A highly accurate numerical solution of a biharmonic equation

A computational solution of a partial differential equation (PDE) involves a discretization procedure by which the continuous equation is replaced by a discrete algebraic equation. The discretization procedure consists of an approximation of the derivatives in the governing PDE by differences of the dependent variables, which are computed only at discrete points (grid or mesh points). The discretization of the continuous problem inevitably introduces an error in computing the derivatives and, as a result, an error in the computational solution. In general, one starts with a given PDE and uses a discretization procedure for developing a finite-difference equation (FDE) that is a linear relation between discrete values of the unknown function computed on grid point. Then, with the aid of a Taylor series expansion about the node at which the derivative is evaluated, the PDE can be rewritten in the following form: PDE = FDE + TE, where the remainder, TE, is

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