Subdivision Schemes Based Collocation Algorithms for Solution of Fourth Order Boundary Value Problems

We present two collocation algorithms based on interpolating and approximating subdivision schemes for the solution of fourth order boundary value problems arising in the mathematical modeling of viscoelastic, and inelastic flows, deformation of beams, arches, and load bearing members like street lights and robotic arms in multipurpose engineering systems. Numerical examples are given to illustrate the algorithms. We conclude that approximating schemes based collocation algorithms give better solution than interpolating schemes based collocation algorithms. Main purpose of this paper is to explore and seek the applications of interpolating and approximating subdivision schemes in the field of boundary value problems along with intrinsic comparison of the results obtained by algorithms based on these schemes. A comparison with other approaches of this type of boundary value problems in order to see the advantages of the proposed methods is also given.


Introduction
Subdivision schemes propose consistent and efficient iterative algorithms to produce smooth curves and surfaces from a discrete set of control points by subdividing them according to some refining rules, recursively.In recent years, subdivision techniques have become an integral part of computer graphics due to their wide range of applications in many areas such as engineering, medical science, graphic visualization, and image processing.The idea of subdivision has been initiated by de Rham [1].Later on, Dyn et al. [2] studied a family of schemes with mask of size four, indexed by a tension parameter.Subdivision schemes can be classified into two important branches, approximating and interpolating ones.Approximating scheme means that the limit curve approximates the initial polygon and that, after subdivision, only the newly generated control points are in the limit curve, while interpolating scheme means that, after subdivision, the control points of the original control polygon and the newly generated control points both lie on the limit curve.Mustafa and Rehman [3] unified existing even-point interpolating and approximating schemes by offering general formula for the mask of (2 + 4)-point even-ary subdivision scheme.Aslam et al. [4] presented an explicit formula which unified the mask of (2 − 1)-point interpolating as well as approximating schemes.Mustafa et al. [5,6] presented an explicit formula for the mask of oddpoints -ary interpolating subdivision schemes.Following are the advantages and disadvantages of interpolating and approximating subdivision schemes in the field of geometric modeling: (i) Interpolating schemes are more useful for engineering applications, especially the schemes with the shape control but approximating schemes do not satisfy the shape control property.
Mathematical Problems in Engineering (ii) Interpolating subdivision schemes have the drawback that, in order to create smoother curves, it is necessary to enlarge the support of the mask.The designers in geometric modeling require subdivision schemes to have their masks with a possibly smaller support and to create good smooth curves.
(iii) Approximating schemes yield smoother curves with smaller support as compared to the interpolating schemes.
In this paper, we want to find the answers of the following questions.Are there attractive characteristics of these schemes in the context of solution of boundary value problems?Do approximating (interpolating) schemes play better role than interpolating (approximation) schemes in this case?To seek answers of these questions, we consider the following interpolating [7][8][9] and approximating [10] subdivision schemes: with order of continuity  4 .Schemes (1) and (2) reproduce polynomial curves of degree nine and three by [11] and [10], respectively.Cardinal supports of these schemes are [−8, 8] and [−6, 6], respectively.We construct collocation algorithms by using the basis functions of the above interpolating and approximating subdivision schemes for the numerical solution of linear fourth order boundary value problems arising in the mathematical modeling of viscoelastic and inelastic flows, deformation of beams, arches, and load bearing members like street lights and robotic arms in multipurpose engineering systems, where elastic members serve as key elements for shedding or transmitting loads and in plate deflection theory and many other areas of engineering and applied mathematics.The mathematical form of these types of problems is given by subject to the boundary condition where () and () are continuous and () ⩾ 0 on [0, 1].
Analytic solution of such type of boundary value problem is possible only in very rare cases.Qu and Agarwal [12,13] solved this type of problems by interpolatory subdivision scheme based collocation algorithm.But they have computed the solution of second order boundary value problems using 6-point interpolating scheme based collocation algorithm.Mustafa and Ejaz [14] solved third order boundary value problems by using 8-point interpolatory subdivision scheme based collocation algorithm.Until now fourth order boundary value problems have not been solved by subdivision based collocation algorithms.This motivates us to find numerical solution of fourth order boundary value problems by interpolating and approximating subdivision schemes based collocation algorithms.The outline of the paper is as follows.In Section 2, we construct subdivision matrices of subdivision schemes (1) and (2) for the computation of eigenvalues and their corresponding (right and left) eigenvectors.Basis functions and their derivatives have been also discussed in this section.In Section 3, subdivision based collocation algorithms for solution of (3) are formulated.Approximation properties of these algorithms are also given in Section 3. In Section 4, numerical examples are presented.Comparison of approximate solutions by interpolating and approximating schemes based collocation algorithms is also given.Conclusion is given in Section 5.

Basic Properties of the Schemes
In this section, we construct subdivision matrices of the schemes defined in (1) and (2) for the computation of eigenvalues and their corresponding eigenvectors.Basis functions of these schemes and their derivatives have also been discussed in this section.

Lemma 1. The fundamental solution (cardinal basis) 𝜙(𝑥) of the subdivision scheme (1) is four times continuously differentiable, supported on [−8, 8], and its derivatives at integers are given by
where    , 0 ⩽  ⩽ 4, are defined in Table 1; the sgn function of a real number  is defined as 's are the column matrices defined as where 0 ≤  ≤ 8 and Lemma 2. The fundamental solution Φ() of subdivision scheme (2) defined in ( 9) is four times continuously differentiable, supported on [−6, 6], and its derivatives at integers are defined as where ]   for 0 ⩽  ⩽ 4 are defined in Table 2; the sgn function of a real number  is defined by (12);   's are the column matrices defined as where   are defined by (14).
From ( 11) and ( 15), we get values of derivatives at the integers given in Tables 3 and 4, respectively.

Description of Numerical Algorithms
In this section, first we formulate two collocation algorithms which are based on interpolating (1) and approximating (2) subdivision schemes for the solution of (3).Then we settle down the boundary conditions to get unique solution.

Collocation Algorithms.
Here we formulate two collocation algorithms based on two subdivision schemes.These collocation algorithms are defined in coming subsections.

Interpolating Collocation Algorithm.
The collocation algorithm based on interpolating scheme (1), say, interpolating collocating algorithm, is given below.In this algorithm, we assume approximate solution  1 () of (3) as where  is the positive integer  ⩾ 8, ℎ = 1/ and   = / = ℎ, and {  } are the unknown to be determined for the solution of (3).The collocation algorithm, together with the boundary conditions to be discussed, is given by with the following type of boundary conditions: where , , , and  are constants.Let   = (  ),   = (  ); then (18) can be written as where Using ( 17) and ( 21) in (20), we get the following  + 1 system of equations: 3.1.2.Approximating Collocation Algorithm.In approximating collocating algorithm (i.e., algorithm based on approximating scheme (2)), we assume approximate solution  2 () of (3) as where  is the positive integer  ⩾ 6, ℎ = 1/ and   = / = ℎ, and {  } are the unknown to be determined for the solution of (3).The collocation algorithm, together with the boundary conditions to be discussed, is given by with the following type of boundary conditions: Equation ( 24) can be written as where Using ( 23) and ( 27) in (26), we get the following  + 1 system of equations: Now we simplify systems ( 22) and (28) in the following theorems.

Boundary Conditions at End Points.
We have two different systems of ( + 1) equations defined by ( 22) and (28).
In order to get unique solution of these systems, we need sixteen more conditions for system (22) and twelve more conditions for system (28).Four conditions can be achieved from boundary conditions given in (4) for both systems of linear equations in which first order derivatives are involved and remaining conditions are achieved by some extrapolation method at the end points.First we find the approximation of the first derivative by difference operators and after that we define the extrapolation method at end points for both systems of linear equations.

Approximation of Derivative Boundary Conditions.
In this section, we approximate the derivative boundary conditions by difference operators.Since approximation order of interpolating scheme (1) and approximating scheme ( 2) is ten and four, respectively, so we approximate derivative boundary conditions at end points with approximation orders ten and four for interpolating and approximating collocation algorithms.
If we use interpolating collocation algorithm for the solution of (3), then approximation of derivative conditions at ends point is defined as and if we use approximating collocation algorithm for the solution of (3), then approximation of derivative conditions at end points is defined as (56)

Adjustment of Boundary Conditions.
Still we need twelve and eight more conditions for systems ( 22) and (28), respectively, to get stable systems for the solution of (3).For this we made some adjustment of boundary conditions for systems ( 22) and (28), which are defined below.
Case 1.If we use interpolating collocation algorithm for the approximate solution of (3), then we define six conditions at left end points and six conditions at the right end points.Since subdivision scheme (1) reproduces nine-degree (i.e., tenth order) polynomials, so we define boundary conditions of order ten for solution of (22).For simplicity only left end points  −7 ,  −6 ,  −5 ,  −4 ,  −3 ,  −2 are discussed and the values of right end points  +2 ,  +3 ,  +4 ,  +5 ,  +6 ,  +7 can be treated similarly.
The values  −7 ,  −6 ,  −5 ,  −4 ,  −3 ,  −2 can be determined by the nonic polynomial () interpolating (  ,   ), 2 ≤  ≤ 7. Precisely, we have where Since by ( 17) by replacing   by −  , we have Hence the following boundary conditions can be employed at the left end: (62) Finally, we get the following new system of ( + for the solution of (3) then we need eight more conditions.So in this case, we define four extra conditions at the left end points and four conditions at the right end points by some extrapolation method.Since the subdivision scheme reproduces cubic (i.e., fourth order) polynomial, so we define quartic polynomial for the adjustment of boundary treatment.The values  −4 ,  −3 ,  −2 ,  −1 are determined by the quartic polynomial () interpolating (  ,   ).This polynomial is defined as where Since by (23)  2 (  ) =   for  = 1, 2, 3, 4, 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  +1 = ( +1 ),  =  + 1,  + 2,  + 3,  + 4, and So we have the following boundary conditions at the right end: Finally, we get a following new system of ( + 13) linear equations with ( + 13) unknowns {  }, in which  + 1 equations are obtained from (41) and (47), four equations from boundary conditions (25), and eight from boundary conditions (66) and (68).

Stable Systems of Linear Equations.
In this section, we present stable systems of linear equations for both interpolating and approximating collocation algorithms.

Numerical Examples and Comparison
In this section, the interpolating and approximating collocation algorithms described in Section 3 are tested on the problems given below.Absolute errors between exact and approximate solutions are also calculated.For the sake of comparisons, we also tabulated the results in this section.Graphical illustrations of solutions are presented.

Numerical Examples.
Here we find the numerical solutions of some of the boundary value problems arising in the mathematical modeling of viscoelastic and inelastic flows and so forth.Here we present the numerical solution of the above problem by interpolating and approximating collocation algorithms.
Solution by Interpolating Collocation Algorithm.In this method, by solving the system of linear equations (73) at  = 10, we obtain the approximate solution (17)

Example 2 .
Consider the following fourth order linear boundary value problem, iv () = ( 4 + 14 3 + 49 2 + 32 − 12)   0 ≤  ≤ 1 (94) with  (0) =  (1) =   (0) =   (1) ,(95)corresponds to the bending of a thin beam clamped at both ends.The unique solution of (94) is () =  2 (1 − ) For above eigenvalues   , the eigenvectors    and    that satisfies  1    =      and      1 =      are called right and left eigenvectors of the matrix  1 , respectively.We can also define the right ]   and left ]   eigenvectors of  2 in a similar way.The normalized left and right eigenvectors corresponding to first five eigenvalues of  1 and  2 are given in Tables

Table 1 :
Eigenvalues and eigenvectors of the matrix  1 .

Table 5 :
Determinants of the matrices.