Fractional Calculus

Andrés L . Granados M.

Fractional Calculus

Andrés L . Granados M.

https://doi.org/10.13140/RG.2.2.29497.75364/10

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Fractional Calculus resume facsimil that reproduces the principal formulas for calculation of derivatives and integrals of fractional order.

FRACTIONAL CALCULUS p p(p − 1)(p − 2) . . . (p − r + 1) = r r! p p(p + 1)(p + 2) . . . (p + r − 1) = r r! (−1)r Equivalent to Newton − Gregory Interpolation Polynomials n 1 (p) r p (−1) f (t − rh) fh (t) = p r h r=0 p a Dt f (t) 1 = Γ(p) −p a Dt f (t) 1 Γ(p) = (p) lim lim n αnr (t) n→∞ nh=t−a −α a Dt f (t) nh=t−a h→0 nh=t−a r=0 t 1 Γ(α) f (x) dxα = α a −α t Db f (t) = 1 Γ(α) h→0 nh=t−a (0 ≤ p ≤ n) (t) Grünwald-Letnikov Derivarives t n n n p−1 lim (t − x)p−1 f (x) dx αr (t) = lim h (rh) f (t − rh) = n→∞ r=0 = lim fh Γ(p) p lim βr = lim p−1 =1 r→∞ r→∞ r r r=0 = f (−α) (t) = (−p) −p a Dt f (t) h→0 nh=t−a βr αnr (t) h→0 nh=t−a Similar to Newton − Cotes Integration Formulas n p (−p) p f (t − rh) fh (t) = h r r=0 = lim fh (t) n −p p p+1 p+1 = = − r r r r−1 a r=0 t a b (x − t)α−1 f (x) dx −α a Dt (Inverse) t ⎧ k t d ⎪ ⎪ f (x) dxα ⎪ ⎨ dtk α a p t a Dt f (t) = ⎪ ⎪ ⎪ f (x) dx|p| ⎩ Cauchy Formula Riemann-Liouville α = a It = α a It Integral (t − x)α−1 f (x) dx (α ∈ R+ ) p ≥ 0 (p + α = k) p < 0 (k ∈ N, p ∈ R) |p| a p q a Dt [ a Dt f (t) ] n a Dt f (t) = = q p a Dt [ a Dt f (t) ] dn f (t) d d · · · f (x) = f (n) (t) = dtn dt dt n times p p p a Dt f (t) g(t) = Ωn (t) − Rn (t) = Commutative + Additive Property t t t −n n f (x) dx · ·· dx ··· f (x) dx = a Dt f (t) = n a a a n times p+q f (t) a Dt n times Ωpn = n p k=0 k f (k) (t) p−k g(t) a Dt (Leibniz Rule) (n ≥ p + 1) t t 1 −p−1 n n (t − x) wf (x) g(x) dx wf (x) = (x − τ )n f (n+1) (τ ) dτ = n! Γ(−p) a x p a Dt f ◦g(t) = Apply diﬀerential “Leibniz Rule” to ‘◦’ product for composition of functions. Rnp (t) p a Dt B(t) A(t) f (t, x) dx = B(t) p a Dt f (t, x) dx + f [t, B(t)] a Dtp B(t) A(t) 1 − f [t, A(t)] a Dtp A(t) Integral with Parameters Note: Regressive Polynomials of Newton-Gregory Pm (p) = h−p (p) fh (t) = Pm (p) + Rm (p) m (−1)n−k k=0 m <p < m + 1 Rm (p) = h −p n−m−1 r=0 p−k−1 ∇k f (a + kh) n−k p−m−1 (−1) ∇m+1 f (a − rh) r r Backward Diﬀerence p p−1 p−1 = + r r r−1 ∇f (t − rh) = f (t − rh) − f (t − (r + 1)h) ∇k f (t − rh) = ∇k−1 f (t − rh) − ∇k−1 f (t − (r + 1)h) lim n→∞ n n−k p−k k ∇ f (a + kh) lim = f (k) (a) h→0 hk =1 n! nz n→∞ z(z + 1) · · · (z + n) Γ(z) = lim lim Pm (p) = h→0 nh=t−a m (t − a)−p+k f (k) (a) Γ(−p + k + 1) k=0 p − m − 1 −m+p lim βr = lim (−1) Γ(−p + m + 1) r =1 r→∞ r→∞ r n→∞ p a Dt f (t) n−m−1 αnr (t) = lim r=0 h→0 nh=t−a n−m−1 h (rh)m−p r=0 (p) h→0 nh=t−a h→0 nh=t−a t lim (−1) n→∞ = (t − x)m−p f (m+1) (x) dx a n−k ∇m+1 f (t − rh) = hm+1 = lim fh (t) = lim Pm (p) + lim Rm (p) h→0 nh=t−a n 1 lim Rm (p) = h→0 Γ(−p + m + 1) nh=t−a r lim nz (z + n) = lim z n→∞ (ℜ(z) > 0) p−k−1 (n − k)p−k n−k 1 Γ(−p + k + 1) t (t − x)m−p f (m+1) (x) dx a p a Dt (t − a)ν = Γ(ν + 1) (t − a)ν−p Γ(−p + ν + 1) t m (t − a)−p+k f (k) (a) 1 + (t − x)m−p f (m+1) (x) dx Γ(−p + k + 1) Γ(−p + m + 1) a k=0 t t 1 dm+1 1 −p−1 m−p = (t − x) f (x) dx = m+1 (t − x) f (x) dx Γ(−p) a dt Γ(−p + m + 1) a = After performing repeatedly integration by parts (cancelling each of the terms in the sum) and finally making opposite diﬀerentiation ( outside (d/dt)m+1 & inside a Dt−m−1 brackets ). It is convenient the change of variables α = m + 1 − p, then the diﬀerential is of order m + 1 = p + α and α < 1 (Riemann-Liouville definition). p q q p =⇒ f (k) (a) = 0, (k = 0, 1, . . . , m − 1) a Dt [ a Dt f (t) ] = a Dt [ a Dt f (t) ] Reference. [1] Podlubny, Igor. Fractional Diﬀerential Equations, An Introduction to Fractional Derivatives, Fractional Diﬀerential Equations, to Methods of their Solution and some of their Applications. Academic Press (San Diego-California, USA), 1999. by Andrés Leonell Granados M. January 10th, 2024 2

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