Quartic Non-polynomial Spline Solution of a Third Order Singularly Perturbed Boundary Value Problem

In this study, the non-polynomial spline function is used to find the numerical solution of the third order singularly perturbed boundary value problems. The convergence analysis is discussed and the method is shown to have second order convergence. The order of convergence is improved up to fourth order using the improved end conditions. Numerical results are given to describe the efficiency of the method and compared with the method developed by Akram (2012), which shows that the present method is better.


INTRODUCTION
The purpose of the study is to develop a new spline method for the solution of third order singularly perturbed boundary value problem. The method depends on a non-polynomial spline function which has a trigonometric part and a polynomial part. The following third order self adjoint singularly perturbed boundary value problem is considered, as: ( where, α 0 , α 1 , α 2 and α 3 are constants and ε is a small positive parameter (0<ε<<1), also f (x) and p (x) are smooth functions and p (x) = p = constant. The spline function has the form T n = {1, x, x 2 , cos (kx), (kx), sin (kx)} It is to be noted that k can be real or imaginary. The theory of singularly perturbed problems frequently occur in many branches of engineering and applied sciences, for instance in geophysics, fluid dynamics, Newtonian fluid mechanics, quantum mechanics, gas dynamics, chemical reactions, optimal control theory etc. The numerical treatment of singularly perturbation problems yields major computational difficulties and the usual numerical methods fail to produce accurate results for all independent values of x when ε is very small, owing to the multi-scale character of the solution of singularly perturbation problems. That is there are thin transition layers, where the solution varies rapidly, while away from the layers the solution behaves frequently and varies slowly. Three principle approaches are frequently used to solve such kind of problems numerically, namely the finite difference methods, the finite element methods and spline approximation methods. In this study the third one, namely, the spline approximation method is used. Howers (1976), Kelevedjiev (2002) and Roos et al. (1996) discussed the existence and uniqueness of singularly perturbed Boundary Value Problems (BVPs). Lie (2008) constructed a computational method for singularly perturbed two point BVP in the form of series in reproducing Kernel space. Rashidinia and Mahmoodi (2007) developed the classes of methods for the numerical solution of singularly perturbed two point BVP using non polynomial cubic spline and the method is second order as well as fourth order accurate. Khan et al. (2006) used sextic spline to solve second order singularly perturbed BVP and the method is fifth order accurate. Yao and Cui (2007) developed a new algorithm for a class of singularly BVPs in the reproducing Kernel space. Akram (2012) presented a quartic spline solution of a third order singularly perturbed BVP and the method is second order accurate. Akram and Mehak (2012) proposed a quintic spline technique to solve fourth order singularly perturbed BVP.

Consistency relations:
To develop the consistency relations the following fourth degree non polynomial spline is considered: Defined on [a, b], where ϵ[ , +1 ] with equally spaced knots, x i = a + ih, i = 0, 1, …, n and ℎ = − .
Using the following notations: The coefficients in (3) are determined as: θ θ θ θ Applying the first and second derivative continuity at knots, that is, ( ) for μ = 1, 2, the following relations are derived: which leads the following consistency relation in terms of T i and y i : Equation (5) can also be written, as: where, and, 3 1 cos 1 2 cos . sin 2 sin It is to be noted that, if α = 0 and = 1 2 then the truncation error of the above equations is − 1 240 ℎ 7 7 . Using Eq. (1) and (6) can be rewritten as: End conditions: The system Eq. (7) consists of (n-2) equations with (n-1) unknowns, so one more equation is required. Following Akram and Siddiqi (2006), the required end condition can be written as: where, a 1 , a 2 , c 0 and b i , i = 0, 1, 2, 3 are arbitrary parameters which are to be calculated using method of undetermined coefficients. The end condition of O (h 5 ) can be calculated, as: Using Eq. (1), Eq. (9) can be rewritten as: Similarly the end condition of O (h 5 ) for the system (2) can be calculated, as: The order of truncation error of end conditions can be improved. Following Akram and Siddiqi (2006), the improved end conditions of O (h 8 ) for the system (1) can be determined, as: 0  1  2  3  0  1  2  3  4  0  3   97  69  1 99  176  309  96  7  15  ,  7  7  28  7  7  7  4  T  T  T T  y  y  y  y  y Similarly end condition of O (h 8 ) for the system (2)

CONVERGENCE OF THE METHOD
The system of Eq. (7) and (14) provides the required quartic non polynomial spline solution of the BVP (1), which can be written in matrix form, as: where, A = (a ij ) is a tetradiagonal matrix of order n-1 :   3  3  3   3  3  3   3  3  3  3   3  3  3  3   3  3  3   97  176 69  309  96  7  7  7  7  7  7  4  3  3   3 3 , From Eq. (17) and (18), it follows that: To determine the error bound the row sums S 1 , S 2 , S n-1 of matrix A are calculated, as: Since the matrix A is observed to be irreducible and monotone, A -1 exists and its elements are non negative, therefore from Eq. (20), it can be written as: Also, from the theory of matrices it can be written as: where, l is a constant independent of h. From Eq. (28), it follows that: Similarly, it can be proved that ‖ ‖ = (ℎ 4 ) for the solution of BVP (2) given by Eq. (7) and (16). These results are summarized in the following theorems.
Theorem 1: The method given by Eq. (7) and (14) for solving the boundary value problem (1) for sufficiently small h gives a fourth order convergent solution.
Theorem 2: The method given by Eq. (7) and (16) for solving the boundary value problem (2) for sufficiently small h gives a fourth order convergent solution.
On the same fashion, truncation errors of the Eq. (10) and (7) where, l is a constant independent of h From Eq. (30), it follows that: Similarly, it can be proved that ‖ ‖ = (ℎ 2 ) for the solution of BVP (2) given by Eq. (7) and (12). These results are summarized in the following theorems.
Theorem 3: The method given by Eq. (7) and (10) for solving the boundary value problem (1) for sufficiently small h gives a second order convergent solution.
Theorem 4: The method given by Eq. (7) and (12) for solving the boundary value problem (2) for sufficiently small h gives a second order convergent solution.
To illustrate the implementation of the method and error analysis of the BVP (1) and (2), two examples are discussed in the following section.

NUMERICAL EXAMPLES
Example 1: Consider the following boundary value problem: (1) ( ) ( ) 81 cos 3 3 sin 3 , The analytical solution of system (32) and (33) Table 1 and 2, respectively. Similarly, the observed maximum errors (in absolute values) associated with y i , for the problem (32) and (33) corresponding to improved end conditions are summarized in Table 3 and 4, respectively.

Remark:
The comparison of Table 1 and 3 with 5 shows that the errors in absolute are better than that developed by Akram (2012), corresponding to the problem (32). Similarly, the comparison of Table 2 and 4 with 6 shows that the errors in absolute are better than that developed by Akram (2012), corresponding to the problem (33).

CONCLUSION
Quartic non polynomial spline function is used to develop a numerical method for solving third order singularly perturbed boundary value problem. The method is second order convergent and fourth order convergent as well using the improved end conditions. The numerical illustration shows that the developed method maintains a very remarkable high accuracy that makes it very encouraging for dealing with the solution of singularly perturbed boundary value problems.