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Spoon is a form of free-form surfaces. In this paper, a method for designing a spoon using partial differential equations (PDE) is discussed. Boundary curves are defined in term of Fourier series. By selecting the appropriate boundary curves, a design of spoon is generated using fourth order elliptic PDE. Then by solving the elliptic PDE, a smooth design of spoon can be generated. In addition, this paper also examined the impact of the choice of number of boundary curves in generating different spoon shapes. [ABSTRACT FROM AUTHOR]

In the paper we study questions about solvability of some boundary value problems with periodic conditions for an inhomogeneous biharmonic equation. The exact conditions for solvability of the problems are found. [ABSTRACT FROM AUTHOR]

In this paper, we consider a model case of stationary vibrations of a thin flat plate, one side of which is embedded, the opposite side is free, and the sides are freely leaned. In mathematical modeling, there is a local boundary value problem for the biharmonic equation in a rectangular domain. Boundary conditions are given on all boundary of the domain. We show that the considered problem is self-adjoint. Herewith, the problem is ill-posed. We show that the stability of solution to the problem is disturbed. Necessary and sufficient conditions of existence of the problem solution are found. Spaces of the ill-posedness of the considered problem are constructed. [ABSTRACT FROM AUTHOR]

In this paper, a generalized third boundary value problem for the biharmonic equation in the unit ball with boundary operators up to third order containing normal derivatives and Laplacian is investigated. The existence and uniqueness theorems are proved. Some particular cases of general problem are considered. [ABSTRACT FROM AUTHOR]

In this paper, we consider the boundary value problem in two-dimensional rectangular domain for the biharmonic equation. The studied ill-posed boundary value problem is reduced to the optimal control problem. In terms of solutions of the adjoint boundary value problem, we set optimality conditions. It is found strong criterion for solvability of ill-posed boundary value problem. [ABSTRACT FROM AUTHOR]

The paper is concerned with a construction of cubic spline wavelet bases on the interval which are adapted to homogeneous Dirichlet boundary conditions for fourth-order problems. The resulting bases generate multiresolution analyses on the unit interval with the desired number of vanishing wavelet moments. Inner wavelets are translated and dilated versions of well-known wavelets designed by Cohen, Daubechies, and Feauveau. The construction of boundary scaling functions and wavelets is a delicate task, because they may significantly worsen conditions of resulting bases as well as condition numbers of corresponding stiffness matrices. We present quantitative properties of the constructed bases and we show superiority of our construction in comparison to some other known spline wavelet bases in an adaptive wavelet method for the partial differential equation with the biharmonic operator. [ABSTRACT FROM AUTHOR]

In this note, in the 2-disk a classical solution of a boundary value problem for the inhomogeneous biharmonic equation is obtained as well as a method of its construction is discussed. [ABSTRACT FROM AUTHOR]