The biharmonic equation
can also be written in mixed formulation; see, e.g., [19, 34] and the references therein.
Wu, "Infinitely many sign-changing solutions for a class of biharmonic equation
with p-Laplacian and Neumann boundary condition," Applied Mathematics Letters, vol.
Tavkhelidze, "Asymptotic behavior of the solution of a biharmonic equation
in the neighborhood of non regular points of the boundary of the domain at infinity," Trudy Moskovskogo Matematiceskogo Obscestva, vol.
Similarly, comparing HMF and some general harmonic system (or even biharmonic equation
) [5-8], the mapping system is more complicated than the nonmapping system due to the curvature flow of the Riemannian manifolds.
Ye, "Weak Galerkin finite element methods for the biharmonic equation
on polytopal meshes," Numerical Methods for Partial Differential Equations, vol.
Using the fundamental solution of the biharmonic equation
[OMEGA](x,y) = [(8[pi]).sup.-1] ([x.sup.2] + [y.sup.2])ln[square root of ([x.sup.2] + [y.sup.2])]), we obtain the solution of equation (1), satisfying (2).
On the first step of the iterative procedure the following biharmonic equation
for a given load q(x, y) is solved:
Higher-order accuracy can be achieved by modifying standard difference formulas; examples are Shortley-Weller discretization  and the fast solution proposed by Mayo in  for solving Poisson or biharmonic equation
on irregular regions with smooth boundary.
We should point out that, for biharmonic equation
, the method is very efficient.
Particularly, for the inhomogeneous biharmonic equation
, analogous results are presented in [16,18].
Thus, we have to solve the biharmonic equation
[[DELTA].sup.2][phi] = 0.
This technic can be extended to apply the method to Poisson equation, biharmonic equation
, and some other types of problems with different types of boundary conditions (Neumann and mixed).