Chapter 1 Introduction
1.9 Organization of Present Work
Chapter 1 consists of brief introduction about IFS, FIF, FIS and brief literature of shape preserving interpolation of univariate and bivariate data. Also, a brief litera- ture review of the numerical approximation methods of BVPs of ordinary differential equations is given.
In Chapter 2, at beginning, we construct the rational cubic FIF that contains two families of shape parameters to interpolate the univariate data. For an interpolation method to be effective, the interpolant corresponding to a data set should converge to the corresponding data defining function. For this purpose, we have derived the uniform error bound between the FIF and the original function of the class C3 that defines the data set. To preserve the geometric properties of the data, namely, positivity, monotonicity and convexity, the scaling factors and shape parameters are restricted so that constructed FIF would preserve these properties. Also, constrained interpolation problem of the developed FIF is also discussed. When all the scaling factors are zero, the fractal interpolation method reduces into the method developed in [124].
Next, we provide the construction of rational cubic FIF that contains three families of shape parameters to interpolate the univariate data. To study the effectiveness of the constructed FIF, the uniform error bound between the constructed FIF and the original function that belongs to the class C2 is derived. The scaling factors and shape parameters are constrained so that the constructed FIF preserves the geometric natures of the univariate data namely positivity, monotonicity and convexity. The presented shape preserving interpolation scheme generalizes the traditional shape preserving in- terpolation scheme studied in [1].
In Chapter 3, to interpolate the bivariate data that lies on the rectangular grid, a FIS is constructed. First, the fractal boundary curves are constructed along the grid lines. Then with the help these fractal boundary curves and the blending functions, the FIS is constructed. On each rectangular patch, the constructed FIS contains four scaling factors and 12 shape parameters. The scaling factors and shape parameters are restricted to get the positivity, monotonicity and convexity preserving FIS. Also
the shape parameters and scaling factors are restricted so that constructed FIS would lie above the plane whenever the surface data lies above the plane. Also to see the effectiveness of the constructed FIS, the uniform error bound between the FIS and the original function that belongs to the class C4 is developed.
In Chapter 4, we consider the self-adjoint singularly perturbed BVPs of the form
−εu00(x) +q(x)u(x) = f(x), x∈(0,1), u(0) =η0, u(1) =η1,
and
−εu00(x) +q(x)u(x) = f(x), x∈(0,1),
−u0(0) =η0, u0(1) =η1,
where 0 < ε ≤ 1, q and f are sufficiently smooth functions in [0,1] and q(x) > 0, x∈[0,1]. These BVPs exhibit boundary layer at both ends of the interval.
Continuity conditions of the fractal cubic spline are used to obtain the numerical methods for these BVPs. To see the efficiency, error analysis of the developed methods are established. The discretized equations given by continuity conditions are second- order and hence the resulting fractal cubic spline methods are also second-order. The numerical results are tabulated and compared with the numerical results of the cubic spline method.
In Chapter 5, we consider the nonself-adjoint singularly perturbed BVPs of the form
εu00(x) =p(x)u0(x) +q(x)u(x) +f(x), x∈(0,1), u(0) =η0, u(1) =η1,
where 0< ε≤1,p,q,f are sufficiently smooth functions,p(x)>0 orp(x)<0,q(x)>0 in [0,1]. Solutions of these BVPs exhibit boundary layer atx= 0 orx= 1 ifp(x)<0 or p(x)>0 respectively. We used the continuity conditions of the fractal cubic spline to get the numerical method for these BVPs and convergence analysis of the developed method is established and it is shown that the proposed method has second-order convergence.
Next, we consider the linear singular BVPs of the form
u00(x) + k
xu0(x)−q(x) = f(x), x∈(0,1), u0(0) = 0, u(1) =η1.
where k = 1,2, the functions q, f are sufficiently smooth, q(x) > 0 in [0,1]. This problem has singularity at x = 0. Second-order convergent numerical method obtained via fractal cubic spline has been used to get the numerical solutions for these BVPs.
Next, we consider the non-linear singular BVPs of the form
u00(x) + k
xu0(x) =F(x, u(x)), x∈(0,1), u0(0) = 0, u(1) =η1,
where k = 1,2. Also for (x, u(x)) ∈ D = {0 ≤ x ≤ 1, −∞ < u(x) < ∞}, the functions F, ∂F/∂u are continuous, ∂F/∂u ≥ 0 on D and ∂F/∂u > 0 on D◦ = {0 <
x < 1, −∞ < u(x) < ∞}. Fractal cubic spline and quasi-linearization technique are used to get the numerical approximations for these BVPs. Convergence analysis of the proposed methods are established and it tells that the constructed method has second-order convergence.
In Chapter 6, we consider the self-adjoint singularly perturbed BVPs and the nonself-adjoint singularly perturbed BVPs. The method developed to get the numerical approximations for the self-adjoint singularly perturbed BVPs in Chapter 4 and the method proposed in Chapter 5 to get the numerical solutions for the nonself-adjoint singularly perturbed BVPs are second-order convergent. In order to obtain higher-order convergent methods, in this chapter we have used the fractal non-polynomial cubic spline to get the numerical solutions for these BVPs. The continuity conditions of the fractal non-polynomial cubic spline are used to construct the numerical method for these BVPs.
Convergence analysis of the developed methods are carried out and it is shown that the proposed methods have fourth-order convergence.
In Chapter 7, at the beginning, we consider the BVPs of the form
−εu00(x) +q u(x) = f(x), x∈(0,1), u(0) =η0, u(1) =η1,
where 0< ε≤1,q >0,f is sufficiently smooth function in [0,1]. Continuity conditions of the fractal quintic spline are used to get the numerical scheme and the developed numerical scheme has fourth-order convergence.
Next, we consider the BVP of the form
u00(x) +F(x, u(x)) = 0, x∈(0,1), u(0) =η0, u(1) =η1.
We assume that for (x, u(x)) ∈ D = {0 ≤ x ≤ 1, −∞ < u(x) < ∞}, the func- tions F and ∂F/∂u are continuous. This problem possess a unique solution pro- vided sup
(x,u)∈D
∂F/∂u < π2 [42]. We assume that ∂F/∂u ≤ 0 on D and ∂F/∂u < 0 on D◦ = {0 < x < 1, −∞ < u(x) < ∞}. Using quasilinearization technique, we con- vert the non-linear BVP into sequence of linear BVPs. Then each of these linear BVPs have been solved using the method obtained from fractal quintic spline. The developed numerical method has fourth-order convergence.
In Chapter 8, we consider the BVPs of the forms
u(4)(x) +q(x)u(x) =f(x), x∈(0,1),
u(0) =η0, u(1) =η1, u00(0) = ˆη0, u00(1) = ˆη1,
and
u(4)(x) +q(x)u(x) = f(x), x∈(0,1),
u(0) =η0, u(1) =η1, u0(0) = ˆη0, u0(1) = ˆη1,
where the functionsq andf are continuous in [0,1]. Continuity conditions of the fractal quintic spline are used to develop the numerical methods for these problems. Truncation errors corresponding to these methods are derived. The developed methods have second- order convergence.
Chapter 2
Shape Preserving Rational Cubic Fractal Interpolation Functions
In this chapter, new FIFs in the field of shape preserving interpolation are constructed.
The developed FIFs areC1-continuous functions and they can be utilized for preserving all the three fundamental shape properties namely positivity, monotonicity and convex- ity. The derivatives of the developed FIFs having irregularity in a finite subset or a dense subset of the interpolation interval. The proposed schemes have some interesting features.
• The proposed methods are best tool to approximate a function which is continuous and its derivatives are irregular.
• When all the scaling factors are zero, FIFs obtained from proposed methods, reduces into a classical rational cubic splines.
• The proposed methods are equally applicable for the data with derivatives or data without derivatives.
• In the developed methods, extra knots are not needed to get the shape preserving interpolants.
• In the proposed schemes, shape preserving fractal interpolant is unique for the fixed scaling factors and the fixed shape parameters. By changing the scaling factors
and the shape parameters, infinitely many shape preserving fractal interpolants can be obtained.
The FIF with two families of shape parameters is constructed and shape preserving aspects of the constructed FIF are analyzed in Section 2.1. To construct the FIF, we consider the IFS that contains the rational functions where the numerator of the rational functions contain cubic polynomials and the denominator of rational functions contain the preassigned quadratic polynomials each containing 2 shape parameters. When all the scaling factors are zero, the developed rational cubic FIF reduces to the classical rational cubic interpolant introduced by Sarfraz et al. [124]. The convergence analysis of the constructed FIF to an original function which belongs to the class C3 is derived.
Sufficient conditions on the shape parameters and the scaling factors are derived to preserve the positivity, monotonicity and convexity nature of a univariate data. Also, the constrained interpolation problem of the constructed FIF is discussed. We consider some numerical examples to test the applicability of the developed interpolation scheme.
In Section 2.2, to preserve the shapes of the univariate data, rational cubic FIF with three families of the shape parameters is constructed. The rational cubic FIF is constructed with the help of the IFS involving rational functions. The numerator of the rational functions contain cubic polynomials and the denominator of rational functions contain the preassigned quadratic polynomials each containing 3 shape parameters. To see effectiveness of the interpolantion scheme, error analysis of the constructed FIF to an original function that belongs to C2 is derived. Parameters are constrained so that the constructed FIF would preserve the positivity, monotonicity and convexity of a prescribed set of data points. The shape preserving aspects of the FIF are implemented through numerical examples. We get the classical rational cubic interpolant introduced by Abbas et al. [1] when all the scaling factors are zero.