A method for coupling and decoupling, utilizing finite difference, is developed to
solve the biharmonic problem on a unit square. This problem is reformulated as a coupled
system involving two second-order partial differential equations. This approach necessitates
solving the original problem through a sequence of boundary value problems for the Poisson
equation. It achieves this using a minimal number of mesh points, distinguishing itself from
the traditional methods employed in prior research to address this particular issue. A compact
finite difference scheme has been introduced for the solution of fourth and sixth-order Poisson
equations. This innovative approach effectively reduces the computational cost of the
proposed algorithm, especially when dealing with large grid numbers, compared to traditional
methods. Simultaneously solving these Poisson equations can be easily programmed. We plan
to apply this method to analyze the fourth-order differential problem of a square clamped
plate subjected to various loads. The biharmonic problem has been examined with a focus on
achieving higher-order accuracy. The outcomes of numerical experiments showcase the
method's optimal global accuracy and reveal super convergence results. Notably, a sixth-order
accuracy is observed at both the boundary nodes and interior points