We present a fast direct solver methodology for the Dirichlet biharmonic problem in a rectangle. The solver is applicable in the case of the second order Stephenson scheme [34] as well as in the case of a new fourth order scheme, which is discussed in this paper. It is based on the capacitance matrix method ([10], [8]). The discrete biharmonic operator is decomposed into two components. The fir...

A VLSI algorithm for solving a special block–five– diagonal system of linear algebraic equations will be presented. The algorithm is considered for the VLSI parallel computational model where both the time of the algorithm and the area of its design are components of the complexity estimations. The linear system arises from the finite– difference approximation of the first biharmonic boundary v...

In general, higher order elliptic equations and boundary value problems like the biharmonic equation or the linear clamped plate boundary value problem do not enjoy neither a maximum principle nor a comparison principle or – equivalently – a positivity preserving property. The problem is rather involved since the clamped boundary conditions prevent the boundary value problem from being written ...

The generalized weak Galerkin (gWG) finite element method is proposed and analyzed for the biharmonic equation. A new discrete second order partial derivative introduced in gWG scheme to allow arbitrary combinations of piecewise polynomial functions defined interior on boundary general polygonal or polyhedral elements. error estimates are established numerical approximation a H2 norm L2 norm. r...

Considering difference equations in discrete space instead of differential equations in Euclidean space, we investigate a probabilistic formula for the solution of the Dirichlet problem for biharmonic functions. This formula involves the expectation of a weighted sum of the pay-offs at the successive times at which the Markov chain is in the complement of the domain. To make the infinite sum co...

This is a supplement to our previous research note [3]. In [3], we gave a characterization of biharmonic curves in Minkowski 3-space. More precisely, we pointed out that every biharmonic curves with nonnull principal normal in Minkowski 3-space is a helix, whose curvature κ and torsion τ satisfy κ2 = τ2. In the classification of biharmonic curves in Minkowski 3-space due to Chen and Ishikawa [1...

Motion of a biharmonic system under action of small periodic force and small damped force is studied. The biharmonic oscillator is a physical system acting under a biharmonic force like: θ θ 2 sin sin b a + . The article contains biharmonic oscillator analysis, phase space research, and analytic solutions for separatrixes. The biharmonic oscillator performs chaotic motion near separatrixes unde...