The evaluation of sums (matrix–vector products) of the solutions of the three-dimensional biharmonic equation can be accelerated using the fast multipole method, while memory requirements can also be significantly reduced. We develop a complete translation theory for these equations. It is shown that translations of elementary solutions of the biharmonic equation can be achieved by considering ...

The aim of this article is to establish the existence of at least three‎ ‎solutions for a perturbed $p$-biharmonic equation depending on two‎ ‎real parameters‎. ‎The approach is based on variational methods‎.

We consider in dimension four weakly convergent sequences of approximate biharmonic maps into sphere with bi-tension fields bounded in L for some p > 1. We prove an energy identity that accounts for the loss of Hessian energies by the sum of Hessian energies over finitely many nontrivial biharmonic maps on R.

We introduce and fully analyze a new commutation relation $\overline{K} L_1 = L_2 K$ between finite convolution integral operator $K$ differential operators $L_1$ $L_{2}$, that has implications for spectral properties of $K$. This work complements our explicit characterization commuting pairs $KL=LK$ provides an exhaustive list kernels admitting or sesquicommuting operators.

In this paper, we study B-focal curves of biharmonic B -general helices according to Bishop frame in the Heisenberg group Heis   Finally, we characterize the B-focal curves of biharmonic B- general helices in terms of Bishop frame in the Heisenberg group Heis        

We consider a finite element method based on biorthogonal or quasi-biorthogonal systems for the biharmonic problem. The method is based on the primal mixed finite element method due to Ciarlet and Raviart for the biharmonic equation. Using different finite element spaces for the stream function and vorticity, this approach leads to a formulation only based on the stream function. We prove optim...

In this article we prove the regularity of weakly biharmonic maps of domains in Euclidean four space into spheres, as well as the corresponding partial regularity result of stationary biharmonic maps of higher-dimensional domains into spheres. c © 1999 John Wiley & Sons, Inc. Introduction In this article we consider the notion of biharmonic maps and begin an analytic study of the regularity pro...

In order to explore a new approach to construct surfaces bounded by geodesics or lines of curvature, a method of surface modeling based on fourth-order partial differential equations (PDEs) is presented. Compared with the free-form surface modeling based on finding control points, PDE-based surface modeling has the following three advantages. First, the corresponding biharmonic surface can natu...