1. Introduction

In this paper, we consider the following class of singular singularly perturbed boundary value problems (BVPs):

(1)

(2)

where is a small parameter, and are smooth functions and are constant. The existence and uniqueness of such type problem is given by ^[23].

As we know the given problem is singular because of regular singularity at the coefficient of derivative terms. These types of problem are different from singular perturbation problem and a very little literature is available for these types of problems. Singular perturbation problem appear in many branches of applied mathematics and so many engineers and researcher works on them. These types of problems also commonly occur in the fluid mechanics, quantum mechanics, optimal control theory geophysics, reaction-diffusion equation etc.

In today’s era of mathematics, there are so many special methods to find the accurate solution of the given problems in terms of ordinary and partial differential equations ^{[8, 14, 19]}. For detail one may refer to the book C.M. Bender and S.A. Orszag ^[2], J.J.H Miller, E. O’ Riordan and G.I. Shiskin ^[13], R.E. O’ Malley ^[17], H.G. Roos, M. Stynes and L. Tobiska ^[22] and the reference therein. There are so many authors who have given their contribution in the field of singular perturbation problem and with spline function ^{[6, 10, 12, 15, 16, 18]}. Spline function is one of the most important methods to find the solution in computational mathematics. Using spline and its types; we can solve our problem in an efficient manner. To understand the spline function we can refer the book Carl De Boor ^[3] and P.M. Prenter ^[20] and the reference therein. M.K. Kadalbajoo and V.K. Aggarwal ^[9] used fitted mesh B spline method to solve a class of singular singularly perturbed Boundary value problem (BVPs). Yogesh Gupta, Pankaj Kumar Srivastava and Manoj Kumar ^[7] gave the application of B-Spline method for the numerical solution of a system of singularly perturbed boundary value problems. Gazala Akram ^[1] used Quartic spline methods to solve third-order SPBVPs. Jincai Chang, Qianli Yang and Long Zhao ^[4] explained the comparison of B-spline method and finite difference method to solve the BVPs of linear ODEs. M. Cui and F. Geng ^[5] used a computational method for solving third-order singularly perturbed boundary problem. M. Kumar and his coworkers ^[11] used cubic spline method to solve initial value technique for solving second order singular perturbation BVPs while J. Rashidnia, R. Mohammadi and M. Ghasemi ^[21] also used cubic spline method to solve their problems. Ikram A. Tirmizi, Fazal-i-Haq and Siraj-ul-islam ^[24] Used Quartic non-polynomial spline function to solve this type of problem.

There are a quite amount of work has been done for development of numerical methods for boundary value problems using spline and its types. The present paper describes the Quartic B-spline method to solve the singular singularly perturbed boundary value problem. Remaining part of this paper is organized as follows:

Section 2 describes the definition of basics of Quartic B-spline and value of its derivatives at nodal points. In Section 3, the Quartic B-spline method for third-order singular singularly perturbed boundary value method is described for solving equation (1) and (2). Section 4 of this paper consists of numerical solution of three problems. Finally paper is concluded in Section 5.

2. Basics of Quartic B-spline Method

Consider equally spaced knots of a partition on . Let be the space of continuously-differentiable, piecewise fourth degree polynomial on , that is is the space of fourth-degree splines on . Consider the Quartic B-spline basis in . The forth-degree B-splines are defined as:

and for

(3)

To solve the singular singularly perturbed third-order boundary value problems are evaluated at the different knots which are summarized in Table 1.

Table 1. Values of B_i(x), B_i'(x), B_i''(x) , and B_i'''(x) at knots

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3. Description of the Method

Consider the singular singularly perturbed third-order boundary value problem of the form:

(4)

(5)

where is a small positive parameter and are constants. and are sufficiently smooth functions.

When we remove the singularity of given equation (4), then the modified form of equation is given as

(6)

Let be the Quartic B-spline function at the nodal points. Then can be written as

(7)

be the approximate solution of boundary value problem (4), where ’s are unknown coefficient and ’s are fourth degree B-spline function. Then let be grid points in the interval [0, 1]. So that we have, , , at the knots. Using the table of B-spline function, we get approximate value of and as

(8)

(9)

(10)

(11)

Putting above values from equations (8)-(11) in equation (4), we get

(12)

which is arranged in the following manner:

(13)

As we know from equation (6),

Equation (13) can be arranged in the form of above equation (6), we get

(14)

The given boundary condition becomes

(15)

and

(16)

and

(17)

Equation (14), (15), (16) and (17) lead to a system with unknowns.

Now, we write the above system of equations in the following form: , where are unknown,

and the coefficient matrix is given by:

(18)

To find the approximate solution of equation (4), we arrange the system of equations in above matrix form and solve them by using MATLAB.

4. Numerical Illustration

In this section, we have solved three examples by using Quartic B-spline Method to show the accuracy of present method. Results of these examples are calculated by Using MATLAB. A good compatibility is found between the exact and approximate solution.

Example 4.1: Consider the following singular singularly perturbed boundary value problem:

having boundary conditions

where

The exact solution of the given problem is

The error analysis for the given problem 4.1 is written into the table (Table 4.1) for different values of and

Table 4.1. Maximum absolute error

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Example 4.2: Consider the following singular singularly perturbed boundary value problem:

having boundary conditions

where

The exact solution of the given problem is

The error analysis for the given problem 4.2 is written into the table (Table 4.2) for different values of and

Table 4.2. Maximum absolute error

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Example 4.3: Consider the following singular singularly perturbed boundary value problem:

having boundary conditions

where

The exact solution of the given problem is

The error analysis for the given problem 4.3 is written into the table (Table 4.3) for different values of and

Table 4.3.1. Maximum absolute error at ε=10^-2

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Table 4.3.2. Maximum absolute error at ε=10^-3

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5. Conclusion

In this paper, Quartic B-spline Method for solving third-order singular singularly perturbed boundary value problem is proposed. There are three examples which are solved by using present method. The results obtained by this method are shown in Table 4.1, 4.2, 4.3.1 and 4.3.2 respectively. It is clear that from the numerical examples of the present method gives better agreement with the exact and approximate values. Hence Quartic B-spline method is very efficient and its implementation is also very easy and accurate to evaluate according to the given problem and boundary conditions.

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