Solution of Second Order Linear and Nonlinear Two Point Boundary Value Problems Using Legendre Operational Matrix of Differentiation

In this paper, an approach using Tau method based on Legendre operational matrix of differentiation [2] & [5] has been addressed to find the solutions of second order linear and nonlinear two point boundary value problems of ordinary differential equations. In the implementation of this approach, the given second order two point boundary problems is converted into a system of algebraic equations, whose solutions are the Legendre coefficients. The validity and efficiency of the method has also been illustrated with numerical examples supported by graphs

Shape Preserving C2 Rational Cubic Spline Interpolation

In this study a piecewise rational function  with cubic numerator and linear denominator involving two shape parameters has been developed to address the problem of constructing positivity preserving curve through positive data, monotonicity preserving curve through monotone data and convexity preserving curve through convex data within one mathematical model. A simple data dependent condition for a single shape parameter has been derived to preserve the positivity, monotonicity and convexity of respectively positive, monotone and convex data. The remaining shape parameter is left free for the user to modify the shape of positive, monotone and convex curves when needs arise. We extended the result of [1] to a piecewise rational cubic function