A Study on Fractal Interpolation in Shape

In Chapter 4, a fractal cubic spline is used to obtain the numerical solutions of the self-adjoint singularly perturbed boundary value problems with Dirichlet and Neumann boundary conditions. In Chapter 7, the numerical solutions of the self-adjoint singularly perturbed boundary value problems are obtained using fractal quintic splines.

Introduction

Overview

Iterated Function Systems

Fractal Interpolation Functions

The operator defined in (1.3) is called the Read-Bajraktarevi´c operator and f satisfies the conditions given in Theorem 1.3.1.

Smooth Fractal Interpolation Functions

The following statement is an immediate consequence of the above statement and states the relationship between the IFS of f and the IFS of ˆf. By repeating Proposition 1.4.1, the conditions for αi and ri of the IFS (1.4) are obtained, so that the corresponding FIF belongs to the class Cr.

Fractal Interpolation Surfaces

Shape Preserving FIF and FIS

Various numerical schemes have been developed to preserve the forms of bivariate data. We can also see in the references various numerical schemes that have been developed to preserve the shapes of bivariate data.

Spline Functions

In general, in the finite element method, we seek an approximate solution u of the differential equation in the form For example, De Boor [52] and Sakai [121] used the cubic spline to obtain numerical approximations of GDPs.

Motivation of Present Work

Unlike traditional interpolation, by controlling the scaling factors inherent to fractal splines, we can obtain a definite derivative of non-differentiable fractal splines on a finite or dense subset of the interval. The order of convergence of the numerical method developed by the fractal splines and the order of convergence of the numerical method developed by the corresponding non-recursive splines are the same.

Organization of Present Work

Continuity conditions of the fractal cubic spline are used to obtain the numerical methods for these BVPs. Continuity conditions of the fractal quintic spline are used to obtain the numerical scheme and the developed numerical scheme has fourth-order convergence.

FIF with Two Families of Shape Parameters

Convergence analysis
Shape preserving aspects of FIF
Numerical examples

Taking the arbitrary scale factors and shape parameters, the FIF from Figure 2.1(a) is drawn. The values of the scale factors and the shape parameters used in drawing various FIFs in Figures 2.5 and 2.6 are given in Table 2.3.

Figure 2.1: Positivity preserving interpolation.

FIF with Three Families of Shape Parameters

Shape preserving interpolation

By perturbing the scale factor α2 with respect to Figure 2.8(b), the FIF given in Figure 2.8(c) is constructed. Taking arbitrary scale factors and shape parameters plots FIF (Figure 2.10(a)), but the FIF does not preserve the monotonicity of the data. Therefore, the parameters are selected based on Theorem 2.2.3. Using these parameters, FIF (Figure 2.10(b)) is plotted.

Accordingly, scale factors and shape parameters are taken according to Theorem 2.2.4, using these parameters, FIF is plotted given in Figure 2.12(b) which is convex. The classical rational cubic spline given in Figure 2.12(e) is plotted using scale factorsαi = 0 for all i= 1,2.

Figure 2.8: Positive preserving interpolation.

Conclusion

In this chapter, we developed a new FIS for preserving the shapes of the bivariate data lying on a rectangular grid. This transfinite interpolation method plays a crucial role in shape-preserving bivariate interpolation due to the fact that the shape properties of the transfinite interpolation surface follow trivially from the corresponding shape properties of the networks of boundary curves [28]. In Section 3.1, the rational cubic FIFs are constructed to approximate the original function along the gridlines of the interpolation domain.

Then, a rational cubic FIS is constructed using rational cubic FIFs and mixing functions. These constraints on scaling factors and shape parameters ensure positivity, monotonicity, and convexity of the constructed FIS.

Fractal Interpolation Surfaces

Construction of fractal boundary curves
Construction of rational cubic FIS
Error analysis

In this section, the uniform error bound between the rational cubic FIS Φ and the original function S ∈ C4(D) is estimated using the classical rational cubic interpolation surface C. Then using error bounds obtained along the grid lines, an error bound between the rational cubic FIS Φ and the original function S. The following statement shows the error limit between the classical rational cubic spline and the original function.

Let Φ and C be the corresponding rational cubic FIS and the classical rational cubic interpolation surface, respectively. Similarly, expanding the classical rational cubic interpolation surface gives C ∈ C1(D) in the Taylor series around the point (x, yj)∈Di,j.

Shape Preserving FIS

Positivity preserving interpolation
Constrained interpolation
Monotonicity preserving interpolation
Convexity preserving interpolation

Then the conditions on the scale factors and the shape parameters for Bj(x) are to satisfy the positivity. Then the conditions on the scale factors and the shape parameters for Bi∗(y) are to satisfy the positivity. Then the conditions on the scale factors and the shape parameters for Bj(x) are monotonically increasing.

Then the conditions for the scaling factors and shape parameters for Bi∗(y) to be monotonically increasing are. Φ is monotonically increasing if the scaling factors and shape parameters satisfy the conditions given in Theorems 3.2.7 and 3.2.8.

Numerical Examples

Form parameters matrices inx direction Figs Form parameters matrices iny direction Figs. b) Positive classical rational cubic interpolation surface. To overcome this, scale factors and shape parameters are taken according to constraints given in Propositions 3.2.4 and 3.2.5, with the help of these parameters rational cubic FIS is constructed which is shown in Figure 3.3(b). The scale factors and the shape parameters are calculated according to Theorems 3.2.7 and 3.2.8, using these, monotonic FIS is constructed which is shown in Figure 3.4(a).

Taking all scaling factors as zero, the classical monotone interpolation surface is constructed and is given in Figure 3.4(b). Therefore, the scaling factors and shape parameters are bounded according to the constraints given in Theorems 3.2.10 and 3.2.11.

Table 3.4: Shape parameter matrices for positive interpolation with u u u = [1.5] 3×4 , u u u ∗ = [1.5] 4×3 vvv = [1] 3×4 and v vv ∗ = [1] 4×3 .

Conclusion

Bawa and Natesan [13] established a numerical method using the cubic spline method for numerical solutions of self-adjoint perturbed BVPs with mixed boundary conditions. The discretization equations given by the continuity conditions are second order and therefore the resulting cubic fractal line method is second order. The present chapter is organized as follows: In Section 4.1, the singularly perturbed BVPs with Dirichlet boundary conditions are solved using the fractal cubic spline method.

In Section 4.2, numerical solutions of singly perturbed BVPs with the Neumann boundary conditions are obtained using the fractal cubic spline method. The obtained numerical results are tabulated and compared with the numerical results of cubic spline method.

Dirichlet Boundary-Value Problem

Numerical results are given to demonstrate the practical applicability of the proposed method, and the numerical results are compared with those of the cubic splicing method. In the following proposition, suitable conditions for h and α are derived to prove the diagonal dominance of the matrix A. Thus, the conditions prescribed in Proposition 4.1.1 guarantee the strict diagonal dominance of the matrix A.

It can be observed that the conditions given in Theorem 4.1.1 ensure that the row sums of the matrix A are positive. In this section, to demonstrate the applicability of the developed method computationally, we consider the two numerical examples whose exact solutions are known.

Table 4.1: Maximum point-wise error and order of convergence corresponding to Exam- Exam-ple 4.1.1.

Neumann Boundary-Value Problem

Using a similar procedure followed to calculate T0(h), we obtain the truncation error associated with the difference equation given in (4.16) as. In the following proposition, h and scaling factor α are suitably bounded so that the matrix A would be strictly diagonally dominant. The matrix A is thus strictly diagonally dominant when h and α fulfill the conditions stated in theorem 4.2.1.

We can see that the conditions stated in Proposition 4.2.1 ensure that the row sums of the matrix A are positive. In this section, the presented numerical method is tested on the following singularly perturbed two-point BVP.

Conclusion

In this chapter, we consider the non-self-adjoint BVPs and numerical solutions of these BVPs are obtained. Aziz and Khan [9] used spline in compression to solve the non-self-adjoint singular perturbed BVPs and this method has quadratic convergence. In this chapter, we have used fractal cubic spline method to solve the non-self-adjoint uniquely perturbed BVPs, and the proposed method has second-order convergence.

For example, Chawla and Katti [40] used three-point finite difference method to solve the singular BVPs. Continuity conditions of the fractal cubic spline are taken as the discretization equations to solve these BVPs.

Singularly Perturbed Nonself-Adjoint Boundary- Value Problems

Error analysis

Many researchers tried to solve the singular BVPs by considering series expansion procedures in the vicinity of the singularity and then solve the regular BVP in the rest of the interval using any numerical method. To test the proposed method, the numerical results are tabulated and compared with the numerical solutions corresponding to the cubic spline method. We substitute the original function valueu(xi) instead of the approximate valueUi in (5.2) and thus obtain the truncation error Ti(h), i = 1,2,.

Thus, we have the second-order convergence method to obtain the numerical solutions of the BVP given in (5.1). The maximum pointwise error and the order of convergence corresponding to the cubic spline are given in Table 5.4.

Linear Singular Boundary-Value Problems

A is invertible because kA−Bk∞= max{C1, C2} and from (5.16) it can be seen that the eigenvalues of A are non-zero when α is sufficiently small. The maximum point error and order of convergence corresponding to the cubic spline method are given in Table 5.8.

Non-Linear Singular Boundary-Value Problems

Error analysis

Using the procedure followed in Section 5.2, it can be seen that, when h is sufficiently small, we find that the matrix B(r) is monotonic. The numerical solutions of the non-linear singular BVPs are calculated as follows: For each fixed N, by taking initial approximation U0(0), U1(0). Once the criterion is satisfied, we consider U(r+1) as the numerical solution U for the nonlinear singular BVPs.

To calculate the numerical solutions of the following nonlinear singular BVPs, we take δ = 10−15 and 8 iterations. The maximum pointwise error and the order of convergence corresponding to the cubic spline method are tabulated in table 5.10.

Conclusion

In this chapter, a fractal non-polynomial cubic spline method is developed to obtain the numerical solutions for both self-adjoint and non-self-adjoint singularly perturbed BVPs. The numerical method developed in Chapter 4 for the self-adjoint, singularly perturbed BVPs and the numerical method developed in Chapter 5 for the non-self-adjoint, singularly perturbed BVPs are of second order convergence. To obtain the numerical approximations for the self-adjoint singularly perturbed BVPs, different types of non-polynomial splines have been used.

Khan and Khandelwal [82] developed a sixth-order numerical convergence method using non-polynomial sextic spline to obtain the numerical solutions for the self-adjoint singularly perturbed BVPs. In Section 6.1 we took the continuity conditions of the fractal non-polynomial cubic spline as the discretized equations for computing the numerical solutions of the self-adjoint singularly perturbed BVPs.

Self-Adjoint Boundary-Value Problems

To obtain the numerical solutions for non-self-adjoint perturbed BVPs, Aziz and Khan [9] developed the second-order convergence method using spline in compression. By modifying the Aziz and Khan method, Bawa [12] developed the fourth-order convergence method using splines in compression. The convergence analysis of the proposed method is determined and the computational efficiency of the method is verified through numerical examples.

In Section 6.2, we solved the non-self-consistent singularly perturbed BVPs by the fractal non-polynomial cubic spline method and the fourth-order convergence is achieved by introducing the parameter in the approximations of derivative values.