Orthogonal Function Solution of Differential Equations

Transcription

1 Royal Holloway Uiversity of Loo Departet of Physics Orthogoal Fuctio Solutio of Differetial Equatios trouctio A give oriary ifferetial equatio will have solutios i ters of its ow fuctios Thus, for eaple, the solutio of the SHO Schröiger equatio is epresse i ters of Herite polyoials ultiplyig a gausia: SHO Schröiger equatio H 16 e / We are talkig here about the eigefuctios of the equatio a their eigevalues ( i this case) Now we are goig to look ito usig the wrog fuctios to solve a ifferetial equatio Why ight we wat to o this? There are a uber of reasos 16 / We coul use the set of H e like a quatu SHO to see how uch the behaviour of the syste was Siilarly, we coul use the set e i0k ωt5 to see how uch the behaviour was like that of free travelig waves These proceures coul be iportat i their ow right, or as part of a perturbatio or approiatio schee The practical cosequece of this proceure is that ifferetial equatios are trasfore ito atri equatios (A of course this is at the heart of the equivalece of the Schröiger a the Heiseberg approaches to quatu echaics The fuaetal ieas The fuctios which arise fro the solutio of a ifferetial equatio ofte have soe iportat properties The equatios which arise i physical probles ca be assue to coprise a coplete orthogoal set of fuctios (Forally, this result follows fro the Stur-Liouville theore) Orthogoal We kow what orthogoal eas For the set of fuctios orthogoality itegral (it ight also have a weight fuctio) < 16A there is a PH130 Matheatical Methos 1

2 Royal Holloway Uiversity of Loo Departet of Physics The itegral ust be evaluate over the correct iterval a it is zero uless = f the fuctios are oralise the the itegral is uity whe = Coplete f the set of fuctios is coplete, this eas that ay fuctio f16ca be epresse as a liear su of the fuctios f = f This epressio for f will be vali over the orthogoality iterval You shoul reeber how to fi the = coefficiets f : f16 f The eoiator is eee whe the fuctios are ot oralise Differetiatio A iportat a highly relevat eaple of the copleteess property is the ifferetiatio of a fuctio Startig fro a fuctio f16, if we ifferetiate it we obtai a ew fuctio f / Now copleteess tells us that ay fuctio f16 ca be epresse as a liear su of the basis fuctios 16 But sice f / is just aother fuctio, this too ca be epresse as a liear su of the basis fuctios We have 16= 16 f f a we siilarly epect that f = 16 g The questio, the, is how to fi the coefficiets g for the erivative Give the basis set < 16Athe fuctio f16 is specifie by the coefficiets f We wat to fi the coefficiets g which correspoigly specify the erivative f / Let s procee i the followig way Start fro f16: a ifferetiate it 16= 16 f f PH130 Matheatical Methos

3 Royal Holloway Uiversity of Loo Departet of Physics f = 16 f Copleteess assures us that each of the 16/ will be a liear fuctio of the basis fuctios For eaple, the erivative of Herite polyoials is epresse as H H 16= 1 16 geeral we will have 16 = 16 D The coefficiets D ee two iices, oe to iicate which basis fuctio we are ifferetiatig (), a the other to iicate how uch of each of the basis fuctios we ee i the erivative () For the Herite polyoial case we the have D δ, Check that you uersta this = + 1 Usig the above result we ca retur to the task of ifferetiatig our fuctio f f = = f D Df = f We frae our questio by sayig that if we write the erivative as the epasio f = 16 g the what are the coefficiets g? We ow have the aswer: g = D f 16 You shoul observe that this has the for of a atri prouct f we represet g a f as colu atrices the g D D D : f g g : 1 3 = D1 D : : f D31 : : : f3 : : : : : 1 PH130 Matheatical Methos 3

4 Royal Holloway Uiversity of Loo Departet of Physics Notatio a 3 vector aalogy With 3 vectors, oce oe has a give basis set = i, j, k B the ay vector ca be epresse as a liear su of these v = vi+ v i jj+ vkk Oce the basis set is ecie upo the ay physical vector v is specifie by givig its coefficiets v i, v j, a v k a these coefficiets ca be asseble ito a colu atri vi v j vk Matheaticias ight call such a colu of ubers a vector We ay eve occasioally succub to this ieactitue Strictly speakig, this colu of ubers is a particular represetatio of the physical vector v if we chage the basis set i, j, k = B the we ee a ifferet colu of coefficiets to represet the sae ol vector v Now cosier fuctios With a give basis set < 16 Aay fuctio f16 ca be epresse as a liear su of these f16 = 16 f Oce the basis set is ecie upo the ay fuctio f16 is specifie by givig its coefficiets;@a f these ubers ca be asseble ito a colu atri f1 f f This colu of coefficiets (a atheaticia s vector) is a particular represetatio of the fuctio f16 f we chage the basis set < 16 Awe were usig the we woul ee a ifferet colu of coefficiets to represet the sae ol fuctio Takig the otatioal aalogy further we coul eote the colu of coefficiets by a vector sybol f That is, we coul ake the associatio f 16~ f As a eaple of usig this otatioal aalogy cosier the ifferetiatio result of before g = D f 3 : Here the f are the coefficiets of the fuctio f16a the g are the coefficiets of its erivative f we eote these colus of coefficiets by the vectors f a f the the ifferetiatio result becoes the vector-atri equatio f = Df PH130 Matheatical Methos 4

5 Royal Holloway Uiversity of Loo Departet of Physics where D represets the atri whose eleets are D This shows us that ultiplyig with the atri D has the effect of ifferetiatio Matri represetatio of ifferetial equatios The above aalogies suggest that ifferetial equatios ay be represete by atri equatios This ay be oe forally i the followig way Startig fro the siple eaple g = f, i ters of the basis set < 16 Athis ay be epresse as 16 g 16 = Df (Reeber that the suatio iices are copletely arbitrary) The trick to tur this ito a atri equatio is to ultiply both sies by oe of the basis fuctios a the perfor a orthogoality itegral Let s ultiply by p 16a itegrate O the left ha sie as varies through its rage of values the orthogoality itegral of 16with p 16will always be zero ecept whe = p Thus we pick out just the p ter fro the su O the right ha sie, as varies through its rage we will siilarly pick out the p ter We the e up with the equatio g = D f, p p or g= Df which is the atri represetatio of the origial equatio Differetial equatios with costat coefficiets Hoogeeous equatios Let s cosier a seco orer (hoogeeous) ifferetial equatio with costat coefficiets f16 A B f Cf 0 16= This is trasfore ito a atri equatio i four steps: 1 Epress f as a liear su of the basis fuctios: f = f PH130 Matheatical Methos 5

6 Royal Holloway Uiversity of Loo Departet of Physics Substitute this ito the ifferetial equatio: A f + B f C f + 16 = 0 3 Multiply by p A & ' p 1 6 a o the orthogoality itegral: ( f B p16 16 ) f C > + p16 16 Cf * &' ( )* + = 0 4 etify the atri for of each ter By copariso with the erivatio of the ifferetiatio atri D above, the ifferetial equatio ay be writte i atri for as ADf+ BDf+ 1f=0 or AD + BD+ 1 f =0 where 1 is the uit atri < A Now AD + BD+ 1 is just aother atri, which we ca eote by M Thus our ifferetial equatio has bee trasfore, with the basis fuctio set < 16A, ito the atri equatio Mf = 0 The hoogeeous ifferetial equatio has bee trasfore ito a hoogeeous atri equatio This woul the be solve by the usual proceures for solvig hoogeeous atri equatios hoogeeous equatios Let us ow cosier a ihoogeeous equatio such as f A + B f + Cf = s where s ust be epresse as liear sus of the basis fuctios: f = f is the source ter, akig the equatio ihoogeeous Now both f a s s 16= 16 s We use the vector otatio aalogy to eote the colu of coefficiets of f16 by the vector sybol f a siilarly the coefficiets of s 16by s 16 f s 16 ~ f ~ s The, usig the sae proceure as i the previous case, we arrive at the ihoogeeous atri equatio PH130 Matheatical Methos 6

7 Royal Holloway Uiversity of Loo Departet of Physics Mf = s The ihoogeeous ifferetial equatio has bee trasfore ito a ihoogeeous atri equatio This woul the be solve by the usual proceures for solvig ihoogeeous atri equatios The foral solutio is particularly straightforwar: we ultiply both equatios, fro the left, by the iverse of M The we have 1 1 M Mf = M s or f = M 1 s This solves the proble; f16is give i ters of s Differetial equatios with variable coefficiets Whe the coefficiets of the ifferetial equatio are variable, that is whe we have a equatio of the for f f A B C 1616 f = 0 thigs are slightly ore coplicate We procee to trasfor this ito a atri equatio usig the sae four steps as before: 1 Epress f as a liear su of the basis fuctios: f = f Substitute this ito the ifferetial equatio: A f B f C f + = Multiply by p 16a o the orthogoality itegral: p1a f p16b & ( f ' ) + * &' ( )* + + > p1616 C16 Cf = 0 4 etify the atri for of each ter this case the we o t have a irect coectio with the ifferetiatio atri D we ha before Nevertheless this still has the for of a atri equatio Mf = 0 where the eleets of the atri M are ow Mp = p A + p B + p1616 C 16 PH130 Matheatical Methos 7