Finite pointset method for fourth order differential equations

Abstract

The aim of this dissertation is to present Finite Pointset Methods for fourth order differential equations. In this thesis, fourth order two point boundary value problem, biharmonic equation, one space dimensional extended Fisher-Kolmogorov equation and two space dimensional extended Fisher-Kolmogorov equation are considered. Finite Pointset Method is applied on these problems and numerical computations are carried over. Further, order of convergence is computed numerically. Finite Pointset Methods is one of the meshfree methods which is also a local iterative method based on the weighted least square approximation technique. The advantage of meshfree methods is that, it will avoid the construction of meshes in the domain, but it will use a set of nodes in the whole process of numerical simulations. Therefore, even for complex geometry, it does not depend on the formation and management of meshes. Further, it is not necessary that these nodes must be assigned uniformly in the domain. In this thesis, Finite Pointset Methods with and without splitting technique are applied to fourth order time-independent/ time-dependent and linear/ non-linear differential equations. In the case of using the splitting technique, the fourth order differential equation is split into two coupled system of second order differential equations by introducing an intermediate function, so that the complexity of handling fourth order derivative can thus be avoided. The advantage of splitting technique is that it will reduce the resultant matrices at each particle to a size of 6_6 newline