scholarly journals Spline Methods for a Class of Singularly Perturbed Boundary Value Problems

2013 ◽  
Vol 2 (1) ◽  
pp. 1-10
Author(s):  
O.M Ogunlaran ◽  
O.A Taiwo
Keyword(s):  
Numerical Methods ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Spline Function ◽  
Boundary Value ◽  
Polynomial Spline ◽  
Second Order ◽  
Singularly Perturbed ◽  
Uniform Grid ◽  
Numerical Examples

In this paper, we develop numerical methods based on a non-polynomial spline function with uniform grid for solving certain class of singularly perturbed boundary value problems. The proposed methods are second-order and fourth-order accurate. Numerical examples are provided to demonstrate the efficiency of the proposed methods.

scholarly journals Solution of Boundary Value Problems by Approaching Spline Techniques

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
P. Kalyani ◽  
P. S. Rama Chandra Rao
Keyword(s):  
Boundary Value Problems ◽  
Spline Function ◽  
Boundary Value ◽  
Second Order ◽  
Point Boundary ◽  
Spline Method ◽  
B Spline ◽  
Numerical Examples ◽  
Specific Choice ◽  
Nodal Points

In the present work a nonpolynomial spline function is used to approximate the solution of the second order two point boundary value problems. The classes of numerical methods of second order, for a specific choice of parameters involved in nonpolynomial spline, have been developed. Numerical examples are presented to illustrate the applications of this method. The solutions of these examples are found at the nodal points with various step sizes and with various parameters (α, β). The absolute errors in each example are estimated, and the comparison of approximate values, exact values, and absolute errors of at the nodal points are shown graphically. Further, shown that nonpolynomial spline produces accurate results in comparison with the results obtained by the B-spline method and finite difference method.

Numerical methods for the solution of special and general sixth-order boundary-value problems, with applications to Bénard layer eigenvalue problems

1990 ◽  
Vol 431 (1883) ◽  
pp. 433-450 ◽  
Keyword(s):  
Numerical Methods ◽  
Boundary Value Problems ◽  
Eigenvalue Problems ◽  
Boundary Value ◽  
Second Order ◽  
Sixth Order ◽  
Asymptotic Estimates ◽  
Eighth Order ◽  
The Family ◽  
Convergent Method

A family of numerical methods is developed for the solution of special nonlinear sixth-order boundary-value problems. Methods with second-, fourth-, sixth- and eighth-order convergence are contained in the family. The problem is also solved by writing the sixth-order differential equation as a system of three second-order differential equations. A family of second- and fourth-order convergent methods is then used to obtain the solution. A second-order convergent method is discussed for the numerical solution of general nonlinear sixth-order boundary-value problems. This method, with modifications where necessary, is applied to the sixth-order eigenvalue problems associated with the onset of instability in a Bénard layer. Numerical results are compared with asymptotic estimates appearing in the literature.

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