In this paper, we construct a family of new isometric immersions with vanishing normal curvature by getting solutions of a system of second order partial differential equations of

hyperbolic type. The definition of the normal curvature [R.sub.n] is given in [3], p.

The approximation of corresponding initial boundary value problem of the system of PDEs is based on the conservative averaging method (CAM), where the new

hyperbolic type splines are used.

Inspired by recent study of the

hyperbolic type potential well [20-28], in which we have found that their solutions can be exactly expressed by the confluent Heun functions [23], in this work we attempt to study the solutions of the Razavy potential.

This amounts to the condition that the uniformizing surface group [GAMMA] of ([summation], G) is the trivial group or a cyclic subgroup of PSL(2, R) of parabolic or

hyperbolic type.

The evaluated schemes are the simple proportional-derivative (PD), denoted as [mathematical expression not reproducible], (arctangent) Atan and (square-root) Sqrt control schemes [38], represented as [mathematical expression not reproducible], respectively, and two

hyperbolic type members of the proposed control family, denoted by [mathematical expression not reproducible], for exponents k =1 and k = 2, respectively.

Because the Maxwell's equation is a

hyperbolic type and the heat transfer equation is parabolic type differential equations, their FDTD stability criteria are completely different from each other.

Sosulski, On neutral partial functional-differential inclusions of

hyperbolic type, Demonstratio Mathematica, XXIII, no.

Colombo and Rizzo, whose credentials are not given, bring together seven chapters that present research on numerical simulation, in relation to

hyperbolic type partial differential equations and kinetic equations of plasma physics; problems encountered when studying the effects of plasma instabilities by using a kinetic or fluid approach; and the symplectic and multi-symplectic methods in simulating the behaviors of nonlinear solitary waves.

Recently it is found that the solutions of the

hyperbolic type potentials [14-21] are given explicitly by the confluent Heun function [22].

For simple engineering calculations conservative averaging method (CAM) with special integral

hyperbolic type splines is chosed.