Riemann-Liouville Fractional Calculus of Blancmange Curve and Cantor Functions

Journal of Applied Mathematics and Computation, 2020, 4(4), 123-129 https://www.hillpublisher.com/journals/JAMC/ ISSN Online: 2576-0653 ISSN Print: 2576-0645 Riemann-Liouville Fractional Calculus of Blancmange Curve and Cantor Functions

Srijanani Anurag Prasad

Department of Mathematics and Statistics, Indian Institute of Technology Tirupati, India.

How to cite this paper: Srijanani Anurag Prasad. (2020) Riemann-Liouville Frac- Abstract tional Calculus of Blancmange Curve and Riemann-Liouville fractional calculus of Blancmange Curve and Cantor Func- Cantor Functions. Journal of Applied Ma- thematics and Computation, 4(4), 123-129. tions are studied in this paper. In this paper, Blancmange Curve and Cantor func- DOI: 10.26855/jamc.2020.12.003 tion defined on the interval is shown to be Fractal Interpolation Functions with appropriate interpolation points and parameters. Then, using the properties of Received: September 15, 2020 Fractal Interpolation Function, the Riemann-Liouville fractional integral of Accepted: October 10, 2020 Published: October 22, 2020 Blancmange Curve and Cantor function are described to be Fractal Interpolation Function passing through a different set of points. Finally, using the conditions for *Corresponding author: Srijanani the fractional derivative of order ν of a FIF, it is shown that the fractional deriva- Anurag Prasad, Department of Mathe- tive of Blancmange Curve and Cantor function is not a FIF for any value of ν. matics, Indian Institute of Technology Tirupati, India. Email: [email protected] Keywords

Fractal, Interpolation, Iterated Function System, fractional integral, fractional derivative, Blancmange Curve, Cantor function

1. Introduction Fractal geometry is a subject in which irregular and complex functions and structures are researched. Fractal Interpola- tion Function (FIF) was introduced by Barnsley using the theory of Iterated Function System (IFS) [1]. Different kinds of FIFs like Hidden-variable FIFs, Coalescence Hidden-variable FIFs, Hermite FIFs, Spline FIFs, Super FIFs have been constructed [2-7]. Unlike classical Newtonian derivatives, the Riemann-Liouville fractional derivative is defined via fractional integral. A lot of works on fractional calculus already exist in the literature [8-11]. Fractional calculus of a FIF has been investigated in [12, 13], and Fractional calculus of a CHFIF was explored [14]. In this paper, Blancmange Curve or Takagi Curve and Cantor function is first shown to be a Fractal Interpolation Function with an appropriate number of interpolation points and relevant parameters. Then, the ν-order Riemann-Liouville fractional integral of Blancmange Curve and Cantor function is derived. Further, using the conditions for the fractional derivative of order ν of a FIF, it is shown that the fractional derivative of Blancmange Curve and Cantor function is not a FIF for any value of ν. The organization of the paper is as follows: Section 2 starts with a brief introduction to the construction of a Fractal Interpolation Function (FIF). A FIF is constructed as the graph of the attractor of a suitably defined Iterated Function System (IFS). The definition of ν-order Riemann-Liouville fractional integral and definition of ν-order Riemann-Liouville fractional derivative is given in Section 3. Then, the generalized results of the fractional integral of FIF on any interval [a, b] are presented. Also, properties relating fractional derivative and fractional integral and the condition for ν-order Rie- mann-Liouville fractional derivative of FIF are also discussed in the same section. In Section 4 and Section 5, it is first shown that Blancmange Curve and Cantor function can be viewed as a Fractal Interpolation Function. Then, the fractional integral of order ν of Blancmange Curve and Cantor function are derived using the generalized results. Further, using the conditions for the fractional derivative of order ν of a FIF, it is shown that the fractional derivative of Blancmange Curve and Cantor function is not a FIF for any value of ν.

DOI: 10.26855/jamc.2020.12.003 123 Journal of Applied Mathematics and Computation

Srijanani Anurag Prasad

2. Construction of Fractal Interpolation Function (FIF) This section begins with a brief introduction of Fractal Interpolation Function (FIF), which is constructed such that it is a fixed point of Read-Bajraktarevic operator on the suitable space of continuous functions. 2 Given any interpolation data {( , ) : = 0,1, … , }, where < 0 < 1 < . . . < < , the largest interval [ 0 , ] and the smaller sub-intervals [ 1, ] for = 1, 2, … , are denoted by and respectively. For 𝑖𝑖 𝑖𝑖 𝑁𝑁 = 1, 2, … , , the contraction maps𝑥𝑥 𝑦𝑦 from∈ ℝ the𝑖𝑖 interval 𝑁𝑁into subintervals−∞ 𝑥𝑥are defined𝑥𝑥 as 𝑥𝑥 ∞ 𝑁𝑁 𝑛𝑛− 𝑛𝑛 𝑛𝑛 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑛𝑛 𝑁𝑁1 𝐼𝐼 𝐼𝐼 𝑛𝑛 𝑁𝑁 ( )𝐿𝐿𝑛𝑛= + = 𝐼𝐼 1 + 𝐼𝐼(𝑛𝑛 0). (1) 0 𝑥𝑥𝑛𝑛 − 𝑥𝑥𝑛𝑛− Suppose the functions ×𝐿𝐿𝑛𝑛 𝑥𝑥 𝑎𝑎𝑛𝑛 for𝑥𝑥 =𝑏𝑏𝑛𝑛1,2, …𝑥𝑥𝑛𝑛,− are defined as 𝑥𝑥 − 𝑥𝑥 𝑥𝑥𝑁𝑁 − 𝑥𝑥 ( , ) = + ( ) (2) 𝐹𝐹𝑛𝑛 ∶ 𝐼𝐼 ℝ ⟼ ℝ 𝑛𝑛 𝑁𝑁 where, are chosen such that | | < 1 and are continuous functions chosen such that 𝐹𝐹𝑛𝑛 𝑥𝑥 𝑦𝑦 𝛾𝛾𝑛𝑛 𝑦𝑦 𝑞𝑞𝑛𝑛 𝑥𝑥 ( 0, 0) = 1 ( , ) = are satisfied. The variables are called free variables, and the above con- 𝑛𝑛 𝑛𝑛 𝑛𝑛 dition is𝛾𝛾 called join-up conditions.𝛾𝛾 The suitable 𝑞𝑞Iterated Function System (IFS) required to construct a Fractal Interpolation 𝑛𝑛 𝑛𝑛− 𝑛𝑛 𝑁𝑁 𝑁𝑁 𝑛𝑛 𝑛𝑛 Function𝐹𝐹 𝑥𝑥 𝑦𝑦 (FIF)𝑦𝑦 is now𝑎𝑎 𝑎𝑎defined𝑎𝑎 𝐹𝐹 𝑥𝑥 as 𝑦𝑦 𝑦𝑦 𝛾𝛾 { × ; , = 1,2, … } (3) where, ( , ) = ( ), ( , ) . It was proved by Barnsley [1] that the IFS constructed with the above chosen free 𝐼𝐼 ℝ 𝜔𝜔𝑛𝑛 𝑛𝑛 𝑁𝑁 parameters and satisfying the join up conditions is a hyperbolic IFS for a metric on 2 equivalent to the Euclidean 𝜔𝜔𝑛𝑛 𝑥𝑥 𝑦𝑦 �𝐿𝐿𝑛𝑛 𝑥𝑥 2 𝐹𝐹𝑛𝑛 𝑥𝑥 𝑦𝑦 � 2 metric. Therefore, on ( ), the space of compact sets in , there exists a unique set∗ such that = =1 ( ), where ( ) = { ( , ): ( , ) }. This fixed set is called the attractor of𝑑𝑑 the aboveℝ hyperbolic IFS constructed𝑁𝑁 𝑘𝑘 𝑘𝑘 with the given interpolation𝐻𝐻 ℝ data. Barnsley showed in the ℝsame paper that this attractor𝐴𝐴 is graph of a continuous𝐴𝐴 ⋃ function𝜔𝜔 𝐴𝐴 𝑘𝑘 𝑘𝑘 : 𝜔𝜔 such𝐴𝐴 that𝜔𝜔 (𝑥𝑥 )𝑦𝑦= 𝑥𝑥 𝑦𝑦for ∈ =𝐴𝐴 0,1, … , , and thus𝐴𝐴 Fractal Interpolation Function (FIF) was defined as: The Fractal Interpolation Function (FIF) for the interpolation data {( , ): = 0,1, … , } is defined as the con- 𝑖𝑖 𝑖𝑖 tinuous𝑓𝑓 𝐼𝐼 → ℝfunction 𝑓𝑓 𝑥𝑥 ,𝑦𝑦 whosegraph𝑖𝑖 is the𝑁𝑁 attractor of IFS { × ; , = 1,2, … , }. 𝑖𝑖 𝑖𝑖 Consider the space ( , ) , where the set is defined by 𝑥𝑥= 𝑦𝑦{ |𝑖𝑖 𝑁𝑁 , ( 0) = 𝑛𝑛 ( ) = 𝑓𝑓 ∶ }𝐼𝐼 → ℝ 𝐼𝐼 ℝ 𝜔𝜔 𝑛𝑛 ( , ) =𝑁𝑁 max | ( ) – ( )| 0 and theℑ maximum metric on the set, is given by . It was shown in [1] that FIF , isℑ a 𝑑𝑑fixed point of Read-Bajraktarevℑ ic operator ℑ defined𝑔𝑔 𝑔𝑔 ∶ 𝐼𝐼 → ℝ 𝑖𝑖𝑖𝑖 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑔𝑔 𝑥𝑥 𝑦𝑦 𝑎𝑎𝑎𝑎𝑎𝑎 𝑔𝑔 𝑥𝑥𝑁𝑁 𝑦𝑦𝑁𝑁 𝑑𝑑ℑ ℑ 𝑑𝑑ℑ 𝑔𝑔 𝑔𝑔� 𝑥𝑥∈ 𝐼𝐼 𝑔𝑔 𝑥𝑥 𝑔𝑔� 𝑥𝑥 by, ( )( ) = 1( ), 1( ) , . Hence, the FIF satisfies the recursive equation 𝑓𝑓 𝑇𝑇 − − 𝑇𝑇 𝑔𝑔 𝑥𝑥 𝐹𝐹𝑛𝑛 �𝐿𝐿𝑛𝑛 𝑥𝑥 𝑔𝑔�𝐿𝐿𝑛𝑛 𝑥𝑥 �� 𝑥𝑥(∈) 𝐼𝐼𝑛𝑛= , ( ) , 𝑓𝑓 = 1,2, … , (4) where, and are as defined in (1) and (2). So, another way to define Fractal Interpolation Function (FIF) is as follows: 𝑓𝑓�𝐿𝐿𝑛𝑛 𝑥𝑥 � 𝐹𝐹𝑛𝑛 �𝑥𝑥 𝑓𝑓 𝑥𝑥 � 𝑥𝑥 ∈ 𝐼𝐼 𝑎𝑎𝑎𝑎𝑎𝑎 𝑛𝑛 𝑁𝑁 The Fractal Interpolation Function (FIF) for the interpolation data {( , ): = 0,1, … , } is defined as the con- 𝑛𝑛 𝑛𝑛 tinuous 𝐿𝐿function𝐹𝐹 which satisfies the recursive equation given by (4). 𝑥𝑥𝑖𝑖 𝑦𝑦𝑖𝑖 𝑖𝑖 𝑁𝑁 3. Fractional Calculus𝑓𝑓 ∶ 𝐼𝐼 → ℝ of FIF In this section, we start with the definition of Riemann-Liouville fractional integral of order > 0 of a function. Then, we shall recall the fact that Riemann-Liouville fractional integral of a FIF is a FIF, which is proved in [12-14]. Next, we shall define Riemann-Liouville fractional derivatives of order > 0 of a function. Finally, the𝜈𝜈 condition under which the -order Riemann-Liouville fractional derivative of a FIF exist is also given. Let < < < < . The Riemann-Liouville fractional𝜈𝜈 integral of order > 0 with lower limit is de- 1 𝜈𝜈 [ , ] ( ) = ( ) 1 ( ) = 0 fined for locally integrable functions as + ( ) . For , it is defined −∞ 𝑎𝑎 𝑥𝑥 𝑏𝑏 ∞ 𝑥𝑥 𝜈𝜈 𝑎𝑎 that ( ) = ( ). 𝜈𝜈 𝜈𝜈 − + 𝑓𝑓 ∶ 𝑎𝑎 𝑏𝑏 → ℝ 𝐼𝐼𝑎𝑎 𝑓𝑓 𝑥𝑥 Γ 𝜈𝜈 ∫𝑎𝑎 𝑥𝑥 − 𝑡𝑡 𝑓𝑓 𝑡𝑡 𝑑𝑑𝑑𝑑 𝜈𝜈 The𝜈𝜈 following proposition is proved in [12-14] and shows that the order Riemann-Liouville fractional integral of a 𝑎𝑎 FIF defined𝐼𝐼 𝑓𝑓 𝑥𝑥 on any𝑓𝑓 interval𝑥𝑥 [ , ] is a FIF. Proposition 1: Let be a FIF passing through the interpolation𝜈𝜈 data− given by {( , ) 2: = 0,1, … , } and constructed using the IFS given𝑎𝑎 𝑏𝑏 by⊂ (3).ℝ Define ( , ) = + ( ) where, 𝑖𝑖 𝑖𝑖 𝑓𝑓 𝜈𝜈 𝜈𝜈 1𝜈𝜈 𝑥𝑥 𝑦𝑦 ∈ ℝ 𝑖𝑖 𝑁𝑁 𝐹𝐹𝑛𝑛 𝑥𝑥 𝑦𝑦 𝑎𝑎𝑛𝑛 𝛾𝛾𝑛𝑛 1𝑦𝑦 𝑞𝑞𝑛𝑛 𝑥𝑥 ( ) = ( ) + 𝑥𝑥𝑛𝑛 − ( ( ) ) 1 ( ) . (5) 0 ( ) 𝜈𝜈 𝜈𝜈 𝜈𝜈 𝜈𝜈 − 𝑛𝑛 𝑛𝑛 𝑥𝑥 𝑛𝑛 0 𝑛𝑛 𝑞𝑞 𝑥𝑥 𝑎𝑎 𝐼𝐼 𝑞𝑞 𝑥𝑥 � 𝐿𝐿 𝑥𝑥 − 𝑡𝑡 𝑓𝑓 𝑡𝑡 𝑑𝑑𝑑𝑑 2 Then, Riemann-Liouville fractional integral of a FIF of orderΓ 𝜈𝜈 is 𝑥𝑥also a FIF passing through the data {( , ) ( ) 𝜈𝜈 = 0,1, … , }, where = 0, = , and = + ( ) = ( ) for = 1, …𝑖𝑖 , 1. 0 1 𝜈𝜈 𝜈𝜈 +1 0 𝑥𝑥 𝑦𝑦𝑖𝑖 ∈ ℝ ∶ 𝜈𝜈 𝜈𝜈 𝑞𝑞𝑁𝑁 𝑥𝑥𝑁𝑁 𝜈𝜈 𝜈𝜈 𝜈𝜈 𝜈𝜈 𝜈𝜈 𝑁𝑁 𝜈𝜈 𝑗𝑗 𝑗𝑗 𝑗𝑗 𝑁𝑁 𝑗𝑗 𝑁𝑁 𝑗𝑗 𝑖𝑖 𝑁𝑁 𝑦𝑦 𝑦𝑦 −𝑎𝑎𝑁𝑁 𝛾𝛾𝑁𝑁 𝑦𝑦 𝑎𝑎 𝛾𝛾 𝑦𝑦 𝑞𝑞 𝑥𝑥 𝑞𝑞 𝑥𝑥 𝑗𝑗 𝑁𝑁 − DOI: 10.26855/jamc.2020.12.003 124 Journal of Applied Mathematics and Computation

Srijanani Anurag Prasad

Proof: For any [ 0, ] and {1,2, … , }, ( ) 𝑥𝑥 ∈ 𝑥𝑥 𝑥𝑥𝑁𝑁 1 𝑘𝑘 ∈ 𝑁𝑁 ( ) = 𝐿𝐿𝑘𝑘 𝑥𝑥 ( ( ) ) 1 ( ) 0 ( ) 𝜈𝜈 𝜈𝜈 − 𝑥𝑥 𝑘𝑘 0 𝑘𝑘 𝐼𝐼 𝑓𝑓�𝐿𝐿 𝑥𝑥 � � 𝐿𝐿 1𝑥𝑥 − 𝑡𝑡 𝑓𝑓 𝑡𝑡 𝑑𝑑𝑑𝑑 ( ) Γ 𝜈𝜈 𝑥𝑥1 = 𝑥𝑥𝑘𝑘− ( ( ) ) 1 ( ) + 𝐿𝐿𝑘𝑘 𝑥𝑥 ( ( ) ) 1 ( ) ( ) 𝜈𝜈 − 𝜈𝜈 − 0 1 � � 𝐿𝐿𝑘𝑘 𝑥𝑥 − 𝑡𝑡 𝑓𝑓 𝑡𝑡 𝑑𝑑𝑑𝑑 � 𝐿𝐿𝑘𝑘 𝑥𝑥 − 𝑡𝑡 𝑓𝑓 𝑡𝑡 𝑑𝑑𝑑𝑑 � Replacing by ( ) in the secondΓ integral𝜈𝜈 𝑥𝑥 and using ( ( )) = 𝑥𝑥(𝑘𝑘−) + ( ) for all = 1,2, … , , it follows that 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑡𝑡 𝐿𝐿 𝑡𝑡 1 𝑓𝑓 𝐿𝐿 𝑡𝑡 𝛾𝛾 𝑓𝑓 𝑡𝑡 𝑞𝑞 𝑡𝑡 𝑘𝑘 𝑁𝑁 1 ( ) = 𝑥𝑥𝑘𝑘− ( ( ) ) 1 ( ) + 𝑥𝑥( ( ) ( )) 1 ( ( )) 0 ( ) 𝜈𝜈 𝜈𝜈 − 𝜈𝜈 − 𝑥𝑥 𝑘𝑘 0 𝑘𝑘 0 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝐼𝐼 𝑓𝑓�𝐿𝐿 𝑥𝑥 � � � 𝐿𝐿 1𝑥𝑥 − 𝑡𝑡 𝑓𝑓 𝑡𝑡 𝑑𝑑𝑑𝑑 � 𝐿𝐿 𝑥𝑥 − 𝐿𝐿 𝑡𝑡 𝑓𝑓 𝐿𝐿 𝑡𝑡 𝑎𝑎 𝑑𝑑𝑑𝑑 � Γ 𝜈𝜈 1𝑥𝑥 𝑥𝑥 = 𝑥𝑥𝑘𝑘− ( ( ) ) 1 ( ) + 𝑥𝑥( ) 1[ ( ) + ( )] ( ) 𝜈𝜈 − 𝜈𝜈 𝜈𝜈 − 0 𝑘𝑘 𝑘𝑘 0 𝑘𝑘 𝑘𝑘 � � 𝐿𝐿 𝑥𝑥 − 𝑡𝑡 𝑓𝑓 𝑡𝑡 𝑑𝑑𝑑𝑑 𝑎𝑎 1� 𝑥𝑥 − 𝑡𝑡 𝛾𝛾 𝑓𝑓 𝑡𝑡 𝑞𝑞 𝑡𝑡 𝑑𝑑𝑑𝑑 � Γ 𝜈𝜈 𝑥𝑥 1 𝑥𝑥 = ( ) + ( ) + 𝑥𝑥𝑘𝑘− ( ( ) ) 1 ( ) (6) 0 0 ( ) 𝜈𝜈 𝜈𝜈 𝜈𝜈 𝜈𝜈 𝜈𝜈 − 𝑘𝑘 𝑘𝑘 𝑥𝑥 𝑘𝑘 𝑥𝑥 𝑘𝑘 0 𝑘𝑘 𝑎𝑎 𝛾𝛾 𝐼𝐼 𝑓𝑓 𝑥𝑥( ) 𝑎𝑎 𝐼𝐼 𝑞𝑞 𝑥𝑥 � � 𝐿𝐿 𝑥𝑥 − 𝑡𝑡 𝑓𝑓 𝑡𝑡 𝑑𝑑𝑑𝑑 � Using (5), it follows _0 = 0 (Γ)𝜈𝜈 + 𝑥𝑥 ( ) = ( , 0 ( )), which implies that order Rie- mann-Liouville fractional 𝜈𝜈integral of the function𝜈𝜈 𝜈𝜈 is a FIF passing𝜈𝜈 through𝜈𝜈 the𝜈𝜈 data {( , ) 2 = 0,1, … , }. □ 𝐼𝐼𝑥𝑥 𝑓𝑓�𝐿𝐿𝑘𝑘 𝑥𝑥 � 𝑎𝑎𝑘𝑘 𝛾𝛾𝑘𝑘 𝐼𝐼𝑥𝑥 𝑓𝑓 𝑥𝑥 𝑞𝑞𝑘𝑘 𝑥𝑥 𝐹𝐹𝑘𝑘 𝑥𝑥 𝐼𝐼𝑥𝑥 𝑓𝑓 𝑥𝑥 𝜈𝜈 − Now, we define Riemann-Liouville fractional derivatives of order > 0 for a function 𝜈𝜈 1([ , ]). 𝑖𝑖 𝑖𝑖 Let < < < < , 0 < and 𝑓𝑓 1([ , ]. Let 𝑥𝑥 𝑦𝑦 ∈ ℝ ∶ 𝑖𝑖 𝑁𝑁 , = { ([ , ]) ( ) ([ , ])} and 𝜈𝜈 1,1, where is the𝑓𝑓 ∈smallest𝐿𝐿 𝑎𝑎 integer𝑏𝑏 greater than 𝑘𝑘 𝑗𝑗 −∞ 𝑎𝑎 𝑥𝑥 𝑏𝑏 ∞ 𝑘𝑘 𝜈𝜈 𝑓𝑓 ∈ 𝐿𝐿 𝑎𝑎 𝑏𝑏 𝑛𝑛−𝜈𝜈 . The Riemann-Liouville𝑗𝑗 fractional derivative𝑗𝑗 of order with lower limit is defined as ( )( ) = ( )( ) 𝑊𝑊 𝑓𝑓 ∶ 𝑓𝑓 ∈ 𝐿𝐿 𝑎𝑎 𝑏𝑏 𝑎𝑎𝑎𝑎𝑎𝑎 𝐷𝐷 𝑓𝑓 ∈ 𝐿𝐿 𝑎𝑎 𝑏𝑏 𝐼𝐼 𝑓𝑓 ∈ 𝑊𝑊 𝑛𝑛 𝑛𝑛 0 𝜈𝜈 𝑑𝑑 𝑛𝑛−𝜈𝜈 and ( )( ) = ( ). 𝑛𝑛 𝑎𝑎 𝑑𝑑𝑥𝑥 𝑎𝑎 𝜈𝜈 The following proposition is proved in [13, 14] and gives𝜈𝜈 the condition 𝑎𝑎under which the 𝐷𝐷 𝑓𝑓 ord𝑥𝑥er Riemann𝐼𝐼-Liouville𝑓𝑓 𝑥𝑥 𝑎𝑎 fractional𝐷𝐷 𝑓𝑓 derivative𝑥𝑥 𝑓𝑓 of𝑥𝑥 a FIF exist. Proposition 2: 𝜈𝜈 − Let be a FIF passing through the interpolation data given by {( , ) 2 = 0,1, … , } and constructed using the IFS given by (3). Suppose < for some fixed > 0. Then Riemann-Liouville fractional derivative of a FIF of 𝑖𝑖 𝑖𝑖 order 𝑓𝑓 is also a FIF provided 𝜈𝜈 𝑥𝑥 𝑦𝑦 ∈ ℝ ∶ 𝑖𝑖 𝑁𝑁 𝑘𝑘 𝑘𝑘 𝛾𝛾 𝑎𝑎 1 𝜈𝜈 1 𝜈𝜈( ) = ( ) + 𝑥𝑥𝑘𝑘− ( )( ( ) ) 1 ( ) 𝑛𝑛 𝑑𝑑𝑑𝑑 −𝜈𝜈 𝜈𝜈 ( ) 𝑛𝑛−𝜈𝜈− 𝑘𝑘 𝑘𝑘 𝑘𝑘 𝑑𝑑 𝑛𝑛 0 𝑘𝑘 𝑞𝑞 𝑥𝑥 𝑎𝑎 𝐷𝐷 𝑞𝑞 𝑥𝑥 � 𝑓𝑓 𝑡𝑡 𝐿𝐿 𝑥𝑥 − 𝑡𝑡1 𝑑𝑑𝑑𝑑 Γ 𝑛𝑛 − 𝜈𝜈 𝑘𝑘 𝑥𝑥 𝑑𝑑�𝐿𝐿 1 𝑥𝑥 � 1 = ( ) + 𝑛𝑛 ( ) 𝑥𝑥𝑘𝑘− ( )( ( ) ) (7) ( ) −𝜈𝜈 𝜈𝜈 =1 −𝜈𝜈− 𝑘𝑘 𝑘𝑘 0 𝑘𝑘 𝑎𝑎 𝐷𝐷 𝑞𝑞 𝑥𝑥 �� 𝑛𝑛 − 𝑗𝑗 − 𝜈𝜈 � � 𝑓𝑓 𝑡𝑡 𝐿𝐿 𝑥𝑥 −[ 𝑡𝑡, ] 𝑑𝑑𝑑𝑑 {1,2, … , } Proof: Using the definition of RiemannΓ -𝑛𝑛Liouville− 𝜈𝜈 𝑗𝑗 fractional derivative,𝑥𝑥 for any 0 and , ( ) = ( )( ( )).Replacing by in(6) and substituting above, we have 0 (𝑛𝑛 ) 0 𝑥𝑥 ∈ 𝑥𝑥 𝑥𝑥𝑁𝑁 𝑘𝑘 ∈ 𝑁𝑁 𝜈𝜈 𝑑𝑑 𝑛𝑛−𝜈𝜈 𝑥𝑥 𝑘𝑘 𝑛𝑛 𝑥𝑥 𝑘𝑘 1 �𝐷𝐷 𝑓𝑓 ��𝐿𝐿 𝑥𝑥 � 𝑑𝑑�𝐿𝐿𝑘𝑘 𝑥𝑥 � 𝐼𝐼 𝑓𝑓 𝐿𝐿 𝑥𝑥 𝜈𝜈 𝑛𝑛 − 𝜈𝜈 1 ( ) = ( ( ) + ( )) + 𝑥𝑥𝑘𝑘− ( ( ) ) 1 ( ) 0 𝑛𝑛 0 ( ) 𝜈𝜈 ( ) 𝑛𝑛−𝜈𝜈 𝑛𝑛−𝜈𝜈 𝑛𝑛−𝜈𝜈 − 𝑥𝑥 𝑘𝑘 𝑑𝑑 𝑛𝑛 𝑘𝑘 𝑥𝑥 𝑘𝑘 𝑘𝑘 0 𝑘𝑘 �𝐷𝐷 𝑓𝑓 ��𝐿𝐿 𝑥𝑥 � �𝑎𝑎 𝐼𝐼 𝛾𝛾 𝑓𝑓 𝑥𝑥 𝑞𝑞 𝑥𝑥 � � 𝐿𝐿1 𝑥𝑥 − 𝑡𝑡 𝑓𝑓 𝑡𝑡 𝑑𝑑𝑑𝑑�� 𝑑𝑑�𝐿𝐿𝑘𝑘 𝑥𝑥 � 1 Γ 𝑛𝑛 − 𝜈𝜈 𝑥𝑥 = ( ( ) + ( )) + 𝑥𝑥𝑘𝑘− ( ( ) ) 1 ( ) 0 0 ( ) 𝑛𝑛 −𝜈𝜈 𝜈𝜈 𝜈𝜈 ( ) 𝑛𝑛−𝜈𝜈 − 𝑘𝑘 𝑥𝑥 𝑘𝑘 𝑥𝑥 𝑘𝑘 𝑑𝑑 𝑛𝑛 0 𝑘𝑘 𝑎𝑎 𝐷𝐷 𝛾𝛾 𝑓𝑓 𝑥𝑥 𝐷𝐷 𝑞𝑞 𝑥𝑥 � � 𝐿𝐿 𝑥𝑥1 − 𝑡𝑡 𝑓𝑓 𝑡𝑡 𝑑𝑑𝑡𝑡� Γ 𝑛𝑛 − 𝜈𝜈 𝑘𝑘 𝑥𝑥 1 𝑑𝑑�𝐿𝐿 𝑥𝑥 � 1 = ( ( ) + ( )) + 𝑛𝑛 𝑥𝑥𝑘𝑘− ( ( ) ) ( ) 0 0 ( ) −𝜈𝜈 𝜈𝜈 𝜈𝜈 −𝜈𝜈 − =1 0 𝑎𝑎𝑘𝑘 𝐷𝐷𝑥𝑥 𝛾𝛾𝑘𝑘 𝑓𝑓 𝑥𝑥 𝐷𝐷𝑥𝑥 𝑞𝑞𝑘𝑘 𝑥𝑥 �� 𝑛𝑛 − 𝑗𝑗 − 𝜈𝜈� � � 𝐿𝐿𝑘𝑘 𝑥𝑥 − 𝑡𝑡 𝑓𝑓 𝑡𝑡 𝑑𝑑𝑑𝑑� Γ 𝑛𝑛 − 𝜈𝜈 𝑗𝑗 𝑥𝑥 DOI: 10.26855/jamc.2020.12.003 125 Journal of Applied Mathematics and Computation

Srijanani Anurag Prasad

Since < it implies that | | < 1. Using (7), it is easily seen that( 0 )( ( )) = 0 ( ) + ( ) 𝜈𝜈 −𝜈𝜈 𝜈𝜈 −𝜈𝜈 𝜈𝜈 and𝑘𝑘 hence𝑘𝑘 the Riemann-Liouville𝑘𝑘 𝑘𝑘 fractional derivative of a FIF of order is also𝑥𝑥 a FIF.𝑘𝑘 □ 𝑘𝑘 𝑘𝑘 𝑥𝑥 𝑑𝑑𝑑𝑑 𝛾𝛾 𝑎𝑎 𝑎𝑎 𝛾𝛾 𝐷𝐷 𝑓𝑓 𝐿𝐿 𝑥𝑥 𝑎𝑎 𝛾𝛾 𝐷𝐷 𝑓𝑓 𝑥𝑥 4.𝑞𝑞𝑘𝑘 Blancmange𝑥𝑥 function 𝜈𝜈 In this section, we shall first show that the Blancmange function or otherwise called the Takagi fractal curve is a Fractal Interpolation Function for appropriate interpolation points. Then, we shall derive the fractional integral of order of Blancmange Curve using the results from Section 3. Finally, we show that the fractional derivative of Blancmange Curve is not a FIF for any value of ν. 𝝂𝝂 ( ) The Blancmange Curve (Figure 1) is defined on [0, 1] as ( ) = [ , ], where, ( ) = = 𝒏𝒏 ∞ 𝒔𝒔 𝟐𝟐 𝒙𝒙 | |, . It is observed that the Blancmange function satisfies the recursive𝒏𝒏 equation given by 𝑩𝑩 𝒙𝒙 ∑𝒏𝒏 𝟎𝟎 𝟐𝟐 𝒙𝒙 ∈ 𝟎𝟎 𝟏𝟏 𝒔𝒔 𝒚𝒚 + + ( ) 𝐦𝐦𝐦𝐦𝐧𝐧𝒎𝒎∈ℤ 𝒚𝒚 − 𝒎𝒎 𝒚𝒚 ∈ ℝ = ( ) + , [ , ] , = , . ( ) 𝒌𝒌−𝟏𝟏 𝒙𝒙 𝒌𝒌 − 𝟏𝟏 𝟏𝟏 𝒌𝒌 − 𝟏𝟏 −𝟏𝟏 𝒙𝒙 𝑩𝑩 � � 𝑩𝑩 𝒙𝒙 𝒙𝒙 ∈ 𝟎𝟎 𝟏𝟏 𝒇𝒇𝒇𝒇𝒇𝒇 𝒌𝒌 𝟏𝟏 𝟐𝟐 𝟖𝟖 𝟐𝟐 𝟐𝟐 𝟐𝟐

Figure 1. Blancmange Curve or Takagi Curve. Theorem 1: Blancmange Curve is a Fractal Interpolation Function on [0, 1]. Proof: Consider = ( , ), , , ( , ) . Let = [ , ]. Then, the IFS required to construct the Blancmange 𝟏𝟏 𝟏𝟏 + +( ) Curve is given by{ × ; , = ,𝟐𝟐 𝟐𝟐} where, ( , ) = , + . It is easily seen that the above 𝚲𝚲 � 𝟎𝟎 𝟎𝟎 � � 𝟏𝟏 𝟎𝟎 � 𝑰𝑰 𝟎𝟎 𝟏𝟏 𝒌𝒌−𝟏𝟏 IFS is hyperbolic, and the attractor of the above IFS is a graph of𝒙𝒙 𝒌𝒌a− continuous𝟏𝟏 𝟏𝟏 𝒌𝒌 −function𝟏𝟏 −𝟏𝟏 that𝒙𝒙 satisfies the recursive equation 𝒌𝒌 𝒌𝒌 𝟐𝟐 𝟐𝟐 𝟐𝟐 given by (8). Hence,𝑰𝑰 byℝ the𝝎𝝎 uniqueness𝒌𝒌 𝟏𝟏 𝟐𝟐 of FIF, 𝝎𝝎the 𝒙𝒙Blancmange𝒚𝒚 � curve𝒚𝒚 is a Fractal Interpolation� Function on [0, 1] constructed from the interpolation data given by and using the IFS given above. □ Now, we shall describe the Riemann-Liouville fractional integral of order ν of the Blancmange curve defined on [0, 1]. Theorem 2: Riemann-Liouville fractional 𝚲𝚲integral of order ν of Blancmange curve defined on [0, 1] is a FIF passing + through the data {( , ) ^ = , , }, where = , = and = . + ( + ) + 𝝂𝝂 𝝂𝝂 𝝂𝝂 𝝂𝝂 𝟐𝟐 𝝂𝝂 𝟏𝟏 𝝂𝝂 𝟏𝟏 ( ) Proof: For =𝒊𝒊 , 𝒊𝒊 , define ( , ) = +𝟎𝟎 ( ) 𝟏𝟏where,𝟐𝟐 𝚪𝚪 (𝝂𝝂 )𝟐𝟐 = 𝟐𝟐 𝝂𝝂 (𝟏𝟏 ) + + 𝒙𝒙 𝒚𝒚 ∈ ℝ 𝟐𝟐 ∶ 𝒊𝒊 𝟎𝟎 𝟏𝟏 𝟐𝟐 + 𝒚𝒚 𝟎𝟎 𝒚𝒚 𝒚𝒚+ 𝝂𝝂( + ) +𝒌𝒌−𝟏𝟏 𝝂𝝂 𝟏𝟏 𝝂𝝂 𝝂𝝂 𝒙𝒙 −𝟏𝟏 𝒙𝒙 + 𝝂𝝂 𝟏𝟏 𝝂𝝂 𝟏𝟏 𝒌𝒌 𝟐𝟐 𝒌𝒌 𝒌𝒌 𝟐𝟐 𝚪𝚪 𝟏𝟏 𝝂𝝂 𝟏𝟏 𝝂𝝂 𝒌𝒌 𝟏𝟏 𝟐𝟐 ( ) . It is𝑭𝑭 observed𝒙𝒙 𝒚𝒚 that 𝒚𝒚 ,𝒒𝒒 𝒙𝒙 = + 𝒒𝒒 𝒙𝒙 + + � =𝒌𝒌 − 𝟏𝟏( , ). Therefore,� ( ) 𝝂𝝂−𝟏𝟏 ( + ) ( + ) ( + ) 𝟏𝟏 𝒙𝒙𝒌𝒌−𝟏𝟏 𝒙𝒙 𝒌𝒌−𝟏𝟏 𝝂𝝂 𝟏𝟏 𝟏𝟏 𝟏𝟏 𝝂𝝂 by Proposition 1, the ν-order Riemann-Liouville fractional integral of Blancmange𝝂𝝂 𝟏𝟏 curve𝝂𝝂 is𝟏𝟏 also a FIF, and it satisfies the 𝚪𝚪 𝝂𝝂 ∫𝒙𝒙𝟎𝟎 � 𝟐𝟐 − 𝒕𝒕� 𝑩𝑩 𝒕𝒕 𝒅𝒅𝒅𝒅 𝑭𝑭𝟏𝟏 �𝟏𝟏 𝚪𝚪 𝝂𝝂 𝟏𝟏 � 𝟐𝟐 𝚪𝚪 𝟏𝟏 𝝂𝝂 𝟐𝟐 𝚪𝚪 𝟐𝟐 𝝂𝝂 𝑭𝑭𝟐𝟐 𝟎𝟎 𝟎𝟎 recursive equation

DOI: 10.26855/jamc.2020.12.003 126 Journal of Applied Mathematics and Computation

Srijanani Anurag Prasad

+ ( ) + = ( ) + ( ) + + 𝒙𝒙𝒌𝒌−𝟏𝟏 ( ) + + (𝝂𝝂 + ) +𝒌𝒌−𝟏𝟏 ( ) 𝝂𝝂−𝟏𝟏 𝝂𝝂 𝒙𝒙 𝒌𝒌 − 𝟏𝟏 𝟏𝟏 𝝂𝝂 𝒙𝒙 −𝟏𝟏 𝒙𝒙 𝟏𝟏 𝒙𝒙 𝒌𝒌 − 𝟏𝟏 𝑰𝑰 𝑩𝑩 � � 𝝂𝝂 𝟏𝟏 𝑰𝑰 𝑩𝑩 𝒙𝒙 𝝂𝝂 𝟏𝟏 � 𝒌𝒌 − 𝟏𝟏 � � � − 𝒕𝒕� 𝑩𝑩 𝒕𝒕 𝒅𝒅𝒅𝒅 𝟐𝟐 𝟐𝟐 𝟐𝟐 𝚪𝚪 𝟏𝟏 𝝂𝝂 𝟏𝟏 𝝂𝝂 𝚪𝚪 𝝂𝝂 𝒙𝒙𝟎𝟎 𝟐𝟐 = + ( ) + ( ). Thus, Riemann-Liouville𝟏𝟏 fractional𝝂𝝂 integral𝝂𝝂 of order ν of Blancmange curve defined on [0, 1] is a FIF passing through the 𝝂𝝂 𝟏𝟏 𝑰𝑰 𝑩𝑩 𝒙𝒙 𝒒𝒒𝒌𝒌 𝒙𝒙 data {( , ) ^ 𝟐𝟐 = , , }. □ Having described𝝂𝝂 that the Riemann-Liouville fractional integral of the Blancmange curve of order ν is a FIF, it is natural 𝒊𝒊 𝒊𝒊 to seek 𝒙𝒙the𝒚𝒚 Riemann∈ ℝ 𝟐𝟐-Liouville∶ 𝒊𝒊 𝟎𝟎 𝟏𝟏fractional𝟐𝟐 derivative of Blancmange curve of order ν to be a FIF as earlier. However, we shall show that the Riemann-Liouville fractional derivative of the Blancmange curve defined on [0, 1] is not a FIF for any ν. Theorem 3: Riemann-Liouville fractional derivative of the Blancmange curve defined on [0, 1] is not a FIF for any ν. Proof: By Proposition 2, the ν-order Riemann-Liouville fractional derivative of Blancmange curve is a FIF provided < and equation (7) is satisfied. Since = = , the value of ν can lie only between and . 𝝂𝝂 𝟏𝟏 Therefore,𝒌𝒌 for 𝒌𝒌 < < 1 , ( ) = 𝒌𝒌( 𝒌𝒌) ( ) = ( ) ( ) . Now, 𝜸𝜸 𝒂𝒂 ( ) 𝒂𝒂 𝜸𝜸 𝟐𝟐 ( ) 𝟎𝟎 𝟏𝟏 𝒅𝒅 𝟏𝟏 𝒙𝒙 −𝝂𝝂 𝒙𝒙 +( ) 𝝂𝝂 −𝝂𝝂 −𝝂𝝂−𝟏𝟏 putting ( ) = for 𝒌𝒌 [ , ], we𝒅𝒅𝒅𝒅 see𝚪𝚪 𝟏𝟏− that𝝂𝝂 𝟎𝟎 𝒌𝒌 𝚪𝚪 𝟏𝟏−𝝂𝝂 𝟎𝟎 𝒌𝒌 𝟎𝟎 𝜈𝜈 𝒌𝒌−𝟏𝟏 𝑫𝑫 𝒒𝒒 𝒙𝒙 � ∫ 𝒙𝒙 − 𝒕𝒕 𝒒𝒒 𝒕𝒕 𝒅𝒅𝒅𝒅� ∫ 𝒙𝒙 − 𝒕𝒕 𝒒𝒒 𝒕𝒕 𝒅𝒅𝒅𝒅 𝒌𝒌−𝟏𝟏 −𝟏𝟏 𝒕𝒕 𝒌𝒌 𝟐𝟐 + ( ) 𝒒𝒒( 𝒕𝒕) = 𝒙𝒙( )𝒕𝒕 ∈ 𝟎𝟎 𝟏𝟏 = ( ) + ( ) ( ) 𝒌𝒌−𝟏𝟏 ( ) ( 𝟏𝟏−𝝂𝝂 ) 𝝂𝝂 −𝝂𝝂−𝟏𝟏 −𝝂𝝂 𝒌𝒌−𝟏𝟏 𝒌𝒌 −𝝂𝝂 𝒌𝒌 − 𝟏𝟏 −𝟏𝟏 𝒕𝒕 𝟏𝟏 𝒙𝒙 𝑫𝑫 𝒒𝒒 𝒙𝒙 � 𝒙𝒙 −(𝒕𝒕 ) � � 𝒅𝒅𝒅𝒅 � 𝒌𝒌 − 𝟏𝟏 𝒙𝒙 −𝟏𝟏 � As , the𝚪𝚪 function𝟏𝟏 − 𝝂𝝂 𝒙𝒙 𝟎𝟎 goes to . Hence,𝟐𝟐 the Riemann-𝟐𝟐Liouville𝟐𝟐 𝟏𝟏 − 𝝂𝝂 fractional derivative of the Blancmange𝟏𝟏 − 𝝂𝝂 curve defined on [0, 1] is not a𝝂𝝂 FIF for any value of ν. □ 𝒙𝒙 → 𝟎𝟎 𝑫𝑫 𝒒𝒒𝟐𝟐 𝒙𝒙 ∞ 5. Cantor function In this section, we shall first show that the Cantor function or otherwise called the devil’s staircase is a Fractal Interpo- lation Function for appropriate interpolation points. Then, we shall derive the fractional integral of order of Cantor’s function using the results from Section 3. Finally, we show thatthefractional derivative of Cantor function is not a FIF for any value of ν. 𝝂𝝂

Figure 2. Cantor Function or Devil’s Staircase.

DOI: 10.26855/jamc.2020.12.003 127 Journal of Applied Mathematics and Computation

Srijanani Anurag Prasad

The Cantor function (Figure 2) is defined on [0, 1] as ( ) = ( ) [ , ], where,

. ( 𝒏𝒏→)∞ 𝒏𝒏 𝑪𝑪 𝒙𝒙 𝐥𝐥𝐥𝐥𝐥𝐥 𝒇𝒇 𝒙𝒙 𝒙𝒙 ∈ 𝟎𝟎𝟏𝟏𝟏𝟏 ( ) = , ( ) = . 𝒏𝒏 < < + ⎧ 𝟎𝟎 𝟓𝟓 𝒇𝒇 𝟑𝟑𝟑𝟑 𝒊𝒊𝒊𝒊 𝟎𝟎 ≤ 𝒙𝒙 ≤ 𝟑𝟑 . ⎪ 𝟏𝟏 𝟐𝟐 𝒇𝒇𝟎𝟎 𝒙𝒙 𝒙𝒙 𝒇𝒇𝒏𝒏 𝟏𝟏 𝒙𝒙 . 𝟎𝟎 𝟓𝟓 ( ) 𝒊𝒊 𝒊𝒊 𝟑𝟑 𝑥𝑥 𝟑𝟑 Theorem 4: Cantor function is a Fractal Interpolation Function⎨ on [0, 1]. 𝟐𝟐 ⎪𝟎𝟎 𝟓𝟓 𝒇𝒇𝒏𝒏 𝟑𝟑𝟑𝟑 − 𝟐𝟐 𝒊𝒊𝒊𝒊 𝟑𝟑 ≤ 𝒙𝒙 ≤ 𝟏𝟏 Proof: Consider = ( , ), , , , ( , ) and⎩ = [ , ]. Then, the IFS required to construct Cantor func- 𝟏𝟏 𝟏𝟏 𝟐𝟐 𝟏𝟏 + tion is given by 𝚫𝚫{ ×� ;𝟎𝟎 𝟎𝟎, �=𝟑𝟑 𝟐𝟐�, ,�𝟑𝟑 } 𝟐𝟐 �where,𝟏𝟏 𝟎𝟎 � ( 𝑰𝑰, ) =𝟎𝟎 𝟏𝟏 , + ( ) with = = ( ) = 𝒙𝒙 𝒌𝒌−𝟏𝟏 ( ) = 𝑰𝑰 𝑰𝑰 𝝎𝝎𝒌𝒌 𝒌𝒌 𝟏𝟏 𝟐𝟐 𝟑𝟑 𝝎𝝎𝒌𝒌 𝒙𝒙 𝒚𝒚 � 𝟑𝟑 𝜸𝜸𝒌𝒌𝒚𝒚 𝒒𝒒𝒌𝒌 𝒙𝒙 � 𝜸𝜸𝟏𝟏 𝜸𝜸𝟑𝟑 𝒒𝒒𝟐𝟐 𝒙𝒙 𝟏𝟏 and𝟑𝟑 = ( ) = . It is observed that that the graph of Cantor function is the attractor of the given IFS and satisfies the 𝒒𝒒 𝒙𝒙 𝟐𝟐 + recursive𝟐𝟐 equation𝟏𝟏 = ( ) + ( ). Hence, by the uniqueness of FIF, the Cantor function is a Fractal 𝜸𝜸 𝒒𝒒 𝒙𝒙 𝟎𝟎 𝒙𝒙 𝒌𝒌−𝟏𝟏 Interpolation Function on [0, 1] constructed from the interpolation data given by and using the given IFS. □ 𝟑𝟑 𝒌𝒌 𝒌𝒌 Now, we shall describe𝑪𝑪 � the� Riemann𝜸𝜸 𝑪𝑪 -𝒙𝒙Liouville𝒒𝒒 fractional𝒙𝒙 integral of order ν of Cantor function defined on [0, 1]. Theorem 5: Riemann-Liouville fractional integral of order ν of Cantor function𝚫𝚫 defined on [0, 1] is a FIF passing through the data {( , ) = , , , } , where = , = , = and = .( . ) ( + ) 𝝂𝝂(−+𝟏𝟏 ) 𝝂𝝂 𝟐𝟐 𝝂𝝂 𝝂𝝂 𝟏𝟏 𝝂𝝂 𝟐𝟐 𝝂𝝂 𝝂𝝂 𝝂𝝂 . 𝒊𝒊 𝒊𝒊 𝟎𝟎 𝟏𝟏 𝟐𝟐 𝟐𝟐 𝟑𝟑 −𝟏𝟏 𝚪𝚪 𝝂𝝂 𝟏𝟏 𝟐𝟐 𝟑𝟑 𝚪𝚪 𝝂𝝂 𝟏𝟏 𝟑𝟑 ( . )𝝂𝝂 ( + ) 𝒙𝒙 𝒚𝒚 ∈ ℝ ∶ 𝒊𝒊 𝟎𝟎 𝟏𝟏 𝟐𝟐 𝟑𝟑 𝒚𝒚 𝟎𝟎 𝒚𝒚 𝒚𝒚 𝒚𝒚 𝟑𝟑 𝝂𝝂 : For = , , , define ( , ) = + ( ) where, = = , = = , and 𝟐𝟐 𝟑𝟑Proof−𝟏𝟏 𝚪𝚪 𝝂𝝂 𝟏𝟏 . 𝝂𝝂 𝝂𝝂 𝝂𝝂 𝝂𝝂 𝝂𝝂 𝟏𝟏 𝝂𝝂 𝝂𝝂 𝝂𝝂 𝒌𝒌 𝒌𝒌 𝒌𝒌 𝟏𝟏 𝟑𝟑 𝟐𝟐 𝟑𝟑 𝟐𝟐 𝟏𝟏 𝒌𝒌 𝟏𝟏 𝟐𝟐 𝟑𝟑 𝑭𝑭+ (𝒙𝒙 𝒚𝒚 ) 𝜸𝜸 𝒚𝒚 𝒒𝒒 𝟏𝟏𝒙𝒙 + 𝜸𝜸 𝜸𝜸 𝜸𝜸 𝒒𝒒 𝟎𝟎 𝟑𝟑 ( ) = 𝝂𝝂 + ( ) , = , . × ( + ) ( ) 𝝂𝝂−𝟏𝟏 𝝂𝝂 �𝒙𝒙 𝒌𝒌 − 𝟐𝟐 � 𝟏𝟏 𝒙𝒙 𝒌𝒌 − 𝟏𝟏 𝒒𝒒𝒌𝒌 𝒙𝒙 𝝂𝝂 � � − 𝒕𝒕� 𝑪𝑪 𝒕𝒕 𝒅𝒅𝒅𝒅 𝒌𝒌 𝟐𝟐 𝟑𝟑 It is observed that , = = ( ) = ( , ) and , = 𝟐𝟐 ( . 𝟑𝟑 𝚪𝚪)𝝂𝝂 𝟏𝟏 ( +𝝂𝝂) 𝚪𝚪 .(𝝂𝝂. 𝟎𝟎 ) ( +𝟐𝟐) ( . )𝝂𝝂 ( + ) 𝝂𝝂 𝟑𝟑 𝟏𝟏 𝝂𝝂 𝝂𝝂 𝝂𝝂 𝟑𝟑 = ( ) = 𝟏𝟏 ( , )𝝂𝝂. Therefore, by Proposition𝝂𝝂 1, the 𝒌𝒌ν-order Riemann𝟐𝟐 -Liouville𝟐𝟐 fractional𝝂𝝂 integral of .( . ) ( + ) 𝑭𝑭 �𝟏𝟏 𝟐𝟐 𝟑𝟑 −𝟏𝟏 𝚪𝚪 𝝂𝝂 𝟏𝟏 � 𝟐𝟐 𝟐𝟐 𝟑𝟑 −𝟏𝟏 𝚪𝚪 𝝂𝝂 𝟏𝟏 𝒒𝒒 𝟎𝟎 𝑭𝑭 𝟎𝟎 𝟎𝟎 𝑭𝑭 �𝟏𝟏 𝟐𝟐 𝟑𝟑 −𝟏𝟏 𝚪𝚪 𝝂𝝂 𝟏𝟏 � 𝟏𝟏 𝝂𝝂 𝝂𝝂 𝝂𝝂 + Cantor𝟐𝟐 𝟐𝟐 𝟑𝟑 − 𝟏𝟏function𝚪𝚪 𝝂𝝂 𝟏𝟏 is 𝒒𝒒also𝟑𝟑 𝟎𝟎 a FIF,𝑭𝑭𝟑𝟑 and𝟎𝟎 𝟎𝟎 it satisfies the recursive equation = ( ) + ( ) . Thus, Rie- mann-Liouville fractional integral of order ν of Cantor function defined𝝂𝝂 on𝒙𝒙 [0,𝒌𝒌− 𝟏𝟏1] is a FIF𝝂𝝂 passing𝝂𝝂 through the data 𝑰𝑰 𝑪𝑪 � 𝟑𝟑 � 𝜸𝜸𝒌𝒌𝑰𝑰 𝑪𝑪 𝒙𝒙 𝒒𝒒𝒌𝒌 𝒙𝒙 {( , ) = , , , }. Having𝝂𝝂 described𝟑𝟑 that the Riemann-Liouville fractional integral of Cantor function of order ν is a FIF, it is natural to seek 𝒊𝒊 𝒊𝒊 Riemann𝒙𝒙 𝒚𝒚 -∈Liouvilleℝ ∶ 𝒊𝒊 fractional𝟎𝟎 𝟏𝟏 𝟐𝟐 𝟑𝟑derivative of Cantor function of order ν to be a FIF as earlier. However, we shall show that the Riemann-Liouville fractional derivative of Cantor function defined on [0, 1] is also not a FIF for any ν. □ Theorem 6: Riemann-Liouville fractional derivative of Cantor function defined on [0, 1] is not a FIF for any ν. Proof: By Proposition 2, the ν-order Riemann-Liouville fractional derivative of Cantor function is a FIF if < and 𝝂𝝂 equation (7) is satisfied. Since = , and = = , = .the value of ν can lie only between 𝒌𝒌 and𝒌𝒌 . 𝟏𝟏 𝟏𝟏 𝜸𝜸 𝒂𝒂 𝐥𝐥𝐥𝐥𝐥𝐥 𝟐𝟐 Therefore, for < < < 1,𝒌𝒌 ( ) = 𝟏𝟏 𝟑𝟑 ( 𝟐𝟐 ) ( ) = ( ) ( ) . 𝒂𝒂 𝟑𝟑 𝜸𝜸 𝜸𝜸( )𝟐𝟐 𝜸𝜸 𝟎𝟎 ( ) 𝟎𝟎 𝐥𝐥𝐥𝐥𝐥𝐥 𝟑𝟑 𝐥𝐥𝐥𝐥𝐥𝐥 𝟐𝟐 𝝂𝝂 𝒅𝒅 𝟏𝟏 𝒙𝒙 −𝝂𝝂 −𝝂𝝂 𝒙𝒙 −𝝂𝝂−𝟏𝟏 Now, putting ( ) = 𝐥𝐥𝐥𝐥𝐥𝐥 𝟑𝟑( ) = for 𝒌𝒌 [ , 𝒅𝒅],𝒅𝒅 we𝚪𝚪 𝟏𝟏see−𝝂𝝂 that𝟎𝟎 ( )𝒌𝒌= (𝚪𝚪 𝟏𝟏−𝝂𝝂 )𝟎𝟎 =𝒌𝒌 . 𝟎𝟎 𝜈𝜈 𝑫𝑫 𝒒𝒒 𝒙𝒙 � ∫ 𝒙𝒙 − 𝒕𝒕 𝒒𝒒 𝒕𝒕 (𝒅𝒅𝒅𝒅�) ∫ 𝒙𝒙 − 𝒕𝒕 𝒒𝒒 𝒕𝒕(−𝒅𝒅𝝂𝝂 𝒅𝒅) 𝟏𝟏 −𝝂𝝂 𝒙𝒙 𝟏𝟏 𝒙𝒙 As , the function ( ) and ( ) goes to . Hence,𝝂𝝂 the Riemann-Liouville−𝝂𝝂 fractional−𝟏𝟏 derivative of the 𝒒𝒒𝟐𝟐 𝒕𝒕 𝒒𝒒𝟑𝟑 𝒕𝒕 𝟐𝟐 𝒕𝒕 ∈ 𝟎𝟎 𝟏𝟏 𝑫𝑫 𝒒𝒒𝒌𝒌 𝒙𝒙 𝚪𝚪 𝟏𝟏−𝝂𝝂 ∫𝟎𝟎 𝒙𝒙 − 𝒕𝒕 �𝟐𝟐� 𝒅𝒅𝒅𝒅 𝟐𝟐𝟐𝟐 𝟏𝟏−𝝂𝝂 Cantor function defined on [0,𝝂𝝂 1] is not a FIF𝝂𝝂 for any value of ν. □ 𝒙𝒙 → 𝟎𝟎 𝑫𝑫 𝒒𝒒𝟐𝟐 𝒙𝒙 𝑫𝑫 𝒒𝒒𝟑𝟑 𝒙𝒙 ∞ 6. Conclusion In this paper, Riemann-Liouville fractional calculus of Blancmange Curve and Cantor Functions are studied. It is first shown that Blancmange Curve and Cantor Function defined on the interval [0, 1] is a Fractal Interpolation Function for appropriate interpolation points. Then, using the properties of FIF, it is proved that Riemann-Liouville fractional integral of order of Blancmange Curve and Cantor Function is a FIF. However, the Riemann-Liouville fractional derivative of Blancmange Curve and Cantor Function is shown to be not a FIF for any value of . 𝜈𝜈 𝜈𝜈

DOI: 10.26855/jamc.2020.12.003 128 Journal of Applied Mathematics and Computation

Srijanani Anurag Prasad

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DOI: 10.26855/jamc.2020.12.003 129 Journal of Applied Mathematics and Computation