THEORETICAL AND NUMERICAL STUDIES OF TWO POINT BOUNDARY VALUE PROBLEMS USING TRIGONOMETRIC AND EXPONENTIAL CUBIC B-SPLINES

A collocation method with different types of B-spline functions such as trigonometric and exponential cubic B-splines functions are proposed from many researchers for studying the numerical solutions of many physical nonlinear partial differential equations and ordinary differential equations. In this paper, a collocation finite difference scheme based on trigonometric and exponential cubic B-splines functions are investigated and applied for the numerical solution of two-point boundary value problems. The convergence of trigonometric and exponential cubic B-splines is proved using diagonally dominant matrices. The numerical examples show that our method is very effective and the maximum absolute error is acceptable.


Description of cubic B-spline methods
Firstly, we assume that the problem domain [a, b] is equally divided into n subintervals [x j , x j+1 ] , j = 0, 1, 2, . . ., n − 1 by the knots x j = a + jh where a = x 0 < x 1 < • • • < x n = b and the step size h = b−a n .Then cubic B-spline collocation methods for solving a two point value problems (1) numerically are presented.

Trigonometric cubic B-spline method
On the above partition together with additional knots x −3 , x −2 , x −1 , x n+1 , x n+2 , x n+3 outside the problem domain, the trigonometric cubic B-spline T CB j (x) is defined as follows where An approximate solution v(x) to the unknown V (x), of the problem ( 1) and ( 2), is written in terms of the expansion of the T CB j (x) as where constants τ j 's are be determined from the collocation points x j , j = 0 , 1 , 2 ... , n and the boundary and initial conditions.The values of the trigonometric cubic B-spline functions T CB j (x) and its principle two derivatives T CB j (x) and T CB j (x) at the knots and their values are summarized in Table 1.
Table 1.Values of T CB j (x) and its principle two derivatives at the knot points where .
Using equations ( 3) and ( 4), the values of v j and their derivatives up to second order at the konts are Substituting from (5) in Eq. ( 1) and Eq. ( 2) we find, and the boundary conditions (2) are given as Solving the system of equations (7) in τ −1 and τ n+1 , the linear algebraic system of equations ( 6) can be converted to the following matrix form ; where A is an (n + 1) × (n + 1) matrix, τ is an (n + 1) dimensional vector with components τ j and the right hand side g(τ ) is an (n + 1) dimensional vector.
For appropriately enough small h, the matrix A is diagonally dominant if and α (x j ) γ (x j ) < 0, j = 0, 1, . . ., n. Furthermore, where .∞ indicates the maximum norm in [a, b].Since D is a linear second order differential operator at the knots, we write, where v(x) be the trigonometric cubic B-spline of interpolation to the unique solution of the problems ( 1) and ( 2).(x j ) mean that error function.Thus, the equation ( 8) becomes where A is an (n + 1) × (n + 1) matrix, τ is an (n + 1) dimensional vector with components τ j and the right hand side g(τ ) and are an (n + 1) dimensional vector.
Let the Lipschitz condition on the forcing function is of the form: where the constant L is independent of x.
So that e = A −1 + A −1 LN e, when A −1 exists and bounded as in (11), then where M is abounded on and we have Then the following theorem shows the convergence of the trigonometric cubic B-spline.Theorem 2.1 Let a two point boundary value problem have the form (1) and ( 2) where the coefficients α (x) , β (x) , γ(x) and the forcing function g(x, V ) are satisfied the conditions (10), ( 14) and (19), then the trigonometric cubic B-spline v(x) interpolating to v(x) converges to V (x) in the maximum error over [a, b] with error norm as (20).

Exponential cubic B-spline method
The exponential cubic B-spline is where We let be the solution of the problem (1) and ( 2) as a cubic B-spline of the interpolation to the true solution V (x), where constants ε j 's are be determined.To solve a two point value boundary value problems, we find ECB j (x), ECB j (x) and ECB j (x) at the knots and their values are summarized in Table 2. Table 2. Values of ECB j (x), ECB j (x) and ECB j (x)at the knot points . where Using equations ( 21) and ( 22), the values of v j and their derivatives up to second order at the konts are Substituting from (23) in Eq. ( 1) and Eq. ( 2) we find, and the boundary conditions (2) are given as From the equations ( 24) and ( 25), we obtain where A is an (n + 1) × (n + 1) matrix, ε is an (n + 1) dimensional vector with components ε j and the right hand side g(ε) is an (n + 1) dimensional vector.
For appropriately enough small h the matrix A is diagonally dominant if and α (x j ) γ (x j ) < 0, 0 ≤ j ≤ n. Furthermore, where .∞ indicates the maximum norm in [a, b], since D is a linear second order differential operator at the knots, we write, where v(x) be the exponential cubic B-spline of interpolation to the unique solution of the problems (1) and ( 2).(x j ) mean that error function.Thus, the equation ( 26) becomes where A is an (n + 1) × (n + 1) matrix, ε is an (n + 1) dimensional vector with components, ε j and the right hand side g(ε) and are an (n + 1) dimensional vector.
Let the Lipschitz condition on the forcing function: where the constant L is independent of x.
So that e = A −1 + A −1 LN e, when A −1 exists and bounded as in (29), then where M is abounded on and we have Then the following theorem shows the convergence of the trigonometric cubic B-spline.Theorem 2.2 Let a two point boundary value problems have the form (1) and ( 2) where the coefficients α (x) , β (x) , γ(x) and the forcing function g(x, V ) are satisfied the conditions (28), (32) and (37), then the exponential cubic B-spline v(x) interpolating to v(x) converges to V (x) in the maximum error over [a, b] with error norm as (38).

Numerical examples
In this section, we present some examples of two point boundary value problems.We take different values of h on the interval [0, 1] and the results are generated with Mathematica using FindRoot function to solve the emerging algebraic equations.At each h, we evaluated the difference between approximate solution and exact solution, and then take the absolute errors of this difference.
Example 1.We consider linear boundary value problem [9] V and the exact solution is given by V (x) = Cosh(10x − 5)/Cosh (5).Thus, we can compare our numerical estimates with this solution to obtain the exact maximum errors of the approximation which summarized in Table 3.  Example 2. We present another linear boundary value problem [9] V (x) − 4V (x) − 4Cosh(1) = 0, and the exact solution is given by V (x) = Cosh (2x − 1) − Cosh(1).Thus, we can compare our numerical estimates with this solution to obtain the exact maximum errors of the approximation which summarized in Table 4.
and the exact solution is given by V (x) = ln (0.5) + 2ln c Sec c(x−0.5)

2
, c = 1.3360556949.Thus, we can compare our numerical estimates with this solution to obtain the exact maximum errors of the approximation which summarized in Table 5.  V (x) − 0.5(V (x) + x + 1) and the exact solution is given by V (x) = 2 2−x − x − 1.Thus, we can compare our numerical estimates with this solution to obtain the exact maximum errors of the approximation which summarized in Table 6.

Conclusion
Numerical treatment of two point boundary value problems is carried out in the paper using trigonometric and exponential cubic B-splines.From the computational results, the proposed methods give reliable solutions of the considered problem if compared with the existing results n literature.The numerical experiments are compared with the analytic solutions by finding the maximum absolute errors and are compared with numerical results in [9] as shown in tables 3, 4, 5 and 6 where we can see that the numerical accuracy is similar to results obtained in [9].

Table 6 .
Comparison of maximum absolute errors for Example 4. h Trigonometric Cubic Exponential Cubic Cubic B-spline B

Table 3 .
Comparison of maximum absolute errors for Example 1.

Table 4 .
Comparison of maximum absolute errors for Example 2.
h Trigonometric Cubic Exponential Cubic Cubic B-spline B-spline errors B-spline errors errors

Table 5 .
Comparison of maximum absolute errors for Example 3.