**International Journal of Partial Differential Equations and Applications**

## Employment Higher Degree B-Spline Function for Solving Higher Order Differential Equations

**Karwan H.F.Jwamer**^{1,}, **Najim Abdullah I.**^{2,}

^{1}Mathematics Department, School of Science, University of Sulaimani, Sulaimani, Kurdistan Region, Iraq

^{2}Mathematics Department, College of Science, University of Garmian, Kalar, Kurdistan Region, Iraq

Abstract | |

1. | Introduction |

2. | Construction of B-Spline |

3. | Quintic B-spline |

4. | Description of the Method |

5. | Numerical Result |

6. | Conclusion |

References |

### Abstract

As the B-spline method was developed for solving higher order differential equations, we present a brief survey to construct a higher degree B-spline. The new technique has been given in this field, accordingly a numerical illustration used to solve boundary value problems by employ quintic B-spline function. An example has been given for calculating maximum absolute error through *n *nodes.

**Keywords:** B-spline, boundary value problems, approximate solution, absolute error

**Copyright**© 2016 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Karwan H.F.Jwamer, Najim Abdullah I.. Employment Higher Degree B-Spline Function for Solving Higher Order Differential Equations.
*International Journal of Partial Differential Equations and Applications*. Vol. 4, No. 1, 2016, pp 16-19. http://pubs.sciepub.com/ijpdea/4/1/3

- H.F.Jwamer, Karwan, and Najim Abdullah I.. "Employment Higher Degree B-Spline Function for Solving Higher Order Differential Equations."
*International Journal of Partial Differential Equations and Applications*4.1 (2016): 16-19.

- H.F.Jwamer, K. , & I., N. A. (2016). Employment Higher Degree B-Spline Function for Solving Higher Order Differential Equations.
*International Journal of Partial Differential Equations and Applications*,*4*(1), 16-19.

- H.F.Jwamer, Karwan, and Najim Abdullah I.. "Employment Higher Degree B-Spline Function for Solving Higher Order Differential Equations."
*International Journal of Partial Differential Equations and Applications*4, no. 1 (2016): 16-19.

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### 1. Introduction

B-spline is a spline function that has minimal support with respect to given degree, smoothness, and domain partition ^{[8]}, and named B-splines because they formed a basis for all splines ^{[3]}. Quartic spline solution of third order singularly perturbed B V P has been studied by ^{[2]}. Theoretical background for stable computation by using B-splines with their derivatives studied by ^{[7]}. ^{[4]} employ quartic B-spline collocation method for solving one-dimensional hyperbolic telegraph equation and exploitation. Quintic B-spline for the numerical solution of fourth order parabolic partial differential equations to find maximum error given by ^{[5]} while ^{[1]} discussed quartic B-spline differential quadrature method, and ^{[6]} employs quartic B-spline method to solve the self-adjoint boundary value problems. ^{[9]} in his paper approximate errors calculated by using cubic B-spline function.

As for us in this paper we construct a higher degree B-spline by two different method for solving self adjoint boundary value problems, in the following section we display deriving methods. Section 3 as example define a quintic B-spline. Section 4 describes the definition of Quintic B-spline. Finally Section 5 consists of a computer procedure to compute maximum error for several nodes.

### 2. Construction of B-Spline

If (c_{i})^{n}_{i=1}is a sequence of control points and x= is (n+d-1) knots for spline of degree d; we have seen that a typical spline can be written as

where B_{I,0 }is written as

(1) |

Any spline of degree 0 can be expressed as a linear combination of the B-spline B_{i,0}.

And f(x)= is piecewise constant function and x_{d+1}x …x_{n+1 }although the end knots allowed to coincide.

So higher B-spline is generate fromlower degree of B-splines by

(2) |

The B_{i,k} functions as defined in (2) are called B-spline of degree *k*.

Another method to generate higher degree B-spline and it is valid only equidistant points:

The B-spline of order m is defined as follows:

Where

### 3. Quintic B-spline

Let be a uniform partition of the interval [0,1] such that a=x_{0}<x_{1}<…<x_{n}=b where h=x_{i+1}-x_{i} or h= then

(x) is the B-spline basis function of 5^{th} degree which also called quintic B-spline vanish outside interval. Each quintic B-spline cover five elements. The basis function is non-zero on five knot spans. The set of quintic B-splines {B_{-3}, B_{-2}, B_{-1}, … , B_{N}, B_{N+1},B_{N+2}} form a basis for the functions over interval [0, 1].

Now let s(x) be the B-spline interpolating function at the nodal points. Then s(x) can be written as s(x)= where c_{j}'s are unknown coefficients and B_{i}(x)'s are quintic B-spline functions. The value of at the nodal points can be obtained and its differentiating with respect to x, which are summarized in Table 2.

#### Table 4.1. We found the coefficients of quintic B-spline and its derivative at nodal points from the definition of our B-spline

### 4. Description of the Method

Consider the self-adjoin fourth-order singularly perturbed boundary value problem of the form:

(3) |

(4) |

Where , , and are constants and is a small positive parameter a(x), and f(x) are sufficiently smooth functions. In this survey, we take a(x)=a= constant. Let u(x)= s(x)= be the approximate solution of boundary value problem (3). Then let x_{0},x_{1},…,x_{n} be n+1 grid points in the interval [0,1]. So that we have, x_{i}=x_{0}+ih, x_{0}=0, x_{n}=1, i=1, 2, …, n; h= at the knots, we get

(5) |

(6) |

(7) |

(8) |

(9) |

Putting the value of equations (5)-(9) in equation (3), we get

(10) |

And the boundary condition becomes,

(11) |

(12) |

(13) |

(14) |

The values of the spline function at the knots are determined using table (4.1) and substituting in equations (10)-(14) a system of (n+4) equations with (n+4) unknown. Now, we can write the above system of equations in the following form

where X_{n}=(c_{-3},c_{-2},c_{-1},…,c_{0},c_{1},…,c_{n+2})^{T} are unknowns,

From equation (10):

and boundary condition(11-14),

, we get the following:

If i=0,then

(15) |

For i=1, we obtain

(16) |

For i=2, then we have

(17) |

For i=3, then

(18) |

If i=4, thus

(19) |

If i=5, then

(20) |

*For i=6 ,*

(21) |

For i-n-4, then

(22) |

For i=n-3, then

(23) |

For i-n-2, then

(24) |

*For i=n-1, then *

(25) |

Finally for i-n, we obtain that

(26) |

And boundary conditions(11)-(14) gives:

(27) |

(29) |

(30) |

### 5. Numerical Result

In this section we solve higher order B. V. Ps. By using quintic B-spline interpolation as follows:

For order four B. V. Ps. Take the following

Example 1: Consider the fourth order boundary value problem:

The maximum error bound gives by the following table:

For order three B. V. Ps. Take the following

Example 2: Consider the following third order singular perturbation problem :

The maximum error bound gives by the following table:

For order two B. V. Ps. Take the following

Example 3: Consider the second order boundary value problem with singular perturbation form: -y''+y=x^{2} ,and subject to the boundary conditions

The maximum error bound gives by the following table:

### 6. Conclusion

In this paper, we design higher order B –Spline to solve second, third, and fourth order singular perturbed boundary value problems. Also there examples are presented with different values of n and ∈ and they showed the efficiency and of our design .

### References

[1] | AlperKorkmaz, A. Murat Aksoy andIdrisDağ, Quartic B-spline Differential Quadrature Method, International Journal of Nonlinear Science, Vol. 11, (2011) No.4,pp.403-411. | ||

In article | |||

[2] | GhazalaAkram, Quantic Spline Solution of a third Singularly Perturbed b. v. p., ANZIMTJ, 53, 2012, 44-53. | ||

In article | |||

[3] | JalilRashidinia and Sharifi, Survey of B-spline Function to Approximate the solution of mathematical problems. | ||

In article | |||

[4] | MarziehDosti, ALirezaNazemi Quantic B-SplineCollocation Method for Solving One-dimensional Hyperbolic Telegraph Equation, Department of Mathematics school of Mathematical. Science, Sharhood University of Technology, sharhood, Iran, Journal of Information and Computing Science Vol.7 No2, 2012, PP. 083-090. | ||

In article | |||

[5] | Shahid S. Siddiqi and SaimaArshed Quantic B-Spline for the Numerical Solution of 4^{th} Order Perturbed Partial Differential equation Department of Mathematics, University of Punjab, Lahore54590,Pakstan. Wall Applied Science Journal 23(12): 115-112, 2013. | ||

In article | |||

[6] | SonaliSaini and Hradyesh Kumar Mishra a New Quartic B-Spline Method for Third-Order Self-Adjoint Singularly Perturbed Boundary Value Problems Applied Mathematic Science Vol 9, 2015 No.8 399-408. | ||

In article | |||

[7] | Prenter, P.M, Splines and Variational Methods, John-Wily, New York(1975). | ||

In article | |||

[8] | VijayDahiya Exploring B-Spline functions for Numerical Solution of Mathematic Problems International Journal of Multidisciplinary Research and Development 2015; 2 (1): 452-458. | ||

In article | |||

[9] | Yogesh Gupta and PankajKumrSrivastava, Application of B-Spline to Numerical solution of a system of Singularly Perturbed ProplemsMathematicaAterna , Vol.1, 2011, No.06, 405-415. | ||

In article | |||