Five-diagonal finite difference methods based on mixed-type interpolation for a certain fourth-order two-point boundary-value problem (Q1203685)

This paper derives various mixed-type interpolational formulas based on a linear combination of a polynomial of degree \(q-2\) and a (hyperbolic) cosine and (hyperbolic) sine function of the same frequency \((k)\) for solving fourth order, two-point boundary value problems directly. Such methods are of order \(q+1\) and their implementation leads to the solution of 5-diagonal linear systems of equations. The frequency value is treated as a free parameter. In the case \(k=0\) the methods reduce to those of \textit{R. A. Usmani} [Indian J. Pure Appl. Math. 14, 398-411 (1983; Zbl 0522.65054)]. The authors prove that truncation errors are of the form \(k^ 2y^{(q+3)}(\xi)+y^{(q+5)}(\xi)\) and suggest choosing \(k^ 2\) so that this term is zero. This is done by approximating these derivatives by second or fourth order schemes. Numerical results indicate an \(O(h^ 2)\) improvement for these methods over Usmani's methods of the same order.