Fractional Calculus of Fractal Interpolation Function on [0,𝑏](𝑏>0)

Abstract

The paper researches the continuity of fractal interpolation function’s fractional order integral on [0,+∞) and judges whether fractional order integral of fractal interpolation function is still a fractal interpolation function on [0,𝑏](𝑏>0) or not. Relevant theorems of iterated function system and Riemann-Liouville fractional order calculus are used to prove the above researched content. The conclusion indicates that fractional order integral of fractal interpolation function is a continuous function on [0,+∞) and fractional order integral of fractal interpolation is still a fractal interpolation function on the interval [0,𝑏].

1. Introduction

Fractal geometry is a subject in which very irregular and complex phenomena and pictures in nature are researched. From the process of fractal development, there have been some effective methods used in studying fractals so far. For example, Mandelbrot [1–5] applied concept of fractal dimension to describe the roughness of fractal curves and fractal surfaces. Barnsley [6, 7] and Massopust [8, 9] proposed that fractal interpolation curve (refer to Figure 1) and fractal interpolation surface (refer to Figure 2) can be applied in fitting and analyzing the shape of the natural graphs. Feng et al. [10–12] proposed concept and principle of fractal variation and used it in estimating the Minkowski dimension of fractal surface and describing the roughness of fractal surface. Li and Wu [13] applied wavelet analysis in researching fractal geometry. Ran and Tan [14] and Mark [15] discussed the relationship between Fourier analysis and wavelets analysis. Generally, researchers always attempt to research fractals through classical integer order calculus. However, it is very rare that fractals are researched through fractional order calculus. Because classical integer order calculus researches smooth curves and surfaces, it almost cannot be applied in analyzing and dealing with fractal problems, while fractional calculus is regarded as an important and effective tool applied in researching fractal interpolation function.

Figure 1 

Fractal interpolation curve.

Figure 2 

Fractal interpolation surface.

In order to discuss property of fractal interpolation function’s fractional order integral, the following content is discussed that the continuity of fractal interpolation function's fractional integral on [0,+∞) and judge whether fractional integral of fractal interpolation function is still a fractal interpolation function on [0,𝑏](𝑏>0) or not. So iterated function system, concepts, and theorems about Riemann-Liouville fractional order integral are used to prove the above problems. The results indicate that fractional order integral of self-affine transformation’s fractal interpolation function is continuous on [0,+∞) and it is still a fractal interpolation function on [0,𝑏](𝑏>0).

2. Main Concepts and Lemmas

Definition 1 (see [16]). Let 𝑣>0, and function 𝑓 is a continuous function on interval (0,+∞) and can be integral on any bounded subinterval included in [0,+∞); then the following formula 𝐼^−𝑣𝑓(𝑥)=1Γ(𝑣)^𝑥₀(𝑥−𝑡)^𝑣−1𝑓(𝑡)𝑑𝑡 (1) is called 𝑣-order Riemann-Liouville fractional integral of 𝑓.

Definition 2 (see [17]). A “hyperbolic” iterated function system consists of a complete metric space (𝑋,𝑑) together with a finite set of contraction mappings 𝑤_𝑖∶𝑋→𝑋, with respective contractivity mappings factors 𝑠_𝑖, for 𝑖=1,2,…,𝑛. The abbreviation “IFS” is used for “iterated function system.” The notation for the IFS just announced is {𝑋;𝑤_𝑖,𝑖=1,2,…,𝑛} and contractivity factor is 𝑠=max{𝑠_𝑖∶𝑖=1,2,…,𝑛}.

Definition 3 (see [7]). Let {(𝑥_𝑖,𝑦_𝑖)∈𝑅²;𝑖=0,1,2,…,𝑛} be a set of points, where 𝑥₀<𝑥₁<𝑥₂<⋯<𝑥_𝑛. An interpolation function corresponding to this set of data is a continuous function 𝑓∶[𝑥₀,𝑥_𝑛]→𝑅 such that 𝑓𝑥_𝑖=𝑦_𝑖,𝑖=1,2,…,𝑛. (2)

The points (𝑥_𝑖,𝑦_𝑖) are called the interpolation points. It is called that the function of 𝑓 interpolates the data and that the graph of 𝑓 passes through the interpolation points.

Lemma 4 (see [17]). Let 𝑛 be a positive integer greater than 1. Let {𝑅²;𝑤_𝑖,𝑖=1,2,…,𝑛} denote the IFS defined above, associated with the data set 𝑥_𝑖,𝑦_𝑖∈𝑅²;𝑖=0,1,2,…,𝑛. (3) Let the vertical scaling factor 𝑑_𝑖 obey 0≤𝑑_𝑖<1 for 𝑖=1,2,…,𝑛. Then there is a metric 𝑑 on 𝑅², equivalent to the Euclidean metric, such that the IFS is hyperbolic with respect to 𝑑. In particular, there is a unique nonempty compact set 𝐺⊂𝑅², such that 𝐺=^𝑛_𝑖=1𝑤_𝑖(𝐺). (4)

In particular, an IFS of the form {𝑅²;𝑤_𝑖,𝑖=1,2,…,𝑛} is considered, where the mapping is an affine transformation of the special structure𝑤_𝑖𝑥𝑦=𝑎_𝑖0𝑐_𝑖𝑑_𝑖𝑥𝑦+𝑒_𝑖𝑓_𝑖. (5)

The transformations are constrained by the data according to 𝑤_𝑖𝑥₀𝑦₀=𝑥_𝑖−1𝑦_𝑖−1,𝑤_𝑖𝑥_𝑛𝑦_𝑛=𝑥_𝑖𝑦_𝑖for𝑖=1,2,…,𝑛, (6)

and 𝑎_𝑖, 𝑒_𝑖, 𝑐_𝑖, 𝑓_𝑖 can be solved from (5) and (6) in terms of the data and vertical scaling factor 𝑑_𝑖 as follows: 𝑎_𝑖=𝑥_𝑖−𝑥_𝑖−1𝑥_𝑛−𝑥₀,𝑒_𝑖=𝑥_𝑛𝑥_𝑖−1−𝑥₀𝑥_𝑖𝑥_𝑛−𝑥₀,𝑐_𝑖=𝑦_𝑖−𝑦_𝑖−1𝑥_𝑛−𝑥₀−𝑑_𝑖𝑦_𝑛−𝑦₀𝑥_𝑛−𝑥₀,𝑓_𝑖=𝑥_𝑛𝑦_𝑖−1−𝑥₀𝑦_𝑖𝑥_𝑛−𝑥₀−𝑑_𝑖𝑥_𝑛𝑦₀−𝑥₀𝑦_𝑛𝑥_𝑛−𝑥₀. (7)

Lemma 5 (see [18]). Suppose 𝐹 is a set of continuous functions which satisfy 𝑓∶[𝑥₀,𝑥_𝑛]→𝑅 and 𝑓(𝑥₀)=𝑦₀,𝑓(𝑥_𝑛)=𝑦_𝑛. The metric is defined by the following formula: 𝑑𝑓,𝑔=max𝑓(𝑥)−𝑔(𝑥),𝑥∈𝑥₀,𝑥_𝑛,∀𝑓,𝑔∈𝐹. (8)
Then (𝐹,𝑑) is a complete metric space. Let the real numbers 𝑎_𝑖, 𝑐_𝑖, 𝑒_𝑖, 𝑓_𝑖, be defined by (7). Define a mapping 𝑇∶𝐹→𝐹 by 𝑇𝑓(𝑥)=𝑐_𝑖𝐿_𝑖⁻¹(𝑥)+𝑑_𝑖𝑓𝐿_𝑖⁻¹(𝑥)+𝑓_𝑖,𝑥∈𝑥_𝑖−1,𝑥_𝑖,𝑖=1,2,…,𝑛, (9) where 𝐿_𝑖∶[𝑥₀,𝑥_𝑛]→[𝑥_𝑖−1,𝑥_𝑖] is the invertible transformation 𝐿_𝑖(𝑥)=𝑎_𝑖𝑥+𝑒_𝑖,𝐿_𝑖⁻¹(𝑥)=𝑥−𝑒_𝑖𝑎_𝑖,𝐿_𝑖⁻¹𝑥_𝑖−1=𝑥₀,𝐿_𝑖⁻¹𝑥_𝑖=𝑥_𝑛, (10) and then 𝑇𝑓 is continuous on the interval [𝑥_𝑖−1,𝑥_𝑖] and 𝑇 is a contraction mapping on (𝐹,𝑑), so 𝑇 possesses a unique fixed point in 𝐹. That is, there exists a function 𝑓∈𝐹 such that 𝑇𝑓=𝑓,∀𝑓∈𝐹. (11)

3. The Continuity of Fractal Interpolation Function’s Fractional Order Integral on the Interval [0,+∞)

Lemma 6. If 𝑓 is a continuous function on the interval [0,+∞) and 0<𝑣<1, then 𝐼^−𝑣𝑓(𝑥) is a continuous function on [0,+∞) too.

Proof. Since 𝐼^−𝑣𝑓(𝑥+Δ𝑥)−𝐼^−𝑣𝑓(𝑥)=1Γ(𝑣)₀^𝑥+Δ𝑥(𝑥+Δ𝑥−𝑡)^𝑣−1𝑓(𝑡)−1𝑑𝑡Γ(𝑣)^𝑥₀(𝑥−𝑡)^𝑣−1𝑓(𝑡)=1𝑑𝑡Γ(𝑣)₀^Δ𝑥(𝑥+Δ𝑥−𝑡)^𝑣−1𝑓(𝑡)+𝑑𝑡_{𝑥+Δ𝑥Δ𝑥}(𝑥+Δ𝑥−𝑡)^𝑣−1𝑓(𝑡)−𝑑𝑡_𝑥0(𝑥−𝑡)^𝑣−1𝑓(𝑡)=1𝑑𝑡Γ(𝑣)₀^Δ𝑥[𝑥+(Δ𝑥−𝑡)]^𝑣−1𝑓(𝑡)+𝑑𝑡_𝑥0𝑥−𝑦^𝑣−1𝑓−𝑦+Δ𝑥𝑑𝑦_𝑥0(𝑥−𝑡)^𝑣−1𝑓(𝑡)𝑑𝑡, (12) then 𝐼^−𝑣𝑓(𝑥+Δ𝑥)−𝐼^−𝑣𝑓(𝑥≤1)|Γ(𝑣)|₀^Δ𝑥𝑥^𝑣−1+𝑀𝑑𝑡_𝑥0(𝑥−𝑡)^𝑣−1𝑓()(𝑡≤1𝑡+Δ𝑥−𝑓)𝑑𝑡|Γ(𝑣𝑥)|^𝑣−1𝑀Δ𝑥+𝜀^𝑥₀(𝑥−𝑡)^𝑣−1=1𝑑𝑡|Γ(𝑣𝑥)|^𝑣−1𝜀𝑀Δ𝑥+𝑣𝑥^𝑣⟶0, (13) where 𝑀=max_{𝑡∈[0,Δ𝑥]}|𝑓(𝑡)|, so 𝐼^−𝑣𝑓(𝑥) is a continuous function on the interval [0,+∞).

Corollary 7. Suppose 𝑓(𝑥) is a fractal interpolation function on the interval [0,+∞); then 𝐼^−𝑣𝑓(𝑥) is continuous on [0,+∞) too.

Proof. Since fractal interpolation function of affine transformation is a continuous function on [0,+∞), 𝑓(𝑥) is continuous on [0,+∞). According to Lemma 6, 𝐼^−𝑣𝑓(𝑥) is a continuous function on [0,+∞) too.

Corollary 8. Suppose 𝑓(𝑥) is a fractal interpolation function of affine transformation on the interval [𝑎,𝑏](0<𝑎<𝑏<+∞); then 𝐼^−𝑣𝑓(𝑥) can be integrated on [𝑎,𝑏].

Proof. From Corollary 7, and since continuous function on finite closed interval is an integrated function, the result of Corollary 8 is right.

4. Judgement Theorem of Fractal Interpolation Function’s Fractional Integral on [0,𝑏](𝑏>0)

Theorem 9. If 𝑓(𝑥) is a fractal interpolation function of affine transformation on the interval [0,𝑏](𝑏>0), then 𝐼^−𝑣𝑓(𝑥) is a fractal interpolation function of affine transformation on the interval [0,𝑏] too.

Proof. Forall𝑏∈(0,+∞), consider the interval [0,𝑏], for 0<𝑣<1, 𝑥∈[𝑥_𝑖−1,𝑥_𝑖](𝑥_𝑖=(𝑖/𝑛)𝑏,𝑖=1,2,…,𝑛). Let iterated function system (IFS) be 𝐿_𝑖(𝑥)=1𝑛𝑥+𝑖−1𝑛𝐹𝑏,_𝑖𝑥,𝑦=𝑐_𝑖𝑥+𝑑_𝑖𝑦+𝑓_𝑖, (14) so 𝐿_𝑖⁻¹(𝑥)()𝐼=𝑛𝑥−𝑖−1𝑏,_−𝑣𝑓(𝑥)=1Γ(𝑣)^𝑥₀(𝑥−𝑡)^𝑣−1𝑓(𝑡)=1𝑑𝑡Γ(𝑣)^𝑘−1_𝑖=1^𝑥𝑖_𝑥𝑖−1(𝑥−𝑡)^𝑣−1𝑓(𝑡)+1𝑑𝑡Γ(𝑣)^𝑥_𝑥𝑘−1(𝑥−𝑡)^𝑣−1𝑓(𝑡)=1𝑑𝑡Γ(𝑣)^𝑘−1_𝑖=1^𝑏₀1𝑥−𝑛𝑦−𝑖−1𝑛𝑏^𝑣−1×𝑓𝐿_𝑖𝑦1𝑛𝑑𝑦Let𝑡=𝐿_𝑖𝑦+1Γ(𝑣)^𝐿𝑘−1₀^(𝑥)1𝑥−𝑛𝑦−𝑘−1𝑛𝑏^𝑣−1×𝑓𝐿_𝑘𝑦1𝑛=1𝑑𝑦Γ(𝑣)𝑛^𝑣𝑘−1_𝑖=1^𝑏₀𝑛𝑥−𝑦−(𝑖−1)𝑏^𝑣−1×𝑐_𝑖𝑦+𝑑_𝑖𝑓𝑦+𝑓_𝑖+1𝑑𝑦Γ(𝑣)𝑛^{𝑣𝐿𝑘−1}₀^(𝑥)𝑛𝑥−𝑦−(𝑘−1)𝑏^𝑣−1×𝑐_𝑘𝑦+𝑑_𝑘𝑓𝑦+𝑓_𝑘=1𝑑𝑦Γ(𝑣)𝑛^𝑣𝑘−1_𝑖=1^𝑏₀𝐿_𝑖⁻¹(𝑥)−𝑦^𝑣−1×𝑐_𝑖𝑦+𝑑_𝑖𝑓𝑦+𝑓_𝑖+𝑑𝑑𝑦_𝑘Γ(𝑣)𝑛^{𝑣𝐿𝑘−1}₀^(𝑥)𝐿_𝑘⁻¹(𝑥)−𝑦^𝑣−1𝑓𝑦+1𝑑𝑦Γ(𝑣)𝑛^{𝑣𝐿𝑘−1}₀^(𝑥)𝐿_𝑘⁻¹(𝑥)−𝑦^𝑣−1×𝑐_𝑘𝑦+𝑓_𝑘=1𝑑𝑦Γ(𝑣)𝑛^𝑣𝑘−1_𝑖=1^𝑏₀𝐿_𝑖⁻¹(𝑥)−𝑦^𝑣−1×𝑐_𝑖𝑦+𝑑_𝑖𝑓𝑦+𝑓_𝑖+1𝑑𝑦Γ(𝑣)𝑛^{𝑣𝐿𝑘−1}₀^(𝑥)𝐿_𝑘⁻¹(𝑥)−𝑦^𝑣−1×𝑐_𝑘𝑦+𝑓_𝑘𝑑𝑑𝑦+^𝑘𝑛^𝑣=1𝑦,Γ(𝑣)𝑛^𝑣×^𝑘−1_𝑖=1𝑑_𝑖^𝑏₀𝐿_𝑖⁻¹(𝑥)−𝑦^𝑣−1𝑓𝑦−𝑑𝑦𝑏𝑐_𝑖𝑣𝐿_𝑖⁻¹(𝑥)−𝑏^𝑣−𝑐_𝑖𝑣()×𝐿𝑣+1_𝑖−1(𝑥)−𝑏^𝑣+1−𝐿_𝑖⁻¹(𝑥)^𝑣+1+𝑏𝑓_𝑖+𝑑_𝑘Γ(𝑣)𝑛^{𝑣𝐿𝑘−1}₀^(𝑥)𝐿_𝑘⁻¹(𝑥)−𝑦^𝑣−1𝑓𝑦+𝐿𝑑𝑦_𝑘−1(𝑥)^𝑣Γ(𝑣)𝑛^𝑣𝑐_𝑘𝑣()𝐿𝑣+1^𝑘−1(𝑥)+𝑓_𝑘𝑣. (15) So 𝐼^−𝑣𝑓(𝑥) is a fractal interpolation function of affine transformation on the interval [0, 𝑏] and its iterated function system (IFS): 𝐿_𝑖(𝑥)=1𝑛𝑥+𝑖−1𝑛𝐹𝑏,𝑖=1,2,…,𝑛,_𝑖=1𝑥,𝑦Γ(𝑣)𝑛^𝑣×^𝑘−1_𝑖=1𝑑_𝑖^𝑏₀𝐿_𝑖⁻¹(𝑥)−𝑦^𝑣−1𝑓𝑦−𝑑𝑦𝑏𝑐_𝑖𝑣𝐿_𝑖⁻¹(𝑥)−𝑏^𝑣−𝑐_𝑖𝑣()×𝐿𝑣+1_𝑖−1(𝑥)−𝑏^𝑣+1−𝐿_𝑖⁻¹(𝑥)^𝑣+1+𝑏𝑓_𝑖+𝑑_𝑘Γ(𝑣)𝑛^{𝑣𝐿𝑘−1}₀^(𝑥)𝐿_𝑘⁻¹(𝑥)−𝑦^𝑣−1𝑓𝑦+𝐿𝑑𝑦_𝑘−1(𝑥)^𝑣Γ(𝑣)𝑛^𝑣𝑐_𝑘𝑣()𝐿𝑣+1^𝑘−1(𝑥)+𝑓_𝑘𝑣, (16) where 𝑦=𝐿_𝑖(𝑥),𝑥∈[0,𝑏]; then ∀𝑥∈[𝑥_𝑖−1,𝑥_𝑖], 𝐼^−𝑣𝑓(𝑥)=𝐹_𝑖𝐿_𝑖⁻¹(𝑥),𝐼^−𝑣𝑓𝐿_𝑖⁻¹(𝑥). (17) So 𝐼^−𝑣𝑓(𝑥) is a fractal interpolation function on the interval [0,𝑏].

5. Conclusion

There are three acquired results from the above content in this paper. Firstly, the fractional order integral of fractal interpolation function is continuous on the interval [0,+∞). Secondly, the fractional order integral of fractal interpolation function can be integrated on any closed interval [𝑎,𝑏]⊂[0,+∞). Finally, the fractional order integral of fractal interpolation function is still a fractal interpolation function on the interval [0,𝑏](𝑏>0).

The fractional order integral’s differentiability of fractal interpolation function and its boxing dimension will be researched in the future.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the Scientific Research Innovation Foundation for Graduate Students of Jiangsu Province (no. CXZZ13_0686) and the Nanjing Normal University Taizhou College Youth Fund Project (no. Q201234).