HYBRID CUBIC B-SPLINE METHOD FOR SOLVING NON-LINEAR TWO-POINT BOUNDARY VALUE PROBLEMS

Abstract: A method based on hybrid cubic B-spline method (HCBSM) is developed, analyzed and applied to solve second-order non-linear two-point boundary value problems. In this method, a free parameter, γ, plays an important role in producing accurate results. Tests on four examples and a comparison of the results with the results obtained using cubic Bspline, extended cubic B-spline and shooting methods indicated that HCBSM is a feasible and accurate method.


Introduction
Consider the non-linear second-order two-point boundary value problem: where f is continuous on the set D = {(x, y, y ′ )|a ≤ x ≤ b, y, y ′ ∈ R} and a, b, α, and β are real numbers.
Boundary value problems arise in many real life applications in science and engineering such as theoretical physics, chemistry and control theory.In general, analytical methods which yield exact analytical solutions are not available for problem (1) and (2) and the only recourse is to use numerical or approximate analytical methods.Nonlinear two-point boundary value problems have been solved approximately with shooting, finite difference, variational approach, Adomian decomposition, homotopy perturbation being amongst the more frequently used methods.Details on these methods as well as others can be found in [1,2,3,4,5,6,7,8,9,10,11,12,13].
Linear and nonlinear two-point boundary value problems had already been treated by using cubic spline polynomial, cubic B-spline, cubic trigonometric Bspline, extended cubic B-spline, quartic B-spline and B-spline collocation methods [14,15,16,17,18,19].From the literature extended cubic B-spline method seems to be more suitable than other spline methods.B-spline interpolation is the best to interpolate any smooth functions compared with finite difference, finite element, finite volume methods [15].Cubic trigonometric B-spline produces more accurate results compared to cubic B-spline if the problems have trigonometric functions [16].The advantage of using extended B-spline is that it possesses a free parameter, λ, that can be optimized to give more accurate results [17].The hybrid cubic B-spline also contains a free parameter, γ.The main purpose of this study is to apply a hybrid cubic B-spline function in solving equation (1).The hybrid cubic B-spline method used is basically that of [20] for solving a generalized nonlinear Klien-Gordon equation for which the results are more accurate compared with some other methods.Therefore we seek to apply HCBSM for solving equation (1).

Features Shooting method
Quite general, applicable to wide variety of differential equations but has limitations and is sensitive to initial conditions.Finite difference method Gives highly accurate results, but these results can be found at the chosen knots only.Conversely, for some other methods, the results can be obtained at any point in the range.

Variational approach
The results which are obtained by variational approach are remarkable simple with high accuracy.Moreover, this method can be easily applied to many nonlinear problems.

Adomian decomposition and reproducing kernel
Using this method can avoid additional calculation in finding unknown parameters; and it can solve singular BVPs easily.Homotopy perturbation method HPM is better than most of the other methods in the literature and gives more accurate results with rapid convergence than variety of the other methods.Shooting reproducing kernel Hilbert space Simplicity and applicability.

Reproducing kernel and least squares
Feasible, effective and gives more accuracy.

Reproducing kernel space method
This method satisfies all boundary conditions, it has important properties for computation and gives more accurate results.

Uniform haar wavelets method
The essential advantage of the Haar wavelet is its efficiency and applicability for a many boundary conditions.Newton type method Successful for resolving of the nonlinear two-point BVPs for ODEs with mixed linear boundary conditions.Variable mesh method Successful to solve linear and nonlinear BVPs and gives more accurate solutions when compared with a family of variable mesh method, a non-uniform mesh cubic spline, and TAGE iterative algorithms.Piece-wise continuous method Gives high accurate results and can apply to linear and nonlinear problems without any additional calculations.New shooting method By using a new type of shooting method, we can convert initial value problems to boundary value problems.Also this method gives a good results compared with other methods.
Generally for k = 0, 1, 2, ..., we can write Then, equation ( 1) can be written as: where with boundary conditions The iterations can be stopped by setting the prescribed absolute error criterion |y for all i that appears.

Hybrid Cubic B-Spline Method
Hybrid cubic B-spline is a combination of B-spline and trigonometric cubic Bspline with one free parameter, γ.The value of γ giving the least error for the numerical solution is specified.

Hybrid Cubic B-Spline
Consider the following linear two-point boundary value problem: with boundary conditions where p(x), q(x), and r(x) are smooth functions.For a finite interval [a,b], suppose that {x i } n i=0 is a uniform partition of a finite interval [a,b] with n ∈ Z + such that a = x 0 < x 1 < ... < x n = b.The partition can be extended by using Hybrid cubic B-spline basis function is established by using a linear combination of the cubic B-spline basis function and trigonometric cubic B-spline basis [20].
Here, blending function of degree four, H 4 i is used and the resulting function is displayed in (6).
) sin( ) sin( ) sin( ) sin( where θ = sin( h 2 ) sin(h) sin( 3h 2 ) and γ ∈ R. The values of γ have an essential role and is influential in the hybrid cubic basis function.It is known that if γ = 0, the basis function is the same as cubic trigonometric B-spline basis function and if γ = 1, the basis function is the same as B-spline basis function.Consequently, this work focuses on the value of γ > 1.A plot of a family of H 4 i (x) together with cubic B-spline and trigonometric cubic B-spline is shown in Figure 1.

Hybrid Cubic B-Spline for a Nonlinear Two-Point BVPs
Suppose the estimate solution S(x) to the exact solution y(x) at point x i respectively, be defined as: where C i are unknown real coefficients and H 4 i (x) are hybrid cubic B-spline basis functions.The values of H i and its derivatives H ′ i , H ′′ i at the nodal points are tabulated in Table 1. where , , .
From equations ( 6) and ( 7), the values of S(x), S ′ (x), and S ′′ (x), at the knots x i are specified in the conditions of C i as

Solution of Nonlinear Two-Point BVPs
In this part, a collocation approach based on hybrid cubic B-spline basis functions is used to obtain a numerical solution of the nonlinear two-point BVPs (1)-( 2).It is required that the approximate solution (5) satisfy the differential equation at the points x = x i by putting ( 7) in ( 4), as a follows: with boundary conditions The values in Table 1 are substituted into equations ( 9)-( 10).Then a system of (n + 3) equations with (n + 3) unknowns C −3 , C −2 , . . ., C n−1 are obtained.This system can be written in the matrix-vector form where T , and M is a (n + 3) × (n + 3) matrix given by The elements of the matrix M are specified below for i = 0, 1, ..., n. ].
To find the values of C −3 , C −2 , , C n−1 in vector Y, we can solve the equation ( 11) as follows: Finally, substituting the values of C i , for i = −3, −2, ..., n − 1 in equation ( 7) we obtain the approximated analytical solution of equation (11).The numerical solution can be calculated after obtaining the values of γ by trial and error [20].

Stability
The system (8) obtained by using HCBSM cannot be solved exactly.However we can calculate the solutions that are close to the exact solution when small errors are introduced in the input functions.Specifically, we need to prove that the difference between these solutions always depends on the coefficients of the linear system; that means, the method is stable [21].Let ζ N and η N be small perturbations in M and Z.Let α be the solution to the system Suppose that M is nonsingular and Then, M + ζ N is nonsingular and From ( 11) and ( 13), we obtain Multiplying ( 14) by (M + ζ N ), we obtain Substituting Z = M Y in (15), we get Multiplying both side of ( 16) by Also, since M is strictly diagonally dominant [21], we know that from equation (17), we obtain This shows the stability of the system.

Results and Discussions
Four examples are presented to demonstrate the accuracy of the proposed method.The results are compared with that of shooting method [1], cubic B-spline, and extended cubic B-spline methods [18], and analytical approximation.The results are also presented with different values of n.Numerical errors are calculated using infinite and two norms, as follows: Example 1.Consider the following nonlinear boundary value problem [1]    The analytic solution is y(x) = x+ 1 x .The results for γ = 2.5 are shown in Table 2.It is clear that the HCBSM is acceptable and more accurate than shooting method [1].The numerical results obtained by HCBSM are shown in Figs.  3. It is clear that the HCBSM is acceptable and more accurate than cubic B-spline method (CBSM) [18] and extended cubic B-spline method (ECBSM) [18].The numerical results obtained by HCBSM are shown in Figs.
The analytic solution is y(x) = log( 1 x+1 ).The results for γ = 2.525 are shown in Table 5.Again, it is clear that the HCBSM is acceptable and more accurate than cubic B-spline method (CBSM) [18] and extended cubic B-spline method (ECBSM) [18].The numerical results obtained by hybrid cubic B-spline method are shown in Figs.5.

Conclusions
In this study, the HCBSM has been developed, analysed and applied to solve second-order nonlinear two-point boundary value problems.It was shown that the linear system (11) is stable.Numerical tests on four nonlinear two point boundary value problems indicate that the method is a feasible and accurate method.

Example 4 .
Consider the following nonlinear boundary value problem[22]

Table 1 :
Coefficient of H i , H ′i and H ′′