The integration of a biharmonic equation by an implicit scheme

The paper presents a step-by-step construction of a finite-difference scheme for a heterogeneous biharmonic equation under zero boundary conditions superimposed on the desired function and its first-order partial derivatives. The finite-difference scheme is based on a square twenty-five-point pattern and has an implicit character. On analytical grid, the error of approximation of the biharmonic operator by the difference analog and the error of approximation of boundary conditions imposed on the first order partial derivatives are calculated by the expansion of the function in the Taylor series with the remainder term in the form of a Lagrange. The boundary conditions imposed on the sought function are satisfied precisely. A finite-difference scheme approximates a boundary value problem with a second order of accuracy over the mesh step. With the help of the Maple computer algebra system the solutions of the problem for different grid steps are obtained. The dependence of the minimum function and calculation time on the number of significant digits is revealed...