Cubic B-Spline Method And its Application

Abstract

Chapter 1 is introductory in nature of the method of collocation which is utilize to find the solution of a differential equation with the initial or boundary constraints in the form of linear combination. of coordinate function with linear coefficients which leads to B-Spline function. In this we discussed how B-Spline functions of different degree obtained by using Cox de- Boor recursion formula. In this we discussed about Cubic B-Spline function in detail and properties of B-Spline basis function.

In chapter 2, A second order singularly perturbed linear differential equation with boundary constraint is considered as- � �𝜑’’(𝑥)+𝑟(𝑥)𝜑’(𝑥)+𝑠(𝑥)𝜑(𝑥)=𝑡(𝑥 ), 𝑥∈[𝛼,𝛽] . Here, ε is consider as a small positive parameter having property 0 < ε < 1, 𝑟(𝑥) , 𝑠(𝑥) and 𝑡(𝑥 ) are sufficiently continuously differentiable function. The given differential equation with boundary value constraint under this consideration will have a unique solution. Generally, when 𝜀 approaches to zero then the solution 𝜑(𝑥) of given differential problem creates a boundary layer having exponential property, at the left side of the given interval. Means there is a narrow region which contains in the domain of the differential equation where the derivatives of solution are very high. In the given chapter, a numerical. approach for solving a system of singularly perturbed differential equation having boundary value constraint using cubic-B-spline functions is developed.

In chapter 3, we used the Cubic B-Spline method to obtain an acceptable solution to second-order singularly perturbed linear differential equations with the boundary conditions and observed that the solution obtained by using cubic B spline method is good and also have negligible maximum absolute error.