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Fractional Calculus Generalised Fractional Calculus Penelope Drastik Supervised by Marianito Rodrigo September 2018 Outline 1. Integration 2. Fractional Calculus 3. My Research Riemann sum: P tagged partition, f :[a; b] ! R n X S(f ; P) = f (ti )(xi − xi−1) i=1 The Riemann sum Partition of [a; b]: Non-overlapping intervals which cover [a; b] a = x0 < x1 < ··· < xi−1 < xi < ··· xn = b Tagged partition of [a; b]: Intervals in partition paired with tag points n P = f([xi−1; xi ]; ti )gi=1 The Riemann sum Partition of [a; b]: Non-overlapping intervals which cover [a; b] a = x0 < x1 < ··· < xi−1 < xi < ··· xn = b Tagged partition of [a; b]: Intervals in partition paired with tag points n P = f([xi−1; xi ]; ti )gi=1 Riemann sum: P tagged partition, f :[a; b] ! R n X S(f ; P) = f (ti )(xi − xi−1) i=1 The Riemann integral R b a f (x)dx = A means that: For all " > 0, there exists δ" > 0 such that if a tagged partition P satisfies 0 < xi − xi−1 < δ" for i = 1;:::; n then jS(f ; P) − Aj < " The Generalised Riemann integral R b a f (x)dx = A means that: For all " > 0, there exists a strictly positive function δ" :[a; b] ! R+ such that if a tagged partition P satisfies ti 2 [xi−1; xi ] ⊆ [ti − δ(ti ); ti + δ(ti )] for i = 1;:::; n then jS(f ; P) − Aj < " Riemann-Stieltjes integral: R b a f (x)dα(x) = A means that: For all " > 0, there exists δ" > 0 such that if a tagged partition P satisfies 0 < xi − xi−1 < δ" for i = 1;:::; n then jS(f ; P; α) − Aj < " The Riemann-Stieltjes integral Riemann-Stieltjes sum of f : Integrator α : I ! R, partition P n X S(f ; P; α) = f (ti )[α(xi ) − α(xi−1)] i=1 The Riemann-Stieltjes integral Riemann-Stieltjes sum of f : Integrator α : I ! R, partition P n X S(f ; P; α) = f (ti )[α(xi ) − α(xi−1)] i=1 Riemann-Stieltjes integral: R b a f (x)dα(x) = A means that: For all " > 0, there exists δ" > 0 such that if a tagged partition P satisfies 0 < xi − xi−1 < δ" for i = 1;:::; n then jS(f ; P; α) − Aj < " The Generalised Riemann-Stieltjes integral R b a f (x)dα(x) = A means that: For all " > 0, there exists strictly positive function δ" :[a; b] ! R+ such that if a tagged partition P satisfies ti 2 [xi−1; xi ] ⊆ [ti − δ(ti ); ti + δ(ti )] for i = 1;:::; n then jS(f ; P; α) − Aj < " Important points: I δ is a function, not a constant I Use of Riemann-Stieltjes sum Motivation Using this integral we can integrate a wide range of functions... I α(x) = x ) Generalised Riemann integral I Riemann-Stieltjes ) Generalised Riemann-Stieltjes I But not all Generalised Riemann-Stieltjes integrable functions are Riemann-Stieltjes integrable: Example: f ; α : [0; 1] ! R ( 0 if x = 0 f (x) = α(x) = 1 if x 2 (0; 1] Properties I The collection of Generalised Riemann-Stieltjes integrable functions is a vector space I Monotonicity I Convergence Theorems - interchanging an integral with a limit. Analogues of MCT, DCT, Uniform CT, Mean CT, Fatou's Lemma, plus more! I Hake's Theorem - improper integrals I Simplification Theorem I Fundamental Theorem of Calculus?? Products of functions? I f bounded below I g regulated I α Lipschitz and increasing ) fg integrable! Existence Theorems Which functions are integrable? I Step functions I Regulated functions I Continuous functions and monotone functions But...we need α to be Lipschitz continuous! Existence Theorems Which functions are integrable? I Step functions I Regulated functions I Continuous functions and monotone functions But...we need α to be Lipschitz continuous! Products of functions? I f bounded below I g regulated I α Lipschitz and increasing ) fg integrable! The α-derivative Why haven't we seen the Fundamental Theorem of Calculus yet?? Since we are integrating with respect to α, it also makes sense to consider derivatives with respect to α... Definition α continuous and strictly increasing f is α-differentiable at x0 if f (x) − f (x0) Dαf (x0) = lim x!x0 α(x) − α(x0) exists. Properties All the usual results still hold... I Algebra of differentiable functions I Chain rule, product rule, quotient rule I Mean Value Theorems I Characterisation of maxima/minima How does it relate to the ordinary derivative? Theorem 0 f (x0) f ; α nice ) Dαf (x0) = 0 α (x0) The Fundamental Theorem Theorem I α continuous and strictly increasing I F continuous I DαF (x) = f (x) for all x 2 I R b Then f integrable and a fdα = F (b) − F (a) Why is this important? I Integrability as consequence, not hypothesis I Integration by parts I Substitution theorems I New result! Factorials ! Gamma function dk f (x) Γ(n + 1) = xn−k dxk Γ(n − k + 1) Fractional Calculus 2 3 k Motivation: Familiar with Dx f ; Dx f ; Dx f ; :::Dx f . µ What about Dx f for µ 2 R? Many ways to approach this problem... 1) (Lacroix, 1819) For f (x) = xn, we have dk f (x) n! = xn−k dxk (n − k)! Fractional Calculus 2 3 k Motivation: Familiar with Dx f ; Dx f ; Dx f ; :::Dx f . µ What about Dx f for µ 2 R? Many ways to approach this problem... 1) (Lacroix, 1819) For f (x) = xn, we have dk f (x) n! = xn−k dxk (n − k)! Factorials ! Gamma function dk f (x) Γ(n + 1) = xn−k dxk Γ(n − k + 1) For functions 1 X aj x f (x) = cj e j=0 we have 1 µ X µ aj x Dx f (x) = cj aj e j=0 3) (Euler, 1730) Interpolation between integer derivatives 2) (Liouville, 1832) We have k ax k ax Dx e = a e Now replace k ! µ. 3) (Euler, 1730) Interpolation between integer derivatives 2) (Liouville, 1832) We have k ax k ax Dx e = a e Now replace k ! µ. For functions 1 X aj x f (x) = cj e j=0 we have 1 µ X µ aj x Dx f (x) = cj aj e j=0 2) (Liouville, 1832) We have k ax k ax Dx e = a e Now replace k ! µ. For functions 1 X aj x f (x) = cj e j=0 we have 1 µ X µ aj x Dx f (x) = cj aj e j=0 3) (Euler, 1730) Interpolation between integer derivatives Replace factorial with Γ, replace k with µ 1 Z x D−µf (x) = (x − u)µ−1f (u)du Γ(µ) 0 (provided it exists) Today's definitions Fractional integral: Recall Dirichlet's Theorem for n-fold integration: 1 Z x (x − u)n−1f (u)du (n − 1)! 0 Today's definitions Fractional integral: Recall Dirichlet's Theorem for n-fold integration: 1 Z x (x − u)n−1f (u)du (n − 1)! 0 Replace factorial with Γ, replace k with µ 1 Z x D−µf (x) = (x − u)µ−1f (u)du Γ(µ) 0 (provided it exists) Fractional derivatives Riemann-Liouville: (integrate then differentiate) Dµf (x) = Ddµe[D−(dµe−µ)f (x)] Caputo: (differentiate then integrate) Dµf (x) = D−(dµe−µ)[Ddµef (x)] Generalised Fractional Calculus What is it? I Riemann integral ! Generalised Riemann-Stieltjes integral I Ordinary derivative ! α-derivative Motivation: I Easier to prove existence results I Easier to deal with improper integrals (Hake's Theorem!) I Can work with a wider range of functions I ...and more! (Future work) Fractional α integral Z x −µ 1 µ−1 Dα f (x) = (x − u) f (u)dα(u) Γ(µ) 0 Theorem α Lipschitz and increasing, f regulated or bounded below ) integral exists. Caputo derivative wrt α −(dµe−µ) dµe C(f ; α; µ)(x) = Dα Dα f (x) Z x −(dµe−µ) dµe = (x − u) [Dα f (u)]dα(u) 0 Theorem α Lipschitz and increasing, f \C n" ) integral exists. Fractional α derivatives Riemann-Liouville derivative wrt α dµe −(dµe−µ) RL(f ; α; µ) = Dα Dα f (x) Existence? Problem - α-derivative. Very specific α needed. Fractional α derivatives Riemann-Liouville derivative wrt α dµe −(dµe−µ) RL(f ; α; µ) = Dα Dα f (x) Existence? Problem - α-derivative. Very specific α needed. Caputo derivative wrt α −(dµe−µ) dµe C(f ; α; µ)(x) = Dα Dα f (x) Z x −(dµe−µ) dµe = (x − u) [Dα f (u)]dα(u) 0 Theorem α Lipschitz and increasing, f \C n" ) integral exists. Example: Functions that ordinary fractional calculus can't handle! Functions α; f : [0; 1] ! R: ( x if 0 ≤ x ≤ 0:5 α(x) = x2 + 0:5 if 0:5 < x ≤ 1 ( 0 if 0 ≤ x ≤ 0:25 f (x) = 1 if 0:25 < x ≤ 1 I Can't use the Simplification Theorem I But the fractional α-integral exists! Outline 1. Integration 2. Fractional Calculus 3. My Research Next steps? I Existence I Geometric interpretation I Fractional differential equations References I Bartle, R. (2001). A modern theory of integration. Providence (R.I.): American mathematical Society. I Bartle, R. and Sherbert, D. (2011). Introduction to real analysis. [New York]: J. Wiley and Sons. I Widder, D. (1989). Advanced calculus. New York: Dover Publ. I Apostol, T. (1974). Mathematical analysis. 2nd ed. Reading, Mass., etc.: Addison-Wesley. I Bartle, R. (1996). Return to the Riemann Integral. The American Mathematical Monthly, 103(8), p.625.
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