Abstract We show that the Benjamin–Bona–Mahony (BBM) equation admits stable travelling wave solutions representing a sharp transition from constant state to periodic train. The is determined by parameters of train: length, amplitude and phase velocity, satisfies both generalized Rankine–Hugoniot conditions for exact BBM its averaged counterpart. Such shock-like structures exist if velocity trai...

We consider the generalized evolution of compact level sets by functions of their normal vectors and second fundamental forms on a Riemannian manifold M . The level sets of a function u : M → R evolve in such a way whenever u solves an equation ut + F (Du,Du) = 0, for some real function F satisfying a geometric condition. We show existence and uniqueness of viscosity solutions to this equation ...

A simple method for solving Prandtl's integro-differential equation is proposed based on a new reproducing kernel space. Using a transformation and modifying the traditional reproducing kernel method, the singular term is removed and the analytical representation of the exact solution is obtained in the form of series in the new reproducing kernel space. Compared with known investigations, its ...

‎This paper deals with the singular Sturm-Liouville expressions‎ ‎$ ‎ell y =‎ -‎y''+q(x)y=lambda y‎ ‎$‎ ‎on a finite interval‎, ‎where the potential function $q$ is real and‎ ‎has a singularity inside the interval‎. ‎Using the asymptotic estimates of a‎ ‎spectral fundamental system of solutions of Sturm-Liouville‎ ‎equation‎, ‎the asymptotic form of the solution of the‎ ‎equation (0.1) and the ...

This paper generalizes results of Lempert and Szöke on the structure of the singular set of a solution of the homogeneous Monge-Ampère equation on a Stein manifold. Their a priori assumption that the singular set has maximum dimension is shown to be a consequence of regularity of the solution. In addition, their requirement that the square of the solution be smooth everywhere is replaced by a s...

‎In this paper‎, ‎we study a class of boundary value problem‎ ‎involving the p-Laplacian oprator and singular nonlinearities‎. ‎We‎ ‎analyze the existence a critical parameter $lambda^{ast}$ such‎ ‎that the problem has least one solution for‎ ‎$lambdain(0,lambda^{ast})$ and no solution for‎ ‎$lambda>lambda^{ast}.$ We find lower bounds of critical‎ ‎parameter $lambda^{ast}$‎. ‎We use the method ...

in this paper, a numerical scheme for solving singular initial/boundary value problems presented.by applying the reproducing kernel hilbert space method (rkhsm) for solving these problems,this method obtained to approximated solution. numerical examples are given to demonstrate theaccuracy of the present method. the result obtained by the method and the exact solution are foundto be in good agr...