Survey of B-spline functions to approximate the solution of mathematical problems

In the present paper, we describe a survey of B-spline techniques which have been used for numerical solutions of mathematical problems recently. Here, we discussed the definition of B-splines of various degrees by two different approaches to generate the recurrence relation to drive the formulation of B-splines. Cubic B-spline applied on two test equations and absolute errors in interpolation are compared with cubic and quintic splines. Some remarks have been included. Numerical results are tabulated in tables; these tables show that the results obtained by cubic B-spline are considerable and accurate with respect to the cubic spline and more or less similar to the quintic spline.


Introduction
The theory of spline functions is a very attractive field of approximation theory. Usually, a spline is a piecewise polynomial function defined in region D, such that there exists a decomposition of D into subregions in each of which the function is a polynomial of some degree k. Also, the function, as a rule, is continuous in D, together with its derivatives of order up to (k −1) [1][2][3][4][5][6][7]. Generally, the piecewise polynomial is considered, and [ a, b] ⊂ R is a finite interval. We introduce a set of partition n = {x 0 , x 1 , · · · , x n } of [ a, b], where x i (i = 0(1)n) are called nodes of the partition. The set of piecewise polynomial of degree k defined on a partition n is denoted by S k ( n ) in each subinterval; I i =[ x i−1 , x i ] is a kth degree polynomial. Specifically, the type of bases B-spline for our purpose is considered, for which we only use the equidistance partition. Moreover, we extend the set of nodes by taking h = b−a n , x 0 = a, and x i = x 0 + ih where i = ±1, ±2, ±3, · · ·. Let { n } be a partition of [ a, b] ⊂ R. A B-spline of degree k is a spline from S k ( n ) with minimal support and the partition of unity holding.
The B-spline of degree k is denoted by B k i (x), where i ∈ Z, and then we have the following properties: The next section explains the explicit definition of B-splines.

Derivation of B-spline functions
In this section we give an introduction of B-splines. The B-splines were so named because they formed a basis for the set of all splines [4]. Through out this section, we suppose that an infinite set of knots {x i } has been prescribed in such a way that The B-spline to be defined now depends on this set of knots.

Definition 1.
Support of function f is defined as the set of points x when f (x) = 0.

B-spline of degree 0
The B-spline of degree 0 is defined by 0 otherwise. B-spline of degree 0 is characterized by the following: 5. Any spline of degree 0 can be expressed as a linear combination of the B -splines B 0 i .
We generate all the higher degree B-splines by a simple recursive definition [3,4]: The B k i functions as defined by Equation 1 are called B-splines of degree k. Since each B k i is obtained by applying a linear factor to B k−1 i and B k−1 i+1 , we see that degrees actually increased by 1 at each step. Therefore, B 1 i is piecewise linear, B 2 i is piecewise quadratic, and so on.
With the function B 0 i as a starting point and Equation 1, we obtain the higher degree B-splines.

B-spline of degree 1
To illustrate Equation 1, let us determine B 1 i in an alternative form: o t h e r w i s e . http://www.iaumath.com/content/6/1/48 B-spline of degree 1 is characterized by the following:

Quadratic B-spline
We determine B 2 i in an alternative form: At first, we determine B 1 i+1 (x) in an alternative form: Thus, o t h e r w i s e .

Alternative approach to drive the B-spline relations
In this section, we give another approach for driving the B-splines which are more applicable with respect to the recurrence relation for the formulations of B-splines of higher degrees [2]. At first, we recall that the kth forward difference f (x 0 ) of a given function f (x) at x 0 , which is defined recursively by the following: . http://www.iaumath.com/content/6/1/48 In particular, It is well known that with evenly spaced knots x i = x 0 + ih and x i+j = x i+j h, n annihilates all polynomials of degree (n − 1). Theorem 1. The nth forward difference n f (x) with evenly spaced knots of any polynomial of degree n − 1 is identically equal to zero.

Theorem 2. The function (x − t) m
+ is defined as follows: times continuously differentiable both with respect to t and x.
The B-spline of order m is defined as follows: Hence, we apply this approach to obtain the B-spline of order one. Let m = 1; thus, In order to obtain the quadratic B-spline, let m = 2. Thus, In order to obtain the cubic B-spline, let m = 3. Thus, In order to obtain the quartic B-spline which is used by [8], let m = 4: 0 o t h e r w i s e . http://www.iaumath.com/content/6/1/48 In order to obtain the quintic B-spline which is used by [9], let m = 5: At last in order to obtain Sextic B-spline which is used by [10], let m = 6 6 x i+4 < t ≤ x i+5 0 otherwise. http://www.iaumath.com/content/6/1/48 Remark 2. The above B-splines have been used to approximate the solution of recent differential equations [11][12][13] and integral equations [14], as well as for interpolation [15][16][17][18].

Numerical illustration
We applied the cubic B-spline to interpolate the following test problems. The maximum errors in the interpolation are tabulated in Tables 1 and 2. Results obtained by cubic B-spline are compared with the absolute error in the interpolation by cubic and quintic splines given in [19].

Conclusions
These tables show that the results obtained by cubic B-spline are considerable and accurate with respect to the cubic spline and are more or less similar to the quintic spline.