Solution of Boundary Value Problems by Approaching Spline Techniques

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 (u(x)) 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 u(x) 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.


Introduction
There are many linear and nonlinear problems in science and engineering, namely, second order differential equations with various types of boundary conditions, which are solved either analytically or numerically.Numerical simulation in engineering science and in applied mathematics has become a powerful tool to model the physical phenomena, particularly when analytical solutions are not available, then very difficult to obtain.The numerical solution of two-point boundary value problems (BVPs) is of great importance due to its wide application in scientific research.Several authors like Bickley [1] and Khan [2] have considered the applications of cubic spline functions for the solution of two point boundary value problems.Detailed explanation of theory of splines is given in [3,4].Some of already established methods to solve the boundary value problems are shooting method, finite difference method, finite volume method, variational iteration method, and Adomian decomposition method.Chawla and Katti [5] employed finite difference method for a class of singular two-point BVPs; a class of BVPs was solved by using numerical integration [6]; Ravi kanth and Reddy dealt with cubic spline [7]; the variational iteration method was proposed originally by He [8] in 1999; Adomian et al. solved a generalization of Airy's equation by decomposition method [9].In the present communication we apply nonpolynomial spline functions to develop numerical method for obtaining the approximations to the solution of second order two point boundary value problem of the form This type of problem (by missing the term containing ()) is proposed by the authors in [10,11].Numerical solution of (1) based on finite difference, finite element, and finite volume methods has been proposed by Fang et al. [10]; Hikmet Caglar et al. [11] applied B-spline interpolation in twopoint BVPs and compared results with finite difference, finite element, and finite volume methods.
Briefly, outline is as follows.In Section 2, we develop a numerical technique based on nonpolynomial spline function for solving second order linear and nonlinear two point boundary value problems (1).To demonstrate the efficiency of the method some numerical examples have been solved and compared with exact solution and also with other known International Journal of Engineering Mathematics methods [12] in Section 3 and conclusions have been presented in Section 4.

Description of the Method
We consider a uniform mesh Δ with nodal points   on [, ] such that =  + ℎ,  = 0, 1, 2, . . .,  where ℎ = ( − )  . ( A nonpolynomial function  Δ () of class  2 [, ] which interpolates () at the mesh points   , for  = 0, 1, . . ., , depends on a parameter  and reduces to ordinary cubic spline in [, ] as  → 0. The spline function we propose has the following form:  3 = Span{1, , cos , sin }, where  is the frequency of the trigonometric part of the spline function which can be real or pure imaginary and which will be used to raise the accuracy of the method.When correlation between polynomial and nonpolynomial spline basis functions are investigated in the following manner: It follows that lim  → 0  3 = {1, , For each segment [  ,  +1 ],  = 0, 1, 2, . . .,  − 1 the nonpolynomial  Δ () has the following form: where   ,   ,   , and   are constants and  is a free parameter.Let   be an approximation to (  ), obtained by the segment  Δ () of mixed spline function passing through the points (  ,   ) and ( +1 ,  +1 ).To obtain the necessary conditions for the coefficients introduced in (5), we not only require  Δ () to satisfie interpolate conditions at   and  +1 , but also the continuity of first derivative at the common nodes ( , ,   ) to be fulfilled.

Numerical Illustrations
In the present work four linear boundary value problems with ℎ = 0.1 and one nonlinear boundary value problem with ℎ = 0.2 for different values of  and  have been solved, whose exact solutions are known.The approximate solution, exact solutions, and absolute errors at the nodal points are tabulated in Tables 1-7, and comparisons are shown in Figures 1-5.The results of the present work are compared with the exact solution of all the problems and with finite difference method, B-spline method of Example 4.
The analytical solution of the given equation is The approximate values (), exact values (() from (30)), and the absolute errors at the nodal points are summarized in Table 2 and the comparison is given in Figure 2.
an applicable technique and approximates the solution very well, and the numerical solutions are in very good agreement with the exact solution.Moreover non-polynomial spline method has less computational cost over other polynomial spline methods.The implementation of the present method is very easy, acceptable, and valid scheme.

Figure 4 :Figure 5 :
Figure 4: (a) Comparison of approximate values and exact values for Example 4. (b) Comparison of results obtained by our method of Example 4 with the values obtained by other methods (Table6).

Table 3 :
(a) Approximate solution, exact solution, and absolute errors of Example 2 Case 2(a).(b) Approximate solution, exact solution, and absolute errors of Example 2 Case 2(b).

Table 6 :
Comparison of present method (series 3) with B-spline method (series 2), finite difference method (series 1) and with exact solution (series 4) for Example 4.