Numerical Solution of Two-Point Boundary Value Problems by Interpolating Subdivision Schemes
A numerical interpolating algorithm of collocation is formulated, based on 8-point binary interpolating subdivision schemes for the generation of curves, to solve the two-point third order boundary value problems. It is observed that the algorithm produces smooth continuous solutions of the problems. Numerical examples are given to illustrate the algorithm and its convergence. Moreover, the approximation properties of the collocation algorithm have also been discussed.
Subdivision plays an important role in computer aided geometric design. Particularly, subdivision schemes for curve design consist of repeated refinement of control polygons. In particular, the linear schemes are well studied with rules for defining the control points at the finer level as finite linear combinations of the control points in the coarser level.
The concept of subdivision has been initiated by de Rham . Later on, in , Dyn et al. studied a family of schemes with mask of size four, indexed by a tension parameter. In , Dubuc and Deslauriers studied a family of schemes indexed by the size of the mask and the arity (or base) ; that is, at each subdivision step, the scheme (, ) uses masks of points to compute new points corresponding to each old edge. Mustafa and Rehman  unified all existing even-point interpolating and approximating schemes by offering general formula for the mask of -point even-ary subdivision scheme. Aslam et al.  presented an explicit formula which unified the mask of -point interpolating as well as approximating schemes. Mustafa et al.  presented an explicit formula for the mask of odd-points -ary (for any odd ) interpolating subdivision schemes. Higher arity schemes have been introduced in [7, 8]. Mustafa and Ivrissimitzis  have showed that when subdivision is used as a data modelling tool large support, or large arity schemes, which generally produce smoother curves, it may outperform simpler schemes.
An interpolating subdivision scheme approach to the construction of approximate solutions of two-point second order boundary value problems was first time introduced in . In this approach a method of collocation was formulated for linear two-point second order boundary value problems. It is proved that the algorithms produce smooth continuous solutions provided the algorithms are chosen appropriately. Later on, in , they reformulated the collocation method by subdivision scheme in order to compute numerical solutions for two-point boundary-value problems of differential equations with deviating arguments. They demonstrated that their approach has further development to their previous works for solving various types of two-point second order boundary-value problems.
They approximated the derivative boundary conditions by forward difference operator so it is difficult (with no flexibility to improve the solution) to generalize their approach to solve third or higher order boundary value problems. We express derivative boundary conditions at end points by using interpolatory subdivision algorithm; therefore it is easy to generalize our approach to deal with higher order boundary value problems.
In this paper, we have reformulated the collocation algorithm by using 8-point interpolating subdivision scheme to compute the approximate solution of two-point third order boundary value problems. Particularly, collocation algorithm with septic precision treatments at the endpoints has the power of approximation . Our reformulated collocation algorithm treats the following type of two-point third order boundary value problem: where and are continuous and on .
The outline of the paper is as follows. In Section 2, we rewrite general form of interpolating subdivision scheme for curve design  and some related results. The 8-point binary interpolating scheme and derivatives of its basis function have also been discussed in this section. In Section 3, a numerical interpolating algorithm of collocation to solve (1) is formulated and its boundary treatments are discussed. In Section 4, approximation properties of this algorithm are given. In Section 5, numerical examples are presented. Section 6 is devoted for conclusion and the possible future research directions.
2. Interpolating Schemes for Curve Design
A general compact form of symmetric univariate binary interpolating subdivision scheme  which maps polygon to a refined polygon is defined by where is called the degree of the scheme and the constants are given by where denotes the binomial coefficient.
The boundary treatments are necessary to produce smooth curve segments by scheme (2). Normally higher order approximation formulae should be used near the ends of the segments and thus Lagrange formulae of order are recommended.
Remark 1. Let be the limit curve generated from the cardinal data ; that is, is the fundamental solution of the subdivision scheme (2); then Furthermore, satisfies the following two-scale equation:
Lemma 3 (see ). Given a square matrix of order , let the normalized left and right (generalized) eigenvectors of be denoted by . Then for any vector , there exists following Fourier expansion:
Lemma 4 (see ). Suppose , is a regular and curve in , . Let , be the limit curve generated by (2) from the initial data , , . Then, on any finite interval , the following estimates hold: where the number depends only on the derivatives of and .
2.1. 8-Point Interpolating Scheme
For by (2) and (3), we have the following 8-point binary interpolating subdivision scheme for curve design: This scheme is -continuous in  and reproduces polynomial curve of degree seven in . The local subdivision matrix of (8) is denoted and defined bywhere , , , and . The some of its eigenvalues is For an eigenvalue , the eigenvectors and that satisfy and are called right and left eigenvectors of the matrix , respectively. Some of the normalized left and right eigenvectors corresponding to eigenvalues are Since for and otherwise then by using Lemmas 2 and 3, we get the following result.
Lemma 5. The fundamental solution (Cardinal basis) is thrice continuously differentiable and supported on and its derivatives at integers are given by where
The graphical representations of the basis limit function defined on cardinal data and its derivatives up to order three for are shown in Figure 1. Figure 1(a) represents the basis limit function defined in (4). Graphical representations of first, second and third derivatives of basis limit functions obtained from (5) for are shown in Figures 1(b), 1(c) and 1(d) at , 1, −1, respectively. The numeric values of first, second and third derivative are given in (13).
3. Numerical Interpolating Collocation Algorithm
In this section, first we formulate a numerical interpolating collocation algorithm for linear third order two-point boundary value problems. Then we settle down the boundary conditions to get unique solution.
3.1. The Collocation Algorithm
Let be a positive integer , , and , , and set , . Let be the approximate solution to (1), where are the unknown to be determined by (1). The collocation algorithm, together with the boundary conditions to be discussed, is given by setting where Using (14) and (16) in (15), we get following system of equations:
Now we simplify the above system in the following theorems.
Theorem 6. For by (17), one gets where and .
Proof. Substituting in (17), we get For , , this implies Since the support of basis function is , , , and are zero outside the interval ; also by (4) and (13), we get If , then For , we get (18). This completes the proof.
Theorem 7. For the system (17) is equivalent to
3.2. Adjustment of Boundary Conditions
The order of the coefficient matrix (28) is . In order to get unique solution of the system, we need twelve more conditions. Here we consider only two different cases. In coming section we will show that the approximate solution can be improved by adjusting different boundary conditions.
Case 1. If we assume then two conditions can be achieved by using following given boundary conditions i-e:
Still we need ten more conditions to get stable system. Since subdivision scheme reproduces seven degree polynomials, we define boundary conditions of order eight for solution of (27). For simplicity only the left end points are discussed and the values of right end points , , , , can be treated similarly.
The values , , , , can be determined by the septic polynomial interpolating at , . Precisely, we have where Since by (14) for then, by replacing by , we have Hence the following boundary conditions can be employed at the left end: Similarly, for the right end, we can define , , , , , , and So we have the following boundary conditions at the right end: Finally, we get the following new system of linear equations with unknowns , in which equations are obtained from (18) and (23), two equations from boundary conditions (29) and ten from boundary conditions (33) and (35): where the coefficients matrix , is defined by (28), and and are formed by (29), (33), and (35)where the first five rows of come from (33) and the sixth row comes from (29) at . Consider where first row of comes from (29) at remaining rows come from (35) and the matrices and are defined as
Case 2. In this case we express the given boundary condition in the following way.
By using (16) we have As we defined earlier if we put we get ; the above equation can be written as Since by boundary condition , By using (13) we can express above equation as
Finally, we get a following new system of linear equations with unknowns , in which equations are obtained from (18) and (23), two equations from boundary conditions (29) and ten from boundary conditions (33) for , (35), and (43): where the coefficients matrix , is defined by (28), and and formed by (29), (33), (35), and (43) are defined asin first four rows come from (33) for , fifth row comes from (43), and the last row comes from (29).
The matrix is same as defined in Case 1 and the matrices and are defined as The nonsingularity of the coefficients matrix has been discussed in next section.
3.3. Existence of the Solution
In this section, we discuss the nonsingularity of the coefficients matrix . We observe that the coefficients matrix is neither symmetric nor diagonally dominant. However it can be shown that is a nonsingular. Since is almost a band matrix with half band width , numerical complexity for solving the linear system using Gaussian elimination is about multiplications. For large , the matrix is almost symmetric except the first and last six rows and columns due to the boundary conditions. Therefore we first consider the symmetric part of it, that is, square symmetric matrix of order defined asIt can be shown that is always nonsingular for each value of . However, is nonsingular for . We have checked the nonsingularity of matrix by different methods. In first method we observe that the determinants of matrix increase as increases and it is not zero for . So is nonsingular. The determinants of at some values of are shown in Table 1. In second method we observe that for , the eigenvalues of matrix are nonzero so by  matrix is nonsingular. However for matrix may or may not be nonsingular. Therefore we claim that systems (36) and (44) are stable for .
4. Error Estimation
In this section, we discuss the approximation properties of the numerical interpolating collocation algorithm. Since the scheme (8) reproduces polynomial curve of degree seven so by Dyn  scheme has approximation order eight. So the collocation algorithm (14) with septic precision treatments at the endpoints has the power of approximation . Here we present our main result for error estimation.
Proof. Since the order of approximation of subdivision scheme (8) is eight so by using (13), we can write for smooth function and small as
This can be written as
Similarly, we have
If we define error function and error vectors at the nodes by
or equivalently , , then this implies
By subtracting (50) from (51), we get
By Lemma 5, we get the following expression:
By subtracting (1) from (15), we get This implies Using (56), we get where and .
As and , so are nonzero while and are zero because they lie outside the interval . Let us define these (the left and right end) error values as Thus system (59) is equivalent to where is the matrix obtained by deleting the first and last six rows and columns of the matrix , where By using (7) Hence, for small , the coefficients matrix will be invertible and thus using the standard result from linear algebra, we have This completes the proof.
The above discussion suggests that the approximations of the solution computed by the method developed in pervious section are second order accurate approximations. This suggestion is supported by the numerical experiments given in the next section
5. Numerical Examples and Discussions
In this section, the numerical collocation algorithm based on 8-point interpolating subdivision scheme described in Section 3, with the 8th order boundary conditions at the end points, is tested on the two-point third order boundary value problems. Absolute errors in the analytical solutions are also calculated. For the sake of comparisons, we also tabulated the results in this section.
Example 1. Consider the boundary value problem with boundary conditions , , . The analytical solution of this problem is By using the collocation algorithm for , we get following solution of the above problem: , where the values of by using (36) are and by using (44) are By using two different boundary treatments presented in Section 3, we obtained two different solutions which are presented in Table 2 along with their absolute errors. The graphical representations of the analytic and approximate solutions of the above problem are shown in Figure 2. Figure 2(a) represents the comparison of analytic and approximate solutions obtained by (36) while analytic and approximate solutions obtained by (44) are shown in Figure 2(b). From this table and figure, we observe that the solution obtained by (44) is significantly better than the solution obtained by (36). So our claim; that is, the approximate solution can be improved by adjusting boundary treatment, is justified. The maximum absolute errors in the solutions obtained by (36) and (44) at step size 10 are and , respectively.
Example 2. Consider the following third order boundary values problem: