Spline methods for the solution of fourth-order parabolic partial differential equations

In this paper a fourth-order non-homogeneous parabolic partial differential equation, that governs the behaviour of a vibrating beam, is solved by using a new three level method based on parametric quintic spline in space and finite difference discretization in time. Stability analysis of the method has been carried out. It has been shown that by suitably choosing the parameters most of the previous known methods for homogeneous and non-homogeneous cases can be derived from our method. We also obtain two new high accuracy schemes of O(k4, h6) and O(k4, h8) and two new schemes which are analogues of Jain's formula for the non-homogeneous case. Comparison of our results with those of some known methods show the superiority of the present approach.