Septic B-Spline Solution of Fifth-Order Boundary Value Problems

Bin Lin

doi:10.4236/jamp.2016.48149

Abstract

A numerical method based on septic B-spline function is presented for the solution of linear and nonlinear fifth-order boundary value problems. The method is fourth order convergent. We use the quesilinearization technique to reduce the nonlinear problems to linear problems and use B-spline collocation method, which leads to a seven nonzero bands linear system. Illustrative example is included to demonstrate the validity and applicability of the proposed techniques.

Highlights

Consider the following fifth-order boundary value problem.Ly= ( x) y(5) ( x) + p ( x) y= ( x) f ( x), c < x < d (1)With boundary conditions= y (c) α= 0, y(1) (c) α= 1, y(2) (c) α2, (2)= y (d ) α= 3, y(1) (d ) α= 4, y(2) (d ) α5 where αi (i = 0,1, 2,3, 4,5) are known real constants, p ( x) and f ( x) are continuous on [c, d ]
Tijini studied the fifth-order boundary value problem based on splines quasi-interpolants and proved to be second order convergent
Instead of solving nonlinear problem (16) with boundary conditions (17), we solve a sequence of linear problems (19) with boundary conditions (20), we consider yk+1 ( x) as the numerical solution to nonlinear problem

Summary

Introduction

Consider the following fifth-order boundary value problem. = y (d ) α= 3 , y(1) (d ) α= 4 , y(2) (d ) α5 where αi (i = 0,1, 2,3, 4,5) are known real constants, p ( x) and f ( x) are continuous on [c, d ]. Consider the following fifth-order boundary value problem. = y (d ) α= 3 , y(1) (d ) α= 4 , y(2) (d ) α5 where αi (i = 0,1, 2,3, 4,5) are known real constants, p ( x) and f ( x) are continuous on [c, d ] This problem arising in the mathematical modeling of viscoelastic flows [1] [2] has been studied by several authors [3]-[5]. Tijini studied the fifth-order boundary value problem based on splines quasi-interpolants and proved to be second order convergent. (2016) Septic B-Spline Solution of Fifth-Order Boundary Value Problems. The septic B-spline function is used as a basis function and the B-spline collocation method is studied to solve the linear and nonlinear fifth-order boundary value problems. The present method is tested for its efficiency by considering two examples

Septic B-Spline Interpolation

Spline Collocation Method

Numerical Tests

Conclusion