil finite…
The generalized sine-Gordon (sG) equation $u_{tx}=(1+\nu\partial_x^2)\sin\,u$ was derived as an integrable generalization of the sG equation. In a previous paper (Matsuno Y 2010 J. Phys. A: Math. Theor. {\bf 43} 105204) which is referred to…
We prove an existence theorem for positive solutions to Lichnerowicz-type equations on complete manifolds with boundary and nonlinear Neumann conditions. This kind of nonlinear problems arise quite naturally in the study of solutions for…
We consider the asymptotic behavior of the soltion to the wave equation with time-dependent damping and analytic nonlinearity. Our main goal is to prove the convergence of a global solution to an equilibrium as time goes to infinity by…
In this paper we present the full classification of the symmetry-invariant solutions for the Gibbons--Tsarev equation. Then we use these solutions to construct explicit expressions for reductions of Benney's moments equations, to get…
In this article, we establish a Hitchin-Kobayashi type correspondence for generalised Seiberg-Witten monopole equations on Kahler surfaces. We show that the "stability" criterion we obtain, for the existence of solutions, coincides with…
Group theoretical methods are used to study the equations describing \chi^{(2)}:\chi^{(2)} cascading. The equations are shown not to be integrable by inverse scattering techniques. On the other hand, these equations do share some of the…
We will revise Garnier-Okamoto's coalescent diagram of isomonodromic deformations and give a complete coalescent diagram. In our viewpoint, we have ten types of isomonodromic deformations and two of them give the same type of the Painlev\'e…
In this paper, we establish a global $C^2$ estimates to the Neumann problem for a class of fullly nonlinear elliptic equations. By the method of continuity, we establish the existence theorem of $k$-admissible solutions of the Neumann…
We show that Pinney's equation [2] with a constant coefficient can be reduced to its linear part by a simple change of variables. Also, Pinney's original solution is simplified slightly.
A simple closed-form solution to the discrete Lyapunov equation (DLE) is established for certain families of matrices. This solution is expressed in terms of the eigen decomposition (ED) for which closed-form solutions are known for all…
In this work we consider the generalized Navier-Stoke equations with the presence of a damping term in the momentum equation. % The problem studied here derives from the set of equations which govern the isothermal flow of incompressible,…
We examine critically the Gambier equation and show that it is the generic linearisable equation containing, as reductions, all the second-order equations which are integrable through linearisation. We then introduce the general discrete…
The existence of dissipative solutions to the compressible isentropic Navier-Stokes equations was established in this paper. This notion was inspired by the concept of dissipative solutions to the incompressible Euler equations of Lions…
We address the problem of classification of integrable differential-difference equations in 2+1 dimensions with one/two discrete variables. Our approach is based on the method of hydrodynamic reductions and its generalisation to dispersive…