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=1is 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+1x …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=x₀<x₁<…<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

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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 x₀,x₁,…,x_n be n+1 grid points in the interval [0,1]. So that we have, x_i=x₀+ih, x₀=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,…,c₀,c₁,…,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:

Table 1. Absolute maximum errors at given N=10, 20, 40 and ∈ =10^-1, 10^-2, 10^-3

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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:

Table 2. Absolute maximum errors at given N=10, 20, 40 and ∈ =10^-1, 10^-2, 10^-3

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

Example 3: Consider the second order boundary value problem with singular perturbation form: -y''+y=x² ,and subject to the boundary conditions

The maximum error bound gives by the following table:

Table 3. Absolute maximum errors at given N=10, 20, 40 and∈ =10^-1, 10^-2, 10^-3

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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

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