Singularly perturbed convection-diffusion boundary value problems with two small parameters using nonpolynomial spline technique

In this paper, a new nonpolynomial cubic spline method is developed for solving two-parameter singularly perturbed boundary value problems. Convergence analysis is briefly discussed. Numerical examples and computational results illustrate and guarantee a higher accuracy by this technique. Comparisons are made to confirm the reliability and accuracy of the proposed technique.


Introduction
We consider the two-parameter singularly perturbed convection-diffusion boundary value problems of the form: LyðxÞ Ày 00 ðxÞ þ lpðxÞy 0 ðxÞ þ f ðxÞyðxÞ ¼ gðxÞ; x 2 ða; bÞ This type of problem arises in the fields like engineering, mathematical physics, and in many areas of applied mathematics. We often come across boundary value problems in which one or small positive parameter multiplies with the derivatives. A large number of research papers have been found in the literature for single parameter convection-diffusion and reaction-diffusion problems [2,8,9,12,16]. However, only a very few authors have discussed two-parameter singularly perturbed boundary value problems [4,6,7,10,11,14,16,[18][19][20]. The nature of two parameters is asymptotically examined by O' Malley [14]. Different numerical methods have been proposed by various authors for two-parameter singularly perturbed problems such as exponentially fitted cubic spline method [7], finite difference, finite element, and B-spline collocation method [6,11], Haar wavelet method [16], and exponential spline technique [18]. For more information about SPPs, readers are referred to books [13,15] and references therein.
In this paper, we introduce a new nonpolynomial cubic spline method as an alternative to existing methods. The paper is organised into five sections. In Sect. 2, we give a brief derivation of nonpolynomial parameters cubic spline. In Sect. 3, we presented the formulation of the method. Convergence analysis is briefly discussed in Sect. 4. Finally, in Sect. 5, numerical examples and comparison with the existing methods are given that demonstrate the practical applicability and superiority of the proposed method.

Nonpolynomial spline function
We consider a uniform mesh M with nodal points . . .; n; and h ¼ ðbÀaÞ n . A nonpolynomial spline function S M ðxÞ of class C 2 ½a; b which interpolates y(x) at mesh points x i ; i ¼ 0ð1Þn depends on a parameter k, if we take k ! 0, then it reduces to ordinary cubic spline in [a, b].
For each segment ½x i ; x iþ1 ; i ¼ 0; 1; 2. . .n À 1, we consider the nonpolynomial cubic spline S M ðxÞ of the form: ð2:1Þ where a i ; b i ; c i , and d i are unknown coefficients and k is a free parameter which will be used to raise the accuracy of the method. Let y(x) be the exact solution and y i be an approximation to yðx i Þ, obtained by the segment S i ðxÞ of the mixed splines function passing through the points ðx i ; y i Þ and ðx iþ1 ; y iþ1 Þ.

The method
At the grid point x i , the proposed two-parameter singularly perturbed boundary value problem (1.1) can be discretized as follows: Using spline's second derivative, we have ð3:2Þ Finally, we arrive at the following system: where

Convergence analysis
In this section, we investigate the convergence analysis of the proposed method. For this, let . . .; n À 1 be an exact column vectors, where Y; Y; T; and E are exact, approximate, local truncation error, and discretization error, respectively.
We can write the standard matrix equation for the method developed in the following form: where M is a matrix of order ðn À 1Þ with The tridiagonal matrices A 0 ; A 1 , and A 2 have the form:  where E ¼ ðY À YÞ ¼ ½e 1 ; e 2 ; . . .; e nÀ1 T . Clearly, the row sums M 1 ; M 2 ; . . .; M nÀ1 of M are If we choose h sufficiently small, matrix M becomes irreducible and monotone [5].
for some i o between 1 to n À 1.

Numerical examples
To test the viability of the proposed method based on nonpolynomial cubic spline, two numerical examples are considered. All the computations were performed using MATLAB. We also compare our method with the existing methods which shown improvement.

Concluding remarks
In this paper, nonpolynomial cubic spline function is used for finding the numerical solution of two-parameter convection-diffusion singularly perturbed boundary value problems. The computations associated with the examples discussed above were performed using MATLAB. The proposed method is computationally efficient and the algorithm can be easily implemented on a computer. Comparison of the method is also depicted through Tables 1, 2