Non-polynomial splines approach to the solution of sixth-order boundary-value problems

In this chapter, non-polynomial spline functions are applied to develop numerical methods for obtaining smooth approximations for the following BVP:     6 , , , / , D y x f x y a x b D d dx     (3.1) subject to the boundary conditions: 2 4 0 2 4 2 4 0 2 4 , , , , ,. y a A D y a A D y a A y b B D y b B D y b B       (3.2) where   y x and (,) f x y are continuous functions defined in the interval   , x a b . It is assumed that   6 , [ , ] f x y C a b  is real and that i A , , 0, 2, 4, i B i  are finite real numbers. The literature on the numerical solution of sixth-order BVPs is sparse. Such problems are known to arise in astrophysics; the narrow convicting layers bounded by stable layers, which are believed to surround A-type stars, may be modeled by sixth-order BVPs (Toomre et al. [146]). Also in (Glatzmaier [44]) it is given that dynamo action in some stars may be calculated by such equations. (Chandrasekhar [27]) determined that when an infinite horizontal layer of fluid is heated from below and is under the action of rotation, instability sets in. When this instability is an ordinary convection, the ordinary differential equation is sixth order. Theorems, which list the conditions for the existence and uniqueness of solutions of sixth-order BVPs, are thoroughly discussed in the book by Agarwal (Agarwal [4]). Non-numerical techniques for solving such problems are contained in papers (Baldwin [18], Baldwin [19]). Numerical methods of solution are contained implicitly in