Abstract: This paper concerns with the use of B-Splines to approximate a solution of a differential equations by collocation. The effect of knot placement on the accuracy of approximation is considered and numerical examples are given to illustrate the effectiveness of knot sequence.
INTRODUCTION
The numerical solution of boundary value problem is a topic in which active research is currently underway. There are number of methods used to solve boundary value problems. The most important of these probably the collocation method. For discussion referring to collocation method (Reddien, 1979) (Deuflhard, 1979) and (Ascher et al., 1985). For an important collocation computer code, (Ascher et al., 1981) and (Ascher et al., 1981). In this study we use B-splines in the numerical solution of an initial boundary value problems by collocation. This method provides a strategy by which we can attack many problems in applied mathematics. Rayleigh Ritz method or Galerkins method could be made quite effective if one were to give up on using polynomials or other analytic functions as trial function and used piecewise polynomial instead.
Numerical solution technique
Now let us consider how this method works ( Boor et al., 1973). We
look for approximating a function g on [a,b], which is given to us implicitly,
as a solution of the differential equation
(2.1) |
with boundary conditions
(2.2) |
Where
Since (2.1) is nonlinear, (2.1-2.2) may have many solutions. Therefore we require that there be a neighborhood around the specific solution g and we will start our iterative process within this neighborhood in order to converge to this particular solution.
We intent to approximate g by piecewise polynomial (pp) functions using collocation.
That is, we determine a pp function f so that it exactly satisfies the differential
equations at certain points, the collocation points. We look
(2.3) |
(2.4) |
Here Pk, ξ denotes the linear space of pp functions of order k with breakpoint sequence ξ.
We choose the collocation points per subintervals and distributed the same in each subinterval with.-1≤P<P2 <K<Pk≤1. We calculate these points as follows;
We choose P as the zeros of the k-th Legendre polynomial. The reason for such a selection can be given with the following theorem (Boor et al., 1978).
Theorem. Assume that the function F in (2.1) is sufficiently smooth in a neighborhood of the curve [a, b]→Rm+1:x→(x; g(x), K, Dm-1 g(x)).
Assume further that the collocation points
(2.5) |
At the breakpoints, the approximation is of even higher order and satisfies
(2.6) |
Here const depends on F,g and k, but does not depend on ξ .
Since the problem (2.3-2.4) is nonlinear, in general, we need to use some iterative scheme for its solution. We can solve by Newtons method starting with a sufficiently close initial guess f0, that is (2.3-2.4) has a solution
with fr+1 the solution
(2.7) |
where
(2.8) |
and
(2.9) |
The function y in (2.7-2.9) is a linear combination of appropriate B-splines.
Let
Therefore the unknown function y can be written in the form
We can determine y by determining its B coefficient vector α . This gives the linear system
(2.10) |
Where linear differential operator L is defined by
The following theorem gives sufficient condition for the existence of discrete
solutions of boundary value problems. Theorem. Let g(x), F (x; z0,
z1, K, zm-1) and
Let 0 =y be only trivial solution of the homogeneous equation 0 =(m) y satisfying the boundary condition (2.2). If the linear homogeneous equation.
has only trivial solution under the boundary condition (2.2), then there exist
a number σ>0 so that unique solution of the problem (2.1-2.2) can be
found inside the sphere
The Effect of knot placement on the accuracy of the Spline approximation
Construction of the piecewise polynomials depends on partition of the interval which is an important matter since every partition leads to a different approximation. It was suggested by (Boor et al., 1978) that we place the breakpoints ξ2, K, ξm so as to minimize
(3.1) |
For this purpose the following analysis should be considered .
is a continuous function of á and â and monotone, increasing in â and decreasing in α when Dk g is continuous. In order to minimize (3.1) we choose ξ2, K,ξ1 so that
(3.2) |
It is not easy task to find appropriate placement of ξis since we dont know Dkg . Let us rewrite (3.2) as
(3.3) |
The last equality reduces the appropriate determining of ξ2, K, ξ1 such that
This latter problem can be easily solved by replacing the function
is continuous and monotone increasing piecewise linear function. Hence its inverse I-1 is defined. It is required to evaluate the function I-1 at the l-1 points iI (b)/l , I=1, K, l-1.
Then we first determine a piecewise constant approximation h to the function
It is possible to determine the function H(x) as an element belonging to P2,ξ ∩ C by considering
We choose h∈ P1,ξ such that
where we have used the abbreviation fi+1/2=Dk-1fξ on [ξi, ξ I+1], all it
If we sum up the process;
I) | Choose the breakpoints ξis and an initial solution f0 . |
ii) | Obtain fξ by using Newtons method. |
iii) | For better approximation, obtain h∈P1, ∈ξ with the use of fξ . |
iv) | Determine the number l and the breakpoints ξi+1=I-1(iI(b)/l), I= 1, K, l-1 |
v) | Replace f0 by f1 and repeat the process. |
The method discussed above have been applied to the following problems and the results obtained are given below.
Numerical results
In this section the method discussed above were tested on two problems.
Example. 0.005y+y2=1, 0≤x≤1
with the following boundary condition:
y(0) = y(1) = 0
If we linearize the problem about the point y=y0 by Newtons method we obtain
0.005 y+2y0y=1+y20
y(0)= y(1)=0
Let y0=x2-1 be initial solution.
Let f∈P5+2 ∩ C(1). We subdivide the interval
[1 , 0] into five subintervals and select the following points initially;
0.00 0.25 0.5 0.75 1.
In each iteration we have used the most recent approximation to the solution as the current guess fr together with a different knot sequence, which is obtained via fr .
The knot sequences obtained are shown as below.
The best results are obtained using the last knot sequence. The solution changes rapidly in the interval [0.75,1] . Therefore numerical results obtained are given for this interval.
Example: y = ey, 0≤x≤1
with the following conditions
y(0) = y (1)= 0
If we linearize the problem about the point y=y0 by Newtons method we obtain
y-ey0 y=(1-y0)ey0
y(0)=y(1)=0
Let y0=x2-x be initial solution.
Let f∈P4+2 ∩C(1) The interval [1, 0] is divided into five subintervals and we choose the following points initially;
0.00 | 0.20 | 0.40 | 0.60 | 0.80 | 1.00 |
The knot sequences obtained are as follows
0.00 | 0.19640 | 0.39710 | 0.60290 | 0.80360 | 1.00 |
0.00 | 0.19727 | 0.39787 | 0.60213 | 0.80273 | 1.00 |
0.00 | 0.19712 | 0.39753 | 0.60200 | 0.80271 | 1.00 |
0.00 | 0.19707 | 0.39741 | 0.60205 | 0.80279 | 1.00 |
0.00 | 0.19709 | 0.39747 | 0.60207 | 0.80275 | 1.00 |
The following results are found for the last knot sequence given above.
Better results can be obtained by increasing the order of polynomials and acccuracy in the iterative process. But the most effective method beyond these is the repositioning of the breakpoints. As a different approximation to the solution we change the place of the knots in each iteration and we observed that accuracy is increased and the number of iteration is reduced.