Finite difference methods for two-point boundary value problems involving high order differential equations

We discuss the construction of finite difference schemes for the two-point nonlinear boundary value problem:y ⁽²ⁿ⁾+f(x,y)=0,y ^(2j)(a)=A _2j ,y ^(2j)(b)=B _2j ,j=0(1)n–1,n2. In the case of linear differential equations, these finite difference schemes lead to (2n+1)-diagonal linear systems. We consider in detail methods of orders two, four and six for two-point boundary value problems involving a fourth order differential equation; convergence of these methods is established and illustrated by numerical examples.