Journal of Applied Sciences

Year: 2003 | Volume: 3 | Issue: 4 | Page No.: 210-215
DOI: 10.3923/jas.2003.210.215

On the Approximative Solution of Boundary Value Problems by Collocation

Fatih Tasci

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.

Keywords: knot sequence, Collocation methods and B-splines

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 Galerkin’s 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

with boundary conditions

Where is a real valued function on R^m+1 and we will assume it to be sufficiently smooth and w_ij are constants and the points x_i satisfy a≤x₁≤K≤x_m≤b. β₁, β₂, K, β _m are continuous linear functionals in Ca^(m-1) and c₁, c₂, K, c_m are known constants. These boundary conditions are linear, the differential equations is nonlinear. Since we will linearize (2.1) in the computations, we could have made these conditions nonlinear as well (Wittenbrink, 1973).

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 for which

Here P_{k, 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<P_k≤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]→R^m+1:x→(x; g(x), K, D^m-1 g(x)).

Assume further that the collocation points has been chosen such that for every q∈P_s. Then the solution f near g of the approximate problem (2.3-2.4) satisfies

At the breakpoints, the approximation is of even higher order and satisfies

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 Newton’s method starting with a sufficiently close initial guess f₀, that is (2.3-2.4) has a solution

with f_r+1 the solution of the linear problem.

where

and

The function y in (2.7-2.9) is a linear combination of appropriate B-splines. Let be the nondecreasing sequence which contains each of ξ₁ and ξ_i+1 k+m times and each interiorbreak point ξ₂ , K, ξ₁, k times. Then n= kl+m and

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

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, z_m-1) and , (x;z₀,z₁,K,z_m-1be functions defined and continuous for

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, ξ_m so as to minimize

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 D^k g is continuous. In order to minimize (3.1) we choose ξ₂, K,ξ₁ so that

It is not easy task to find appropriate placement of ξ_i’s since we don’t know D^kg . Let us rewrite (3.2) as

The last equality reduces the appropriate determining of ξ₂, K, ξ₁ such that

This latter problem can be easily solved by replacing the function by some piecewise constant function . Then

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 makes no difference whether we construct the piecewise constant function h or the continuous piecewise linear 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 f_i+1/2=D_k-1f_ξ on [ξ_i, ξ _I+1], all it

If we sum up 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 Newton’s 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 f_r together with a different knot sequence, which is obtained via f_r .

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” = e^y, 0≤x≤1

with the following conditions

y(0) = y (1)= 0

If we linearize the problem about the point y=y₀ by Newton’s method we obtain

y”-e^y0 y=(1-y₀)e^y0
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;

The knot sequences obtained are as follows

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.

REFERENCES