AC : USING FINITE DIFFERENCE METHODS INSTEAD OF STANDARD CALCULUS IN TEACHING PHYSICS

Transcription

1 AC 7-353: USNG FNE DFFERENCE MEHODS NSEAD OF SANDARD CALCULUS N EACHNG PHYSCS Rada Belu Waye Sae Uversy Aleadru Caal Belu Waye Sae Uversy Aeadu Caal Belu graduaed wh MSc degree Appled Mahmacs rom Waye Sae Uversy. He hold a secod MSc soware egeerg rom he Uvrsy o Weser Oaro Lodo Caada Amerca Socey or Egeerg Educao 7

2 Usg Fe Derece Mehods sead o Sadard Calculus eachg Physcs. roduco Physcs s he bass o umerable echologcal applcaos. has shaped he ace o coemporary socey ad represes he paradgm o all eac sceces. For a proessor mag physcs more accessble o sudes s o oly o ereme mporace bu also oe o he mos challegg ad rewardg ass. he mehod o eachg physcs has goe remarably uchaged or decades eve whe he coes o he subjec are up o dae. Somemes moder eachg echologes are adoped bu mos oe oly laboraores lecure delvery ad preseaos. he las ew decades several auhors have sressed he mporace o usg umercal mehods roducory physcs courses due o he creasg avalably o persoal compuers ad compuer algebra sysems. A rgorous udersadg o physcs presumes a rgorous udersadg o sadard calculus. Classcal physcs however ca be reormulaed usg e derece calculus sead o sadard calculus. hs reormulao s rgorous ad he case o classcal mechacs avods assumg ha space-me s dereable ad hus s cocepually more cosse wh he rsc dscree aure o me ad space 676. O he oher had physcs s mae aure s a dscree scece ad mos egeerg ad physcs problems have o aalyc soluos hs rases he queso o wheher physcs ca be udersood whou usg sadard calculus. aswerg hs queso we oce ha a alerave o sadard calculus s e derece appromaos whch has bee wdely used o solve physcs ad egeerg problems especally whe we are dealg wh problems ha have o aalyc soluos. From a educaoal po o vew hs ca dramacally elarge he base o he eamples used o suppor courses especally mechacs acouscs elecrodyamcs lud mechacs ad moder physcs. hs movaes our eress he reormulao o classcal mechacs elecrodyamcs ad acouscs by usg e derece approach sead o sadard calculus. hs reormulao based o e dereces ogeher wh a dscusso o some o he educaoal aspecs s preseed hs paper. he e derece echques are uve modelg echques easly udersood ad appled by he vas majory o sudes. addo o he sgh ha ca be gaed o classcal mechacs acouscs elecrodyamcs as wh oher braches o physcs here s also he possbly o sudyg real-world physcs ad egeerg problems ha cao be resolved aalycally -4. Also hs approach accords he bee o eplorg may oher areas o physcs ha oherwse requre advaced mahemacal echques ad owledge. Las bu o leas we have o meo ha order o ae he ull advaage o hese poeales he umercal mehods ha should be udersadable o he sudes should be easly programmable preerable by usg compuer algebra sysems ad ece so ha accurae resuls ca be obaed whou ecessve compuaoal resources ad me. he paper s orgazed as ollows. Seco o hs paper s reserved or roduco ad we wll sech he e derece mehods seco ad 3 are reserved or he

3 preseao o he dscree ormulao o mechacs ad elecrodyamcs ad we revew some o he egeerg applcaos o hs approach seco 4. he pedagogcal mplemeao o such ormulaos s dscussed seco 5 ad he las seco s reserved or coclusos dscusso ad uure wor.. Fe Derece Mehods he e derece mehods was developed by A. om he early 9s uder he le mehods o squares o solve o-lear hydrodyamc equaos 93. Sce he he mehod has oud several applcaos solvg dere physcs ad egeerg problems. he e derece mehods are based upo a appromao ha perms he replaceme o dereal equaos by e derece orms. hese e derece appromaos are algebrac orm; hey relae he value o depede varable a a po he soluo rego o he values a some eghborg pos. he e derece mehods have bee appled successully o solve may problems o mechacs acouscs elecrodyamcs lud mechacs ec. Ay appromao o a dervave erms o values a a dscree se o pos s called e derece appromao. he approach used obag e derece appromaos s based o he usg o aylor seres o appromae he dervave o a uco 6-8. he me dervave o parcle poso s epressed by: d d lm rually all-classcal equaos o physcs are deed erms o poso eergy poeal elds or oher quaes ad her rs ad secod order dervaves so rgorous udersadg o classcal mechacs or elecrodyamcs requres owledge o calculus. s decded ha calculus ad advaced mahemacs are ecessary o udersad as well as o prese ad each physcs ha ha s also he opo o he auhors beecal o have alerave approaches mag physcs more accessble o he sudes. Our aemp eachg physcs s based o he use o he e derece calculus sead o sadard calculus. hs volves replacg dervaves o physcal quaes by her e derece couerpars as or eample he me dervave o he parcle poso s replaced by: D where s he smalles me erval. Noce ha he e derece operaor coverges o me dervave as goes o zero. hs replaceme wll lead he case o classcal mechacs o a mor reormulao o eergy momeum ad accelerao whle he dscree elecrodyamcs wll rema que sraghorward. hs provdes he bass or a rgorous mahemacal reame o classcal mechacs or elecrodyamcs ha s more accessble o he sudes. Accordg o Lashmaha ad rgae 988 or Greespa 975 hs operaor has he ollowg properes:

4 [ ] g g D β α β α hs s he well-ow leary propery. [ ] Dg D D g Dg g D 3 Ad Dg g g Dg D g g D 4 he secod-order me derece operaor s deed as: D 5 Smlar operaors ca be ormulaed as we wll show seco 3 he case o dscree elecrodyamcs or ay quaes used classcal physcs. Noce ha we use he rescaled operaor sead o he sadard e-derece operaor. hs operaor coverges o he sadard dervaves as goes o zero. o dee a operaor Σ whch s he verse o D-operaor le: DF.e. [ ] F F /. C s a perodc uco wh C C he [ ] C F D hs meas ha we ca er F up o a arbrary perodc uco C. he rescalg summao operaor Σ s ow deed as [ ] [ ] C C F Σ Λ 6 Ad he summao operaor Σ * o he e derece calculus s gve by:

5 * Σ Λ C / so Σ * [ ] Σ [ ] We ca use he ow summao operaor properes o er he ollowg properes or he rescaled summao operaor Σ: Σ [ βg ] ασ βσg α 7 hs s he well-ow leary propery or he summao operaor. he dscree orm o egral by pars s gve by: [ Dg ] g Σ[ D g Dg ] Σ 8 where we suppressed he arbrary perod uco C. hese ormulae reduce o he sadard calculus ormulae whe goes o zero. We also have [ Σ[ ] C ] d lm cos 9 hs paper wll oly ocus o rescaled summao operaors wh speced lms o egrao so ha we ca drop he arbrary uco C.. Classcal Mechacs Wh he approach preseed he prevous seco we wll reormulae some o he equaos ad heorems o classcal mechacs ollowg he approaches developed reereces 3 4 ad 5.. Newo s Laws he sadard Lagraga ormulao o he classcal mechacs leads he dscree-e case o he ollowg verso o Newo s laws wh sadard oao o mechacs: F m a Dp wh he accelerao usg 5 deed by:

6 a ad wh momeum deed by p m D For he classcal Lagraga L m eergy he eergy uco s deed by D E md D where accou or he poeal he eergy s coserved whe L s o a eplc uco o me. hus all he sadard quaes o classcal mechacs have her couer-pars he dscree mechacs. Moreover several ormulaos o dscree mechacs have bee suded durg he las our decades boh physcally ad mahemacally hese ormulaos were eplc ad or > ad ad N ulzed oe dmeso he ormulae: v m a F a v v v v 3 or. s aoher dscree rom o Newo s equao relaes velocy ad accelerao ad 3 relaes dsace ad velocy. he easbly o hs ormulao s derved rom he proo gve by he classcal coservao law 67 ad rom he applcably o several o-lear problems o physcal eress Newo s Equao ad al-alue Problem a more geeral case ollowg he approach Greespa 975 we cosder a parcle P o mass m beg moo o -as a ad wh a velocy v he he equao v ma F 4 whch a me relaes he orm acg o P o s accelerao s called a dscree Newo s equao. he acual deermao o he moo o P rom dyamcal

7 Equao 4 whe he al values ad v are ow s called a al-value problem. Beore developg addoal physcal coceps le us llusrae he soluos o some o he well-ow dyamcal problems. a Dyamcs gve a cosa orce: Cosder a cosa orce F - /m. hs mples m 5 he soluo s: v m ad v m D For he correspodg problem classcal mechacs: m && whch s he same soluo o ha gve by he calculus-based dyamcs. b he harmoc oscllaor: Cosder a lear orce a y F where parameer a s a cosa. Newo law he case o harmoc oscllaor becomes: a y y md 6 Be deg a y he problem smples o: m he soluo o hs equao s gve by: θ θ θ θ K K K K s cos s cos

8 where θ cos / ν ad v / m / By choosg he comple coeces o K ad K o mae real we ally ge: c cos θ Φ 7 Now we had solved he couous verso o he problem we would have oud [ ] / Φ c cos cosθ Noce ha dscree oscllaor has a hgher requecy ha he couous oscllaor whe θ > [ cos θ ] /. hus dscreeess causes he oscllaor o move a a slghly creased requecy. c Nolear ad damped pedulum: Moo o a olear damped pedulum characerzed by he dyamcal equao: a α v s Λ 8 where α s he dampg coece. he las equao ca be rewre as: v v [ α v 3v s 3s ] Λ 9 Cosderg a parcular al value problem whch a pedulum s released rom a res poso a agle π/4 he: v v π s 4 [ α v 3v s 3s ] Λ v ad [ v v] Λ

9 he moo o he pedulum s geeraed recursvely by usg equaos ad. Moreover oe ca prove heorecally ha he soluo o hs al-value problem ess s uque ad s gve cosrucvely by ad. he mmedae avalably o esece o uqueess ad o he soluo sel whou ay opologcal cosderaos eeds o all al value problems ormulaed erms o 3 ad 4. hs asho we ca hereore resolve olear problems drecly whou learzao usg oly eplc arhmec ormulae..3 Coservao o Eergy ad Momeum For he dscussos he prevous seco he geeral procedure o be used he geeral al-value problem s gve by: v v m v v F 3 F m v F v Λ v v v Λ he wor doe by a orce F o a parcle P movg rom o -D case s gve by W 3 F v F F or m v v m W v v So ha W v v K K m 3 Whch s he dscree orm o coservao law o he eergy ad a smlar way oe ca ge he coservao o momeum. Noce ha hese coceps ca be geeralzed o hgher dmesos by smlar approaches.

10 .4 Dscree Form o he ral heorem E[] deoes he average value o over a log perod o me he he vral heorem saes ha a sysem o parcles: E[ ] E d d m E [ F ] 4 where F s a orce o parcle. hs resul also holds usg he e-derece calculus. Le: [ Dp p Dp D ] p DG G Whe: L m D ad F Dp ad p Dp md hus DG [ F m D ] Requrg DG yelds: E [ ] E m D E[ F ] whch proves he vral heorem. 3. Dscree Elecrodyamcs. he roducory seco o hs paper was argued ha s aural o roduce classcal physcs problems such as eld heory or classcal elecrodyamcs usg a dscree po o vew. hs rus couer o he uo o mos physcss; we argue here ha s aural because we have all bee docraed a uaural approach or reasos amely; our lac o ably o solve realsc problems by had usg dscree equaos ha are o loger releva he world o compuers. As evdece or hs po o vew we poed ou ha eld heores are always roduced dscreely a he very begg. lud mechacs oe begs by cosderg a lud eleme ad coug he momeum mass ad eergy passg or ou hrough s aces. elecrodyamcs oe

11 derves he couy equao or charge by coug he charges passg he aces o a dscree volume eleme. However radoal approach oe mmedaely abados he dscree pcure aes he couum lm ad wres paral dereal equaos. Oe he lears o solve eboo problems usg aalyc echques ad oly much laer oe o a all ormal course wor lears he dscree mehods ha are usually used o solve real-world problems. hs s a dsoro o he aural sequece. he dscree case oe sars wh a pcure ha s much easer or sudes o grasp ha he symbolc mapulaos o vecor calculus; he aural sequece would be o do model calculaos ad he calculaos o realsc sysems usg he dscree equaos ad leave he hghly absrac aalyc mehods or las aer some uo has bee developed or how elecromagec elds behave. he dsored sequece has bee placed upo physcss or he rs couple o ceures o our eperece wh eld heores because dscree calculaos are edous o do by had bu sce compuers become wdely avalable hs dsoro has perssed oly hrough era. Aoher commoly perceved advaage o he couum approach s he esece o several useul egral relaos Ampere s law Gauss law ad Faraday s law symmeres ad he law o coservao o eergy Poyg s heorem whch are o ecessarly rue dscree heory. he dscrezao preseed here has bee chose o have all hese desrable properes. Furhermore he properes are all sraghorward ad rgorously provable usg algebra whereas very ew sudes/physcss ever lear o prove hem he couum approach hey accep hem o ah. he elega egral relaos ha mae elecrodyamcs aracve seems much more elega oe ca prove hem. Oce a resul s obaed he dscree heory s rvally rue he couum lm; learg he correspodg couum lm; learg he correspodg couum resul requres o addoal eor. Mawell s equaos are equaos or deermg he me evoluo o wo vecor elds he elecrc eld E ad he magec eld B. de d c curl B j 5 ε d B d curl E 6 Ad he couy equao d d dv j 7

12 Fgure : Cubc lace he dscree orms o hese equaos cosderg a elemeary cubc lace see Fgure or deals o he compuaoal cell are: j dr dv j c c 8 j B curl c E E ε 9 Fgure : Geomery or elecrc ad magec eld compuao he dscrezao o curl havg a elecrc eld or he dagram Fgure yelds o: j E dr e E curl ] [ 3

13 he curl o E appears he Mawell s equaos or db/d [Equao 6] deed a he edges o he cell Fgure b s dscrezed as: E E [ curle] e 3 he curl o B appearg Equao 4 s dscrezed as: [ curl B ] e B e dr j 3 Equaos 8 3 ad 3 dee he dscree elecromagec sysem. hs s o oly he smples dscrezao bu also has some very ce properes. All he egral relaos ad oher heorems ha are rue o he couum elecrc ad magec elds are eacly rue o hs dscrezao ad ca be proved usg smple algebra. For eample addg up Equao 8 over a se o cells comprsg a rego o space gves he egral orm o he couy equao relag he sum o he charges a rego o he sum o he dscree curres a s surace. he dscree elecrodyamc sysem descrbed above ca be easly smulaed o a compuer. he compuao o elecromagec elds s eeded or a abudace o everyday applcaos such as aeas radars mcrowave devces elecrc maches rasmsso les rado ec. he combao o dscree approach o Mawell s equaos wh he opporuy o play wh such compuer smulaos allows a sude o acqure a physcal udersadg o a wde varey o elecrodyamc pheomea wh a mmum o ormal mahemacs. he eres he algorhm descrbed above s also based o s pedagogcal smplcy. 3. Laplace Equao Oe o he mos used applcaos o e derece elecromagecs s o apply o solve Laplace equao or elecrc poeal - 5. Φ Φ y 33 Applyg he ceral derece ormula ad usg y jy j Λ ad assumg ha y h h s he mesh sze ca be wre as or a compuaoal molecule as Fgure 3:

14 Φ j [ Φ j Φ j Φ j Φ j ] 34 4 s worh o og ha 34 saes ha he value o Ф elecrc poeal a each po s he average o hose a he our surroudg pos. he ve-po compuao molecule or he derece scheme 34 s llusraed Fgure 3. he e dereces are well sued or compug he characersc mpedace phase velocy ad aeuao o dere ypes o rasmsso les such as: blar les coaal cables mcro-srps srples ec. he owledge o he basc characerscs o hese rasmsso les s a paramou mporace desg ad aalyss o elecroc crcus ad sysems. Fgure 3: Compuaoal molecule or Laplace s equao For cocreeess ollowg Sadu 99 cosder a mcro-srp. Fe derece orm o Laplace equao s applcable or hs case due o he ac ha E ad B have o compoes he dreco o propagao so-called EM mode whch s a good appromao he dmesos o he le are much smaller ha he wavelegh. he echques developed here s equally applcable o oher rasmsso les. Equao 34 ca be wre wh he oao o Fgure 3 as: Equao 35 s a geeral ormula o be appled o all odes he ree space ad he delecrc rego o a mcro-srp rasmsso le. A he delecrc boudary Fgure 4 he boudary codo s gve erms o he elecrc dsplaceme vecor: D D 36

15 ad mus be mposed based o he Gauss law. Epressed erms o he geomery Fgure 4 he dscree orm o hs boudary codo ca be epressed as: ε 3 ε ε 4 ε ε ε 4 4 Fgure 4: erace bewee wo meda o delecrc permves. O he le symmery we mpose he codo 37 hs ca be epressed dscree orm as: 4 3 By seg he poeals a ed odes equal o prescrbed values ad applyg he e derece relaoshps descrbed above oe ca deerme he poeals a ree odes. Oce hs s accomplshed he quaes o eress ca be easly compued. 3. rasmsso Les Fe Derece Approach he usual mode o eachg abou rasmsso les s o use a crcu model -3 wh a seres o coeced capacors ducors ad ressors. he sarg po o ay rasmsso le calculao s he wo coupled rasmsso le equaos as:

16 z G C z R L 38 Fgure 5: llusrao o he spaal e-derece grd ad are he me ad space-depede curre ad volage alog he le whle L C R ad G are respecvely he ducace he capacace he seres ressace ad shu coducace per-u o legh. geeral hese parameers ca also deped o poso ad me. he e-derece ormulao o equaos 38 usg he mesh Fgure 5 gves: / / / / / / / / / / / G C E R L 39 Oe ca smply solve equaos 39 or he ewer values o ad wh he resuls: C G z G C L R z R L E o / / / / / / / / / 4

17 Oe calculaes he me-doma respose by assumg some al codos usually ha curres ad volages are all zero a ad he calculag me seps a erave maer. geeral umercal calculaos are much less me cosumg ha he radoally crcu ewor approach s used. 4. Dscussos ad Coclusos hs paper we preseed smple erms he basc coceps o e derece mehods he preseao o classcal mechacs ad elecrodyamcs va several eamples. Algebra ad calculus are wo o he basc ools o mahemacal physcs. Oe o he ma goals o ay educaor s o mamze he learg whle mmzg he eachg. Classcal mechacs ad elecrodyamcs ca be reormulaed usg e derece calculus sead o he radoal approach o usg sadard calculus. hs reormulao s rgorous ad he case o classcal mechacs avods assumg ha space-me s dereable ad hs s cocepually more cosse wh rsc dscree aure o me ad space. he case o eachg elecrcy ad magesm oe ca roduce Mawell s equaos a he begg such ha a elecrc eld s always a dyamcal varable esg a each po o space. hs s geerally o possble a couum approach because he average physcs or egeerg sude s lac o bacgroud advaced calculus would mae he couum o Mawell s equaos meagless. However he dscree Mawell s equaos ca be udersood ad appled o may problems hrough oly basc algebra: problems become o more dcul or absrac ha Coulomb law. deed or a sude who s ucomorable wh he dea o aco a a dsace hey are ac easer o udersad ha Coulomb s law. Usg dscree approach elecrosacs ca be roduced aer Mawell s equaos ad hus becomes jus he sudy o lmg orm o he elecrc eld a log mes. Coulomb s law s a smple cosequece o he dscree Mawell s equaos. oe uses a rase curre ha lows rom y o a sgle cell leavg a charge ha cell he elecrc eld wll eveually sele dow o a sac value.e. Coulomb s law ad he magec eld wll sele dow o zero. Oher advaages o he usg e derece ormulao are: a ca dramacally elarge he umber o eamples ha may be used durg he lecures ad b he opporuy o volvg sudes solvg o-rval real-world problems whch would o oly be very challegg bu also appealg or hem. summary a dscree approach o eachg classcal mechacs ad elecrodyamcs allows oe o covey a he roducory level he cocepual smplcy o Newo s Laws or Mawell s equaos grea syhess o dyamcs or elecrcy ad magesm o cosse heores descrbed by he equaos such as he oes preseed hs paper. Smple algebracally solvable problems ad compuer smulao o ohers mae possble o develop a uve udersadg o he pheomea o classcal mechacs ad elecrodyamcs ha does o deped o he absrac mahemacs o advaced calculus. Whle eachg Romaa oe o he auhors used hs approach eesvely eachg classcal mechacs ad elecromagecs. Laer he used he same approach he Ued Saes eachg egeerg elecromagecs. he respose o he sudes o he e

18 derece approach eachg mechacs ad elecromagecs was surprsgly avorable. 5. Reereces. Belu R.G. A MALAB Program or Calculag rasmsso Le rases as a eachg Ad. J. o El. Eg. Educ o be submed. Cazares C ad D. Faur Advaages ad dsadvaages o varous compuer ools EE educao EEE ras. o Educ. vol pp DeLyser R. R. Usg Mahcad elecromagec educao EEE ras. o Educ. vol pp D ocezo A. ad L. Rea Aalyss o some elemeary umercal mehods mechacs Eur. J. o Physcs vol pp D ocezo A. L. Rea ad P. Roell he oscllaor dscree mechacs Physcs Leers vol. 45B 984 pp D ocezo A. ad L. Rea Some sudes dscree mechacs Eur. J. o Physcs vol pp Greespa D. A ew eplc dscree mechacs wh applcaos J. o he Fral sue 975 pp Greespa D. roduco o Numercal Aalyss ad Applcaos Chcago Marham Lashmaha L. ad D. rgae heory o Derece Equaos New Yor Academc Press Hall.L. ad Z. Cedes. roducg real world desg problems elecromagecs currculum EEE ras. o Educ. vol pp Lamber P. W. ad. aduzer A smple umercal mehod or solvg problems elecrosacs Amer. J. o Phys. vol pp Sadhu M. N. O. Fe derece soluo o elecrodyamcs problems. J. Elecrcal Eg. Educ. vol pp Sadu M. N. O. Elemes o Elecromagecs Hol Rhar ad Wso Scheder M.. Compuao mpedace ad aeuao o EM-le by e derece mehods. EEE ras. Mcro. heory & ech. vol. M pp sscher P. B. Felds ad Elecrodyamcs: A Compuer-Compable roduco Wley New Yor Wol C. ad D. Hodgso Compuer algebra sysems eachg physcs Eur. J. o Physcs vol pp