The Local Point Interpolation Method (LPIM) is a newly developed truly meshless method, based on the idea of Meshless Local Petrov-Galerkin (MLPG) approach. In this paper, a new LPIM formulation is proposed to deal with 4th order boundary-value and initial-value problems for static and dynamic analysis (stability, free vibration and forced vibration) of beams. Local weak forms are developed usi...

We consider the compressible (barotropic) Navier-Stokes system on time-dependent domains, supplemented with slip boundary conditions. Our approach is based on penalization of the boundary behaviour, viscosity, and the pressure in the weak formulation. Global-in-time weak solutions are obtained.

We study strictly parabolic stochastic partial differential equations on R d , d ≥ 1, driven by a Gaussian noise white in time and coloured in space. Assuming that the coefficients of the differential operator are random , we give sufficient conditions on the correlation of the noise ensuring Hölder continuity for the trajectories of the solution of the equation. For self-adjoint operators with...

The weak formulation of moving boundary problems with possibly vanishing specific heat, that is governed by parabolic and/or elliptic differential equations, is developed. The uniqueness of the resulting weak solution is then proved. This approach is used to obtain numerical solutions to some physical examples, which arise in electrochemical machining processes, and in saturated / unsaturated f...

The limit cycle bifurcations of a Z2 equivariant planar Hamiltonian vector field of degree 7 under Z2 equivariant degree 7 perturbation is studied. We prove that the given system can have at least 53 limit cycles. This is an improved lower bound for the weak formulation of Hilbert’s 16th problem for degree 7, i.e., on the possible number of limit cycles that can bifurcate from a degree 7 planar...

encountered in the course of studying weak formulations of differential equations. Unlike the differential expressions, the theory behind the sesquilinear forms (1.2) is not yet fully developed. The most general treatment we have so far is for the case when such forms are semibounded or sectorial [10]. The classical Lax-Milgram theorem which is widely used in treatments involving the bilinear f...

An accurate and yet simple Meshless Local Petrov-Galerkin (MLPG) formulation for analyzing beam problems is presented. In the formulation, simple weight functions are chosen as test functions. The use of these functions shows that the weak form can be integrated with conventional Gaussian integration. The MLPG method was evaluated by applying the formulation to a variety of patch test and thin ...

In this paper we consider the steady water wave problem for waves that possess a merely Lr−integrable vorticity, with r ∈ (1,∞) being arbitrary. We rst establish the equivalence of the three formulations the velocity formulation, the stream function formulation, and the height function formulation in the setting of strong solutions, regardless of the value of r. Based upon this result and using...

The meshless local radial point interpolation (MLRPI) method is applied to examine the magnetohydrodynamic (MHD) ow of third grade uid in a porous medium. The uid saturates the porous space between the two boundaries. Several limiting cases of fundamental ows can be obtained as the special cases of present analysis. The variations of pertinent parameters are addressed.

The paper deals with a path-valued Markov process: the re ecting Brownian snake. It is a particular case of the path-valued process previously introduced by Le Gall. Here the spatial motion is a re ecting Brownian motion in a domain D of R. Using this probabilistic tool, we construct an explicit function v solution of an integral equation which is, under some hypotheses on the regularity of v, ...