This notebook investigates the properties of non-integer differential operators using Fourier analysis.

Transcription

1 Frctionl erivtives.nb Frctionl erivtives by Fourier ecomposition by Eric Thrne 4/9/6 This notebook investigtes the properties of non-integer differentil opertors using Fourier nlysis. In[]:= << Grphics`Legend` We'll strt out on the intervl (-, ). f[x] is normlizble function. Let's pick it to be Gussin for the ske of being concrete. In[]:= := -x F[k] is its Fourier trnsform. In[]:= := EvluteA * Â*k*x xe - h[,,x] is the H ÅÅÅÅÅÅ d dx L derivtive of f[x]. We'll strt by ssuming is rel number greter thn. (< corresponds to integrtion.) We'll lso restrict our ttention to xœrels, for the ske of Mthemtic. In[4]:= h@_, x_ := EvluteA H-ÂL * IntegrteA HkL * F@k * -Â*k*x ÅÅ, 8k, -, <, Assumptions Ø 8x œ Rels, œ Rels, <E * p E Look, M: hypergeometric functions. In[5]:= Out[5]= h@, x ÅÅÅÅ è!!! p JH-ÂL J  HH-L - L x GmmA + ÅÅÅÅ E HypergeometricFA + ÅÅÅÅ, ÅÅÅÅ, -x E + HH-L + L GmmA ÅÅÅ + E HypergeometricFA ÅÅÅ +, ÅÅÅÅ, -x ENN H[,x] is h[,x] explicitly defined to void convergence issues in Mthemtic. i In[6]:= H@_, x_ := ÅÅÅÅÅ è!!! jh-âl i j  HH-L - L x GmmA + ÅÅÅÅ p k k E HypergeometricFA + ÅÅÅÅ, ÅÅÅÅ, -x E + HH-L + L GmmA ÅÅÅ + E HypergeometricFA ÅÅÅ +, ÅÅÅÅ, -x E y z y z {{ In[7]:= H@, x; We here plot the rel prt of these functions becuse Mthemtic clcultes them to finite precision, nd, in doing so, incorrectly gives them tiny (but non-zero) imginry prts. Printed by Mthemtic for Students

2 Frctionl erivtives.nb In[8]:= 8, ê 4, x, ê, x, ê 4, x, x<, 8x, -, <, PlotStyle Ø PlotLegend Ø 8"", " ê4 x ", " ê4 x ", " ê4 x ", " x "<, LegendPosition Ø 8, -.5<, LegendBorderSpce Ø, LegendSpcing Ø, LegendTextSpce Ø 5, LegendLbel Ø "Rel Prt" Rel Prt x ê4 x ê4 x ê4 -.5 x -.75 Out[8]= Ü Grphics Ü Here we plot the frctionl derivtives of f[x] s function of on = (,). Printed by Mthemtic for Students

3 Frctionl erivtives.nb In[9]:= x, 8,, <, 8x, -, <, Mesh Ø True, PlotRnge Ø 8-, <, ColorFunction Ø Hue, ViewPoint Ø 8, -.5,.7< Out[9]= Ü SurfceGrphics Ü This is the sme plot with the rnge extended to the intervl = (,). Printed by Mthemtic for Students

4 We cn keep integrting by prts until the k delt function hs no derivtives cting on it, nd for ll our lbor, the integrnd will still be of the form: Frctionl erivtives.nb 4 In[4]:= Plot@Re@H@, x, 8,, <, 8x, -, <, Mesh Ø True, PlotRnge Ø 8-, <, ColorFunction Ø Hue, ViewPoint Ø 8, -.5,.7< - - Out[4]= Ü SurfceGrphics Ü From the sme method of nlysis, we cn see tht Green's function for the differentil opertor L` = H x L is given by: G Hx-yL = k ÅÅÅÅÅ Â*k*Hx-yL *p HÂ*kL Another interesting item to note in pssing is tht non-integer derivtive of ny polynomil is either or not defined. Proof: Consider the Fourier trnsform of x n. x n *p * H-Ln * ÅÅÅÅ * H k L n Hd HkL L And thus: H x L x n *p * H-Ln * HÂ * kl * ÅÅÅÅ *p * H-Ln- * k IHÂ * kl * ÅÅ * H k L n Hd HkL L M * H k L n- Hd HkL L By integrting by prts, we cn move the (integer) k derivtives from the right, (where they ct on the k delt function,) to the left, where they ct on IHÂ * kl * ÅÅÅÅ M. Speking somewht loosely, the k derivtive cting on this term will give you something of the form: (terms with k rised to some Printed frctionl by Mthemtic power)* for * Students ÅÅÅÅ

5 Frctionl erivtives.nb 5 something of the form: (terms with k rised to some frctionl power)* * ÅÅÅÅ We cn keep integrting by prts until the k delt function hs no derivtives cting on it, nd for ll our lbor, the integrnd will still be of the form: (terms with k rised to some frctionl power)* * ÅÅÅÅ * d HkL At this stge, we cn just do the integrl by removing the k delt function nd setting to k= everywhere else in the integrnd. If ll the "terms with k rised to some frctionl power" re positive powers of k, they will become when we set k=. If ny of the "terms with k rised to some frctionl power" re negtive, then the expression becomes infinite, nd so the frctionl derivtive does not exist. The lst imginble cse is tht mybe there is some term mong the "terms with k rised to some frctionl power" which is k rised to, (i.e. it is constnt.) But upon reflection, we see tht there will never be such term. If is not n integer, then no mtter how mny times we rise or lower the order of k by increments of one (by integrting by prts), we'll never get n integer power of k. Thus, we'll never get term like k s long s is not n integer. This is ll interesting becuse we sw bove tht Gussin, which is expressble--by Tylor expnsion--s n infinite sum of polynomils, hs finite, non-vnishing frctionl derivtives. But here we see tht ny polynomil by itself hs only vnishing or non-existent frctionl derivtives. If you imgine cting with frctionl derivtive on ech term in the Tylor expnsion of the Gussin function, you would find tht you hve n infinite number of infinite terms with oscillting sign. This is telling you tht the Tylor expnsion is not prcticl wy to tke frctionl derivtive of the Gussin--i.e., the method is not well-defined. Rther, you hve to work in the frequency domin in order to mke sense of nything. Keywords: Fourier nlysis, frequency decomposition, frctionl derivtives, non-integer derivtives, Plot Printed by Mthemtic for Students