Riemann-Liouville Fractional Calculus of Coalescence Hidden-variable - - PowerPoint PPT Presentation

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

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Outline

1

Introduction

2

Riemann-Liouville fractional integral

3

Riemann-Liouville fractional derivative

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Outline

1

Introduction

2

Riemann-Liouville fractional integral

3

Riemann-Liouville fractional derivative

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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

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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].

10 20 30 40 50 60 70 80 90 100 20 40 60 80 100 120 10 20 30 40 50 60 70 80 90 100 20 40 60 80 100 120 140 160 180

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Construction of a CHFIF

Given data{(xk, yk) ∈ R2 : k = 0, 1, . . . , N} Generalized data {(xk, yk, zk) ∈ R3 : 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 xN − x0 (x − x0) + xk−1 (1)

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Construction of a CHFIF

Fk : I × R2 → R2 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) ωk : I × R2 → I × R2 ωk(x, y, z) = (Lk(x), Fk(x, y, z)), k = 1, 2, . . . N

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Construction of a CHFIF

Theorem ( [Chand A.K.B. and Kapoor G.P ., 2007])

(1) {I × R2; ωk, k = 1, 2, . . . , N} is a hyperbolic IFS with respect to a metric equivalent to Euclidean metric on R3. (2) The attractor G ⊆ R3 such that G = N

k=1 ωk(G) of the above IFS is

graph of a continuous function f : I → R2 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))}.

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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

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Construction of CHFIF

CHFIF : if xk−1 ≤ x ≤ xk then f1(x) = αk f1(L−1

k (x)) + βk f2(L−1 k (x)) + pk(L−1 k (x))

FIF : if xk−1 ≤ x ≤ xk then f2(x) = γk f2(L−1

k (x)) + qk(L−1 k (x))

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Outline

1

Introduction

2

Riemann-Liouville fractional integral

3

Riemann-Liouville fractional derivative

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Riemann-Liouville fractional integral

Definition

Let −∞ < a < x < b < ∞. The Riemann-Liouville fractional integral of

• rder ν > 0 with lower limit a is defined for locally integrable functions

f : [a, b] → R as Iν

a+ f(x) =

1 Γ(ν)

x

• a

(x − t)ν−1f(t)dt (3) for x > a.

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Riemann-Liouville fractional integral

Given data{(xk, yk) ∈ R2 : k = 0, 1, . . . , N} Generalized data {(xk, yk, zk) ∈ R3 : k = 0, 1, . . . , N} pν

k (x) = aν k Iν x0+ pk(x) +

1 Γ(ν)

xk−1

• x0

(Lk(x) − t)ν−1 f1(t) dt (4) and qν

k (x) = aν k Iν x0+ qk(x) +

1 Γ(ν)

xk−1

• x0

(Lk(x) − t)ν−1 f2(t)dt. (5)

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Riemann-Liouville fractional integral

Fν

k (x, y, z) =

• Fν

k,1(x, y, z), Fν k,2(x, z)

• =
• aν

k αky + aν k βkz + pν k(x), aν k γkz + qν k (x)

• (6)

Define ων

k (x, y, z) = (Lk(x), Fν k (x, y, z)) ;

(7) yν

0 = 0 = zν 0,

zν

N =

qν

N(xN)

1 − aν

NγN

, yν

N =

aν

NβN

1 − aν

NαN

zν

N +

pν

N(xN)

1 − aν

NαN

, zν

k = aν kγkzν N + qν k (xN) = qν k+1(x0)

and yν

k = aν kαkyν N + aν k βkzν N + pν k (xN) = pν k+1(x0), k = 1, 2, . . . , N − 1.

(8)

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Riemann-Liouville fractional integral of FIF

Proposition

Let f2 be a FIF passing through the interpolation data given by {(xk, zk) ∈ R2 : k = 0, 1, . . . , N}. Then, Riemann-Liouville fractional integral of a FIF of order ν is also a FIF passing through the data {(xk, zν

k ) ∈ R2 : k = 0, 1, . . . , N}, where zν k are given by (8).

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Riemann-Liouville fractional integral of FIF

Theorem ( [S.A.P, 2017])

Let f1 be the CHFIF passing through the interpolation data given by {(xk, yk) ∈ R2 : k = 0, 1, . . . , N} and f2 be the corresponding FIF passing through the data {(xk, zk) ∈ R2 : 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 {(xk, yν

k ) ∈ R2 : k = 0, 1, . . . , N}, where yν k are given by (8).

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Riemann-Liouville fractional integral of CHFIF

Sketch of Proof: Let x such that xk−1 < x < xk for some k ∈ {1, 2, . . . , N}. Then, Iν

x0+ f1(x) =

1 Γ(ν)

x

• x0

(x − t)ν−1 f1(t)dt = 1 Γ(ν)     

xk−1

• x0

(x − t)ν−1 f1(t) dt +

x

• xk−1

(x − t)ν−1 f1(t) dt     

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Riemann-Liouville fractional integral of CHFIF

Iν

x0+ f1(x) =

1 Γ(ν)   

xk−1

• x0

(x − t)ν−1 f1(t) dt + aν

k L−1

k

(x)

• x0

(L−1

k (x) − t)ν−1 f1(Lk(t)) dt

     = aν

k αk Iν x0+ f1(L−1 k (x)) + aν k βk Iν x0+ f2(L−1 k (x))

+ aν

k Iν x0+ pk(L−1 k (x)) +

1 Γ(ν)   

xk−1

• x0

(x − t)ν−1 f1(t) dt   

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Outline

1

Introduction

2

Riemann-Liouville fractional integral

3

Riemann-Liouville fractional derivative

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Riemann-Liouville fractional derivative

Definition

Let −∞ < a < x < b < ∞, 0 < ν, f ∈ L1([a, b]) and In−νf ∈ W1,1, where n is the smallest integer greater than ν . The Riemann-Liouville fractional derivative of order ν with lower limit a is defined as (Dν

a+f)(x) = dn

dxn (In−ν

a+

f)(x) and (Dν

a+ f)(x) = f(x) when ν = 0.

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Riemann-Liouville fractional derivative of FIF

qdν

k (x) = a−ν k Dνqk(x) +

a−n

k

Γ(n − ν) dn dxn  

xk−1

• x0

f2(t)(Lk(x) − t)n−ν−1 dt   (9) and pdν

k (x) = a−ν k Dν pk(x) +

a−n

k

Γ(n − ν) dn dxn  

xk−1

• x0

f1(t)(Lk(x) − t)n−ν−1 dt   . (10)

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Riemann-Liouville fractional derivative of FIF

Proposition

Let f2 be a FIF passing through the interpolation data {(xk, zk) ∈ R2 : k = 0, 1, . . . , N} and |γk| < aν

k for some fixed ν > 0. Then

Riemann-Liouville fractional derivative of a FIF of order ν is also a FIF provided (9) is satisfied.

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Riemann-Liouville fractional derivative of CHFIF

Theorem ( [S.A.P, 2017])

Let f1 be the CHFIF passing through the interpolation data given by {(xk, yk) ∈ R2 : k = 0, 1, . . . , N} and f2 be the corresponding FIF passing through the data {(xk, zk) ∈ R2 : k = 0, 1, . . . , N}. For a fixed ν > 0, if the free variables and constrained variables are such that |αk| < aν

k,

|γk| < aν

k and |βk| + |γk| < aν k then Riemann-Liouville fractional

derivative of a CHFIF of order ν is also a CHFIF provided (9) and (10) are satisfied.

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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 ν < log |γk| log ak Riemann-Liouville fractional derivative of f2 of order ν exists and is a FIF provided (9) is satisfied.

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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 ν < min log |αk| log ak , log(|βk| + |γk|) log ak

• Riemann-Liouville fractional derivative of f1 of order ν exists and is a

CHFIF provided (9) and (10) are satisfied.

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Example

Blancmange Curve:[Takagi, 1903] B(x) =

∞

• n=0

s(2nx) 2n x ∈ [0, 1], where, s(y) = min

m∈Z |y − m|, y ∈ R.

B x+k−1

2

• = 1

2B(x) + k−1+(−1)k−1x 2

x ∈ [0, 1] for k = 1, 2.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 1.2 1.4

Figure: Blanchmange Curve

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Example

γk = 1

2 for k = 1, 2.

Lk(x) = 1

2x + k−1 2

qk(x) = k−1+(−1)k−1x

2

ν < log |γk|

log ak = log 1/2 log 1/2 = 1

qdν

1 (x) = 1 21−ν Γ(2−ν)x1−ν

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Example

qdν

2 (x) =

1 21−ν Γ(1 − ν)

• xν −

x1−ν (1 − ν)

• −

1 Γ(1 − ν)

∞

• n=0

1 2n

2n

• m=1

(−1)m−1× ×

• 2n

x+1

2 − m 2n+1

1−ν − x+1

2

− m−1

2n+1

1−ν (1 − ν)

• +

x + 1 2 − m 2n+1 −ν m 2 − A

• −

x + 1 2 − m − 1 2n+1 −ν m − 1 2 − A

• A =
• m/2

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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.

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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.

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Thank You!

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