Everyday experience shows that twisted elastic filaments spontaneously form loops. We model the dynamics of this looping process as a sequence of bifurcations of the solutions to the Kirchhoff equation describing the evolution of thin elastic filament. The control parameter is taken to be the initial twist density in a straight rod. The first bifurcation occurs when the twisted straight rod def...

A meshfree computational method is proposed in this paper to solve Kirchhoff plate problems of various geometries. The deflection of the thin plate is approximated by using a Hermite-type radial basis function approximation technique. The standard Galerkin method is adopted to discretize the governing partial differential equations which were derived from using the Kirchhoff’s plate theory. The...

Abstract: We compare two mathematical theories that address duality between cycles and vertex-cuts of graphs in geometric settings. First, we propose a rigorous definition of a new type of graph, vector graphs. The special case of R2-vector graphs matches the intuitive notion of drawing graphs with edges taken as vectors. This leads to a discussion of Kirchhoff graphs, as originally presented b...

We consider a system of stochastic Allen-Cahn equations on finite network represented by graph. On each edge in the graph multiplicative Gaussian noise driven equation is given with possibly different potential barrier heights supplemented continuity condition and Kirchhoff-type law vertices. Using semigroup approach for evolution Banach spaces we obtain existence uniqueness solutions sample pa...

In this article, we first obtain an embedding result for the Sobolev spaces with variable-order, and then consider following Schrödinger–Kirchhoff type equations a+b∫Ω×Ω|ξ(x)−ξ(y)|p|x−y|N+ps(x,y)dxdyp−1(−Δ)ps(·)ξ+λV(x)|ξ|p−2ξ=f(x,ξ),x∈Ω,ξ=0,x∈∂Ω, where Ω is a bounded Lipschitz domain in RN, 1<p<+∞, a,b>0 are constants, s(·):RN×RN→(0,1) continuous symmetric function N>s(x,y)p all (x,...