Lectures on Fractional Calculus - CARMA

More documents

Recommendations

Info

Scope of Fractional Calculus• Differentiation with respect to arbitrarypowers: they can be negative• Negative differentiation is integration• Integration needs two limits to have a precisemeaning• Fractional derivatives in general need a lowerlimit, and a variable indicated Da,xαα- orderofdifferentiation or integrationa lower limit; x variable
Fractional Differentiation (1)d n x mdx n = m!(m−n)! xm−nm! =Γ(m +1)d n x mdx n = Γ(m+1)Γ(m−n+1) xm−n• This is the Riemann-Liouville derivative.• Can be applied to functions representedby Taylor series• n can be any real or complex number
Page 1 and 2: Three Lectures on
Page 3 and 4: References• A child’s garden of
Page 5: Motivation, History (2)• Initial
Page 9 and 10: Factorial FunctionKey equations for
Page 11 and 12: A Fractional DifferentiationConundr
Page 13 and 14: Integration as NegativeDifferentiat
Page 15 and 16: Differentiation as Negative Integra
Page 17 and 18: Adding up Harmonics• Fourier seri
Page 19 and 20: Momentum space• A lot of physics
Page 21 and 22: The Laplacian (2)Once:n times:∇ 2
Page 23 and 24: The Dirac delta function (2)• The
Page 25 and 26: Green’s functions• A Green’s
Page 27 and 28: The Green’s function in 2DWe star
Page 29 and 30: R ∞0The Green’s function in 2D
Page 31 and 32: A Neat TrickWe know the Green’s f
Page 33: Technical details (2)We write down
Page 36: • So we have deduced:G 2s (r) =

Lectures on Fractional Calculus - CARMA

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?