Abstract: The problem discussed in this paper is about the second-order and fourth-order of
Helmholtz equation of a class of periodic boundary problem and its difference solution in the
transition surface design, which is different from traditional method of PDE in the second-order
and fourth-order partial differential equations. It has more free items than the traditional method,
so more shape control parameters can be adjusted in the surface design. The focused attention is to
discuss the changing effects of equation coefficients and boundary conditions to the surface shape.
If boundary curves and related tangent vectors are given by the designer, then the surface can be
constructed.

Key words: computer application, transition surface design, Helmholtz equation, surface
control