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 x0 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 x0 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 x0 xk−1 x 1 = (x − t)ν−1 f (t) dt + (x − t)ν−1 f (t) dt Γ(ν) 1 1 x0 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 x0 − L 1(x) k ν −1 ν−1 + ak (Lk (x) − t) f1(Lk(t)) dt x0 = 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 x0 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 x0 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 x0 (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.