We introduce a new mixed method for the biharmonic problem. The method is based on a formulation where the biharmonic problem is re-written as a system of four first-order equations. A hybrid form of the method is introduced which allows to reduce the globally coupled degrees of freedom to only those associated with Lagrange multipliers which approximate the solution and its derivative at the f...

In this work, we use a symbolic algebra package to derive a family of nite diierence approximations for the biharmonic equation on a 9 point compact stencil. The solution and its rst derivatives are carried as unknowns at the grid points. Dirichlet boundary conditions are thus incorporatednaturally. Since the approximations use the 9 point compact stencil, no special formulas are needed near th...

In this article, we construct aC0 linear finite element method for two fourth-order eigenvalue problems: the biharmonic and the transmission eigenvalue problems. The basic idea of our construction is to use gradient recovery operator to compute the higher-order derivatives of a C0 piecewise linear function, which do not exist in the classical sense. For the biharmonic eigenvalue problem, the op...

Recommended by Narendra Kumar Govil We first obtain the relations of local univalency, convexity, and linear connectedness between analytic functions and their corresponding affine harmonic mappings. In addition, the paper deals with the regions of variability of values of affine harmonic and biharmonic mappings. The regions their boundaries are determined explicitly and the proofs rely on Schw...

In this study, stress and displacement functions of the three-dimensional theory of elasticity for homogeneous isotropic bodies are derived from first principles from the differential equations of equilibrium, the generalized stress – strain laws and the geometric relations of strain and displacement. It is found that the stress and displacement functions must be biharmonic functions. The deriv...

Let G be a simple graph with order n and size m. The quantity $$M_1(G)=\sum _{i=1}^{n}d^2_{v_i}$$ is called the first Zagreb index of G, where $$d_{v_i}$$ degree vertex $$v_i$$ , for all $$i=1,2,\dots ,n$$ . signless Laplacian matrix $$Q(G)=D(G)+A(G)$$ A(G) D(G) denote, respectively, adjacency diagonal degrees G. $$q_1\ge q_2\ge \dots \ge q_n\ge 0$$ eigenvalues largest eigenvalue $$q_1$$ spectr...