Non-polynomial septic spline method for singularly perturbed two point boundary value problems of order three

ABSTRACT This study introduces a non-polynomial septic spline method for solving singularly perturbed two point boundary value problems of order three. First, the given interval is discretized. Then, the spline coefficients are derived and the consistency relation is obtained by using continuity of second, fourth and fifth derivatives. Further, the obtained fifteen different systems of equations are reduced to a system of equations and boundary equations are developed in order to equate a system of linear equations. The convergence analysis of the obtained hepta-diagonal scheme is investigated. To validate the applicability of the method, two model examples are considered for different values of perturbation parameter and different mesh size h. The proposed method approximates the exact solution very well when . Moreover, the present method is convergent and gives more accurate results than some existing numerical methods reported in the literature.


Introduction
In the demanding development of science and technology, many practical problems such as the mathematical boundary layer theory or approximation of solution of various problems described by differential equations involving large or small parameters, become more complex [1]. Any differential equation in which its highest order derivative is multiplied by a small positive parameter is called perturbation problem and the parameter is known as the perturbation parameter. These problems occur in a number of areas of applied mathematics, science and engineering among them fluid mechanics, elasticity, quantum mechanics, chemical-reactor theory, aerodynamics, plasma dynamics, rarefied-gas dynamics, oceanography, meteorology, modelling of semiconductor devices, diffraction theory and reaction-diffusion processes are some to mention.
In recent years, a considerable amount of numerical methods such as quartic and quantic splines, combination of asymptotic expansion approximations, shooting method and finite difference methods, subdivision collocation methods, and B-splines collocation methods have been developed for solving singularly perturbed boundary value problems using various splines [2][3][4][5][6][7][8][9]. However, as the solution profiles of singular perturbation problems depends on perturbation parameter ε and mesh size h, the numerical treatment of singularly perturbed problems faces major computational difficulties and most of the classical numerical methods fail to provide accurate results for all independent values of x when ε is very small related to the mesh size h (i.e. ε h) [10]. As a result, it is necessary to develop a more accurate numerical method which works nicely for ε h where most of numerical method fails to give good result for singularly perturbed problems.
Hence, the purpose of this study is to develop a spline method for the solution of third order singularly perturbed boundary value problem which is convergent, more accurate than the existing methods and works for the cases where others fails to give good result. The method depends on a non-polynomial spline function which has a trigonometric part and a polynomial part.

Description of the method
Consider the third order singularly perturbed two point boundary value problem of the form: subject to the boundary conditions, where φ 1 , φ 2 and γ are constants, ε is a perturbation parameter 0 < ε 1, u(x) and f (x) are continuous functions.
In order to develop the septic spline approximation for the third-order boundary value problem in Equations (1) and (2), the interval [0, 1] is divided into N equal sub-intervals. For this, we introduce the set of grid points x i = x 0 + ih, i = 0, 1, 2, . . . , N, so that, Let y(x) be the exact solution of the Equations (1) and (2) and y i be an approximation to y(x i ), obtained by the segment S Δ (x) of the spline function passing through the points (x i , y i ) and (x i+1 , y i+1 ). For each ith segment, the non-polynomial septic spline function where a i , b i , c i , d i , e i , f i , g i and r i are constants and k = 0 is the frequency of the trigonometric part of the spline functions which can be real or pure imaginary, and which will be used to raise the accuracy of the method. The arbitrary constants are being chosen to satisfy certain smoothness conditions at the joints. This "non-polynomial spline" belongs to the class C 6 [a, b] and reduces into polynomial splines as parameter k → 0. To derive expression for the coefficients, we first denote: From algebraic manipulation and letting θ = kh, we get the following expression: Using the continuity condition of the fifth, fourth and second derivatives, and substituting the above equations after reducing their indices by one, respectively we have: where α 1 = 1 θ 6 (−θ 5 csc θ − 5θ 3 cot θ − 5θ 3 csc θ − 60θ cot θ − 60θ csc θ + 120), β 1 = 1 θ 6 (10θ 3 csc θ + 120 csc θ + (2θ 4 + 10θ 2 + 120)θ cot θ − 240) In order to eliminate F i s and M i s from Equations (7)-(9), we have replaced i by i + 2, i + 1, i − 1 and i − 2, in Equations (7)- (9), and obtaining the simultaneous solutions with the help of symbolic toolbox by Matlab 2013a. Eliminating F i s and M i s gives the following important relations in terms of y i and third order derivative T i , as where X i s and Z i s for i = 1(1)12 are described in Appendix A. Now, evaluating Equation (1) at the nodal points x i , and using the relation in Equation (5), we get: where Substituting the values of Equation (11) into Equation (10) and simplifying, we get: when k → 0, that is θ → 0, since θ = kh, then  (10) reduces into septic polynomial spline [11]. The relation in Equation (12) gives N − 5 equations in N − 1 unknowns y j , j = 1(1)N − 1. Now, we require four more equations, two at each end of the nodal points.

Convergence analysis
We investigate the convergence analysis for the developed method. The scheme in Equations (12) and (18)-(21) can be written in the matrix-vector form: with Now, considering the above system with the exact solutionȲ = [y(x 1 ), y(x 2 ), . . . , y(x N−1 )] T , we have: where From the above local truncation errors, t k (h) → 0 as h → 0 for k = 1, 2, . . . , N − 1 and this implies that the scheme is consistent.
Subtracting Equation (25) from Equation (28), we obtain the error equation, where Let s i be the i th row sum of the matrix A 0 , then we have: Since 0 < ε 1, we choose h sufficiently small so that the matrix A 0 is irreducible and monotone [12]. Then, it follows that A −1 0 exists and its elements are non-negative.
Hence, from Equation (30), we have: Letā i,j is the (i, j)th element of the matrix A −1 0 , we define: Also, from the theory of matrices, we have: Defining s k * = min It follows that: where And also Equation (32) can be written as which implies ||e i || ≤ | N−1 i=1ā i,j | || T(h)||. From Equations (33) and (35), we get: where N * = max x i−1 ≤ξ i ≤x i+1 ||y 7 (ξ i )|| and ψ = 2365N * 994M k * which is independent of h. It follows that ||E|| = O(h 4 ) and hence the present method is of fourth order convergence.

Numerical examples and results
To The analytical solution of this problem is y(x) = 6εx 3 (1 − x) 5 . Numerical Results are presented in Tables  1 and 2

Conclusion
The non-polynomial septic spline method is developed for the approximate solution of a third order singularly perturbed two-point boundary value problems. The convergence analysis is investigated and shows that the present method is of fourth order convergent. Two examples are considered for numerical illustration of the method. As a result, from Tables 1-4 and Figure 2, one can see that the maximum absolute error decreases as a mesh size h and also perturbation parameter ε decreases, which in turn shows the convergence of the computed solution. Furthermore, the result of the present method is compared with current findings and shows that it is more accurate than some existing numerical methods reported in the literature. The present method approximates the exact solution very well, Figure 1.
Moreover, the study has been analyzed by taking different mesh size h and sufficiently small perturbation parameterε. So, this study developed a better method for solving singularly perturbed boundary value problems for most numerical schemes fail to give good result at small mesh size h and for sufficiently small perturbation parameter ε << h.
Generally, the present method is convergent and more accurate for solving third order singularly perturbed two point boundary value problems.