Riemann-Liouville Fractional Calculus of Coalescence Hidden-variable Fractal Interpolation Functions
Srijanani Anurag Prasad
Department of Applied Sciences, The NorthCap University, Gurgaon
6th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals June 13-17, 2017
S.A.Prasad (NCU) FC June13-17,2017 1/31 Outline
1 Introduction
2 Riemann-Liouville fractional integral
3 Riemann-Liouville fractional derivative
S.A.Prasad (NCU) FC June13-17,2017 2/31 Outline
1 Introduction
2 Riemann-Liouville fractional integral
3 Riemann-Liouville fractional derivative
S.A.Prasad (NCU) FC June13-17,2017 3/31 Fractal Interpolation Function (FIF)
Fractal Interpolation Function (FIF) : [Barnsley M.F., 1986]
Similarities of FIF and traditional methods ∗ Geometrical Character - can be plotted on graph ∗ Represented by formulas
Difference between FIF and traditional methods ∗ Fractal Character
S.A.Prasad (NCU) FC June13-17,2017 4/31 Coalescence Hidden-variable Interpolation Functions
For simulating curves that exhibit self-affine and non-self-affine nature simultaneously, Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) was introduced by [Chand A.K.B. and Kapoor G.P., 2007].
120 180
160 100 140
80 120
100 60 80
40 60
40 20 20
0 0 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100
S.A.Prasad (NCU) FC June13-17,2017 5/31 Construction of a CHFIF
2 Given data{(xk, yk) ∈ R : k = 0, 1,..., N}
3 Generalized data {(xk, yk, zk) ∈ R : k = 0, 1,..., N}
[x0, xN ]= I, [xk−1, xk]= Ik, k = 1, 2,..., N
Lk : I → Ik
Lk(x0)= akx + bk xk − xk−1 = (x − x0)+ xk−1 (1) xN − x0
S.A.Prasad (NCU) FC June13-17,2017 6/31 Construction of a CHFIF
2 2 Fk : I × R → R
Fk(x, y, z)= αky + βkz + pk(x), γkz + qk(x) (2) |αk| < 1 , |γk| < 1 , |βk| + |γk| < 1
Fk(x0, y0, z0) = (yk−1, zk−1)
Fk(xN , yN , ZN ) = (yk, zk)
2 2 ωk : I × R → I × R
ωk(x, y, z) = (Lk(x), Fk(x, y, z)), k = 1, 2,... N
S.A.Prasad (NCU) FC June13-17,2017 7/31 Construction of a CHFIF
Theorem ( [Chand A.K.B. and Kapoor G.P., 2007]) 2 (1) {I × R ; ωk, k = 1, 2,..., N} is a hyperbolic IFS with respect to a metric equivalent to Euclidean metric on R3.
R3 N (2) The attractor G ⊆ such that G = k=1 ωk(G) of the above IFS is 2 graph of a continuous function f : I → R such that f (xk) = (yk, zk) for k = 0, 1,..., N i.e. G = {(x, f (x)) : x ∈ I and f (x) = (y(x), z(x))}.
S.A.Prasad (NCU) FC June13-17,2017 8/31 Construction of CHFIF
Definition The Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) for the given interpolation data {(xk, yk) : k = 0, 1,..., N } is defined as the continuous function f1 : I → R, where f1 is the first component of the continuous function f = (f1, f2), graph of which is attractor of the hyperbolic IFS.
f2 - AFIF (Self-Affine Fractal Interpolation Function)
yk = zk and αk + βk = γk for all k, f1 = f2 is FIF
S.A.Prasad (NCU) FC June13-17,2017 9/31 Construction of CHFIF
CHFIF : if xk−1 ≤ x ≤ xk then
−1 −1 −1 f1(x)= αk f1(Lk (x)) + βk f2(Lk (x)) + pk(Lk (x))
FIF : if xk−1 ≤ x ≤ xk then
−1 −1 f2(x)= γk f2(Lk (x)) + qk(Lk (x))
S.A.Prasad (NCU) FC June13-17,2017 10/31 Outline
1 Introduction
2 Riemann-Liouville fractional integral
3 Riemann-Liouville fractional derivative
S.A.Prasad (NCU) FC June13-17,2017 11/31 Riemann-Liouville fractional integral
Definition Let −∞ < a < x < b < ∞. The Riemann-Liouville fractional integral of order ν > 0 with lower limit a is defined for locally integrable functions f : [a, b] → R as
x 1 Iν f (x)= (x − t)ν−1f (t)dt (3) a+ Γ(ν) a for x > a.
S.A.Prasad (NCU) FC June13-17,2017 12/31 Riemann-Liouville fractional integral
2 Given data{(xk, yk) ∈ R : k = 0, 1,..., N}
3 Generalized data {(xk, yk, zk) ∈ R : k = 0, 1,..., N}
xk−1 1 pν (x)= aν Iν p (x)+ (L (x) − t)ν−1 f (t) dt (4) k k x0+ k Γ(ν) k 1 x 0
and
xk−1 1 qν (x)= aν Iν q (x)+ (L (x) − t)ν−1 f (t)dt. (5) k k x0+ k Γ(ν) k 2 x 0
S.A.Prasad (NCU) FC June13-17,2017 13/31 Riemann-Liouville fractional integral
ν ν ν Fk (x, y, z)= Fk,1(x, y, z), Fk,2(x, z) ν ν ν ν ν = ak αky + ak βkz + pk (x ), ak γkz + qk (x) (6) Define ν ν ωk (x, y, z) = (Lk(x), Fk (x, y, z)) ; (7) ν ν y0 = 0 = z0 , ν ν qN (xN ) zN = ν , 1 − aN γN ν ν ν aN βN ν pN (xN ) yN = ν zN + ν , 1 − aN αN 1 − aN αN ν ν ν ν ν zk = ak γkzN + qk (xN )= qk+1(x0) ν ν ν ν ν ν ν and yk = ak αkyN + ak βkzN + pk (xN )= pk+1(x0), k = 1, 2,..., N − 1. (8)
S.A.Prasad (NCU) FC June13-17,2017 14/31 Riemann-Liouville fractional integral of FIF
Proposition
Let f2 be a FIF passing through the interpolation data given by 2 {(xk, zk) ∈ R : k = 0, 1,..., N}. Then, Riemann-Liouville fractional integral of a FIF of order ν is also a FIF passing through the data ν R2 ν {(xk, zk ) ∈ : k = 0, 1,..., N}, where zk are given by (8).
S.A.Prasad (NCU) FC June13-17,2017 15/31 Riemann-Liouville fractional integral of FIF
Theorem ( [S.A.P, 2017])
Let f1 be the CHFIF passing through the interpolation data given by 2 {(xk, yk) ∈ R : k = 0, 1,..., N} and f2 be the corresponding FIF passing 2 through the data {(xk, zk) ∈ R : k = 0, 1,..., N}. Then, Riemann-Liouville fractional integral of a CHFIF of order ν given by (3) is also a CHFIF passing through the data ν R2 ν {(xk, yk ) ∈ : k = 0, 1,..., N}, where yk are given by (8).
S.A.Prasad (NCU) FC June13-17,2017 16/31 Riemann-Liouville fractional integral of CHFIF
Sketch of Proof:
Let x such that xk−1 < x < xk for some k ∈ {1, 2,..., N}. Then,
x 1 Iν f (x)= (x − t)ν−1 f (t)dt x0+ 1 Γ(ν) 1 x 0
xk−1 x 1 = (x − t)ν−1 f (t) dt + (x − t)ν−1 f (t) dt Γ(ν) 1 1 x 0 xk −1
S.A.Prasad (NCU) FC June13-17,2017 17/31 Riemann-Liouville fractional integral of CHFIF
xk−1 1 Iν f (x)= (x − t)ν−1 f (t) dt x0+ 1 Γ(ν) 1 x 0 − L 1(x) k ν −1 ν−1 + ak (Lk (x) − t) f1(Lk(t)) dt x 0 = aν α Iν f (L−1(x)) + aν β Iν f (L−1(x)) k k x0+ 1 k k k x0+ 2 k xk−1 1 + aν Iν p (L−1(x)) + (x − t)ν−1 f (t) dt k x0+ k k Γ(ν) 1 x 0
S.A.Prasad (NCU) FC June13-17,2017 18/31 Outline
1 Introduction
2 Riemann-Liouville fractional integral
3 Riemann-Liouville fractional derivative
S.A.Prasad (NCU) FC June13-17,2017 19/31 Riemann-Liouville fractional derivative
Definition n−ν 1,1 Let −∞ < a < x < b < ∞, 0 <ν, f ∈ L1([a, b]) and I f ∈ W , where n is the smallest integer greater than ν . The Riemann-Liouville fractional derivative of order ν with lower limit a is defined as dn (Dν f )(x)= (In−ν f )(x) a+ dxn a+
ν and (Da+ f )(x)= f (x) when ν = 0.
S.A.Prasad (NCU) FC June13-17,2017 20/31 Riemann-Liouville fractional derivative of FIF
xk−1 −n n dν −ν ν ak d n−ν−1 q (x)= a D qk(x)+ f2(t)(Lk(x) − t) dt (9) k k Γ(n − ν) dxn x 0 and
xk−1 −n n dν −ν ν ak d n−ν−1 p (x)= a D pk(x)+ f1(t)(Lk(x) − t) dt . k k Γ(n − ν) dxn x 0 (10)
S.A.Prasad (NCU) FC June13-17,2017 21/31 Riemann-Liouville fractional derivative of FIF
Proposition
Let f2 be a FIF passing through the interpolation data R2 ν {(xk, zk) ∈ : k = 0, 1,..., N} and |γk| < ak for some fixed ν > 0. Then Riemann-Liouville fractional derivative of a FIF of order ν is also a FIF provided (9) is satisfied.
S.A.Prasad (NCU) FC June13-17,2017 22/31 Riemann-Liouville fractional derivative of CHFIF
Theorem ( [S.A.P, 2017])
Let f1 be the CHFIF passing through the interpolation data given by 2 {(xk, yk) ∈ R : k = 0, 1,..., N} and f2 be the corresponding FIF passing 2 through the data {(xk, zk) ∈ R : k = 0, 1,..., N}. For a fixed ν > 0, if ν the free variables and constrained variables are such that |αk| < ak , ν ν |γk| < ak and |βk| + |γk| < ak then Riemann-Liouville fractional derivative of a CHFIF of order ν is also a CHFIF provided (9) and (10) are satisfied.
S.A.Prasad (NCU) FC June13-17,2017 23/31 Riemann-Liouville fractional derivative of FIF
Suppose f2 is a FIF passing through interpolation data given by {(xk, zk) : k = 0, 1, 2,..., N} constructed with the free variables γk for k = 1, 2,..., N. Then, for all ν satisfying
log |γ | ν < k log ak
Riemann-Liouville fractional derivative of f2 of order ν exists and is a FIF provided (9) is satisfied.
S.A.Prasad (NCU) FC June13-17,2017 24/31 Riemann-Liouville fractional derivative of CHFIF
Suppose f1 is a CHFIF passing through a interpolation data given by {(xk, yk) : k = 0, 1, 2,..., N} constructed with the free variables αk, γk and constrained variables βk for k = 1, 2,..., N. Then, for all ν satisfying
log |α | log(|β | + |γ |) ν < min k , k k log a log a k k
Riemann-Liouville fractional derivative of f1 of order ν exists and is a CHFIF provided (9) and (10) are satisfied.
S.A.Prasad (NCU) FC June13-17,2017 25/31 Example
Blancmange Curve:[Takagi, 1903] ∞ s(2nx) B(x)= x ∈ [0, 1], 2n n=0 where, s(y)= min |y − m|, y ∈ R. m∈Z − x+k−1 1 k−1+(−1)k 1x B 2 = 2 B(x)+ 2 x ∈ [0, 1] for k = 1, 2. 1.4 1.2
1
0.8
0.6
0.4
0.2
0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure: Blanchmange Curve
S.A.Prasad (NCU) FC June13-17,2017 26/31 Example
1 γk = 2 for k = 1, 2.
1 k−1 Lk(x)= 2 x + 2
k−1+(−1)k−1x qk(x)= 2
ν < log |γk| = log 1/2 = 1 log ak log 1/2
dν 1 1−ν q1 (x)= 21−ν Γ(2−ν) x
S.A.Prasad (NCU) FC June13-17,2017 27/31 Example
1 x1−ν qdν (x)= xν − 2 21−ν Γ(1 − ν) (1 − ν) n 1 ∞ 1 2 − (−1)m−1× Γ(1 − ν) 2n n=0 m=1 1−ν 1−ν x+1 − m − x+1 − m−1 × 2n 2 2n+1 2 2n+1 (1 − ν ) x + 1 m −ν m + − − A 2 2n+1 2 x + 1 m − 1 −ν m − 1 − − − A 2 2n+1 2
m/2 if m is even A = (m − 1)/2 if m is odd
S.A.Prasad (NCU) FC June13-17,2017 28/31 References I
Barnsley M.F. (1986). Fractal functions and interpolation. Constructive Approximation, 2:303–329. Chand A.K.B. and Kapoor G.P. (2007). Smoothness analysis of coalescence hidden variable fractal interpolation functions. International Journal of Non-Linear Science, 3:15–26. Changpin Li and Deliang Qian and Yang Quan Chen (2011). On Riemann-Liouville and Caputo Derivatives. Discrete Dynamics in Nature and Society, Article ID 562494). Hilfer R.(2008). Threefold Introduction to Fractional Derivatives, In. Anomalous Transport: Foundations and Applications, Wiley-VCH,Weinheim.
S.A.Prasad (NCU) FC June13-17,2017 29/31 References II
Prasad S.A.(2017). Fractional calculus of coalescence hidden-variable fractal interpolation functions. Fractals: Complex Geometry, Patterns, and Scaling in Nature and Society, 25(2). Teiji Takagi(1903). A Simple Example of a Continuous Function without Derivative. Proc. Phys. Math. Japan,, 1, 176–177. XueZai Pan(2014). Fractional Calculus of Fractal Interpolation Function on [0, b].. Abstract and Applied Analysis, Article ID: 640628.
S.A.Prasad (NCU) FC June13-17,2017 30/31 Thank You!
S.A.Prasad (NCU) FC June13-17,2017 31/31