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ELSEVIER (AOf �II I'JT�' sdfsdf PRINCIPLESO F MATHEMATICAL MODELING SeconEdd ition CliLv.De y m HarveMyu ddC ollege ClaremoCnatl,if ornia Amsterdam • Boston • Heidelbe•rg London • New York• Oxford Paris • SanD iego• S anF rancisc•o S ingapo• reS ydney• Tokyo ELSEVIER ACADEMIC PRESS sdfsdf AcquisiEtdiiotno Bra:r baHroal land ProjeMcatn ageSra:r aHha jduk EditorAisaslis tTaonmtS :i nger M<lrketiMnagn ageLri:n dBae attie CoveDre sigMni:r iaDmy m CompositNieonw:g eInm agiSnygs te(mPs)L tdC.h,e nn<lIin,d ia PrintTehre:M <lple-VBaoiolkM anufacturing Group ElsevAiceard eimcP ress 200W heelReora dB,u rlingMtAo n0,1 803l,J SA 525B StreSeuti,t 1e9 00S,a nD iegCoa,l ifo9rn2i1a0 1-44U9S5A, 84T heobalRdo·asdL ,o ndoWnC IX8 RR,U K § Thibso oki sp rintoenda cid-fpraepee r. Copyrig©h2t0 04E,l sevIinecrA. l rlgi htrse served. PrinciopflM east hematical Mo1dsEetdl iitnigCo,ln i,vL e.D ym andE liz'lIbveetyh Copyri©g1h9t8 0A,c ademPirce sAsl.rl i ghrtess erved. Nop arotft hipsu bliicoanmt ayb er eproduocret dar nsmititnea dn yf oromr b ya nym eans, elercotniocrm echaniicnacll,u dpihnotgo copyr,e cordoirna gn,yi n fomratiosnt oraagned retriseyvsatle wmi,t hopuetr missiinwo rni tifnrgo mt hepu blsiher. Permissimoanybs e s ought difrreocmEt llseyv ieSrc'ise n8<c ]ee chnolRoiggyh ts DepartmeinnOt x ford,U K:p hone(:+ 441)8 658438f3a0x(,:+ 441)8 6585333e3-,m ail: permissions@else\'YiOelrli.I cloaamlys. I.:ouo km.p leytoeu rre queosntl ivnieta h eE lsevier homepag(eh ttp://elsevbiyse erl.eccot"miC)nu,gs tomSeurp poratn"dt hen "ObtainPienrgm issions." LibraorfCy o ngreCsast aloging-ill-PDuabtlai cation APPLICATISOUNB MITfED BritiLsihb raCrayt aloguiinPn ugb licaDtaitoan A catalorgeuwer fdo trh ibso oki sa vailafbrloemt heB ritiLsihb rary ISBN0:- 12-226551-3 Fora liln formatoinao lnAl c ademPirce spsu blicatviiosoniustr W ebs itaet www.acadclllicpressbooks.com Printientd h eV niteSdt atoefAs m erica 04 05 060 7 08 8 76 5 4 3 2 sdfsdf sdfsdf "���� � 'UU U U iUU J j UlJfJlI(� :U=tJ=]: - nnnn[c= .OOOn( ­ in ·n} ,l :'fUUl l(Ulc:l ::lU::'. l:: ::lM::l:[ J Contents Preface xiii .................................................................... . Acknowledgments . . xvii ................................................... . PARTA :F oundations . 1 ............................................... CHAPTER1 WhatI sM athematicMaold eling?. 3 ........ 11. Why Do We Do MathematicMaold eling? . 4 ................. 11.1. MathemaitcaMlo delinagn dt heS cientific Method ......................................4. ........... ., 11.2. MathematicMaold elign andt heP ractiocfe Engineering . . . . . 5 ............... .. ......................... 12. Princliepso fM athematicMaold eling 6 ........................ 13. Some Methodosf M athematicMaold eling . 8 ............... 13..1 DimensionHaolm ogeneiatnyd C onsistency 9 ...... 13..2 AbstractainodnS calin.g. . . . .. . 9 .. ... .. ..... ........... 13..3 ConservatainodnB alancPer inciples 1 0 .............. 13..4 ConstructLiinnge aMro dels . . . 11 ....................... 14. Summary . . . . . . 11 ........... ............................ ............ 15. Reference.s . . . 12 ................................................... vii viii Contents CHAPTER2 DimensionAanla lysi.s ... ..13 . ............. ... . . 2.1 Dimensioannsd U nits. ....................1.4. .................. 2.2 DimensionHaolm ogeneity. . . . 15 ... ............................ 2.3 Why Do We Do DimensionAanla lysis? .. 16 .................. 2.4 How Do We Do DimensionAanla lysis? . 19 ................... 2.4.1T heB asiMce thodo fD imensionAanla lys.i.s. 2.0. . 2.4.2T heB uckinghaPmiT heoremf or DimensionAanla lysi.s. . . . . 24 . .. . ....................... 2.5 SystemosfUnits . . . .. .. . 28 ................. .... ................... 2.6 Summary 30 ......................................................... 2.7 Reference.s . . . ... .. . . 31 ... ... ................ ...................... 2.8 Problem.s. . . . . . . .. 31 . .................... ................. .. .. . .... CHAPTER3 Scale .... ..... . .. ..3 3 .... .. ........ . ................... .. 3.1 AbstractainodnS cale . . . . 33 ............ .......................... 3.2 Sizea ndS hapeG:e ometric Sc.alin.g . . 35 ............ ......... 3.2.1G eometriScc alinagn dF ligMhuts cle FractioinnBs i rds . .. 36 .................................... 3.2.2L inearaintdyG eometriScc aling . 37 ................... 3.2.3" Log-logP"l otosf G eometriScc alinDga ta 38 ....... 3.3 Sizea ndF unctionB-iIr:d s and Fl.i..g.ht . .4 4 .. . .............. 3.3.1 TheP owerN eedefdo rH overi.n.g. .......4.5. ........ 3.3.2T heP owerA vailabfolreH overi.n.g.. ......4.6. ...... 3.3.3S oT hereI sa HoverinLgi mit..... .........4.7. ......... 3.4 Sizea ndF unctionH-eIaIr:i nagn dS peech. ........4.7. ..... 3.4.1H earinDge pendosn Siz.e. . . . 48 ............... ......... 3.4.2S peechD ependosn Si.ze .... 50 ..... ................... 3.5 Sizea ndL imitSsc:a lien E quations 51 ......................... 3.51. When a ModeIlsN o LongeArp plicable . 52 .......... 3.5.2S calinignE quation.s. .. . .. . 52 ... . ........ ............... 3.5.3C haracteriTismteisc. .........................5..4. ...... 3.6 ConsequencoefsC hoosinag S cale. ... .... 55 . .. . .... ......... 3.6.1S calinagn d DaAtcaq uisition. .. . 55 ........ . ........... 3.6.2S calinagn dt heD esigonf E xperimen.ts 59 .......... 3.6.3 Scalinagn dP erceptioofnP sr esentDeadt a 62 ....... 3.7 Summary . . . .. .. 65 ............................ ...... ................ 3.8 References .. .. . . 66 .............. . .................................. 3.9 Problems . . . 67 ........................... ........................... CHAPTER4 ApproximatainndgV aliadting Models ... . . 71 .......................................... 4.1 TaylorF'osr mul.a . .. . .. . . 71 ............. ......... ................. Contents ix 4..11 TaylorF'osr mulaan dS eries . . 72 ..................... .. 4..12 TayloSre rieosfT rigonometarnidc HyperbolFiucn ctions .. . . ..7 4 ...... ...................... 4..13 BinomiaElx pansio.ns. . . . 78 ..... ........................ 4.2 AlgebraAipcp roximatio.ns . . . ... ..82 .. ...... ... .. ....... .. .... . 4.3 NumericAaplp roximatiSoingsn:i ficFaingtu re.s .. 84 . .. . .. 4.4 ValidattihnegM odel-HIo:w Do We Know theM odelI sO K? . .. . 88 . ........... .......... ..................... 4.4.1C heckinDgi mensioannsd Units . .8 9 .................. 4.4.2C heckinQgu alitatainvdeL imit Beha.vior. 91 ....... . 4.5 ValidattihnegM odel-IHIo:w LargAer et heE rrors? 92 ..... 4.5.1E rror 93 ....................................................... 4.5.2A ccuracayn dP recision. . . . 94 .... ....... .............. .. 4.6 Fitting CutroDv aetsa . . . . 96 ........ .............................. 4.7 ElementaSrtya tis.t.i...c..s. ...................................9 9 4.7.1M ean,M ediana,n dS tandarDde viation . 100 ........ 4.7.2H istogram.s . . .1 02 . ... .................................... 4.8 Summary ... . .. . .. .. . . 106 ... . ........... . .... ........ ............ .. 4.9 AppendiExl:e mentaTrrya nscendenFtuanlc tions. 1 07 .... 4.10R eferences. . . . .. .. 110 ....... .. .............. ...................... 4.11P roblems. ... . . . .1 11 ... . . .. ........... ............................. PARTB :A pplicatio.ns. .. . . ... .1 15 .. ...... . . ................ . ....... CHAPTER5 ExponentGiraolw tha ndD ecay 1 17 ........ 5.1 How Do ThingGse tS o Outo fH and?. .. . .117 . . . ............ . 5.2 ExponentFiuanlc tioannsd T heiDri fferential Equations .. . . .. . .. .122 ..... ...................... ........... ....... 5.2.1C alclua tianngd D isplayiEnxgp onential Functio.n.s . . 122 . . ........................... .............. 5.2.2T heF irst-OrDdieffre rentEiqaualt ion dNI d t- AN 0 ............................1..2 .6. ...... = 5.3 RadioactDievcea y. . . . 127 ........................................ 5.4 Charginagn dD ischargian Cga pacito..r..". ............ 130 5.4.1A CapacitDoirs charges. .. .. . 131 ..... . ........... .. .... 5.4.2A CapacitIosCr h arged . 133 ............................. 5.5 ExponentMioadle lsi nM oneyM atters .1 36 .................. 5.5.1C om poundI ntere.s.t . . .. .. 136 . .............. ..... ...... 5.5.2I nflat.io.n . .. .. . . . . 138 ............... . .. ...... .............. 5.6 A NonlineMaord elo fP opulatiGorno wth. . 141 .. .. ......... 5.7 A CoupleMdo delo fF ightiAnrgm ies. . ..1 44 . ................ 5.8 Summar.y .. .. .. . . . ...1 47 . ................ ..... ... .... .... ........ .

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Science and engineering students depend heavily on concepts of mathematical modeling. In an age where almost everything is done on a computer, author Clive Dym believes that students need to understand and "own" the underlying mathematics that computers are doing on their behalf. His goal for Princi
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Principles of Mathematical Modeling, Second Edition PDF

322 Pages·2004·2.73 MB·English
by  Clive Dym
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ELSEVIER (AOf �II I'JT�' sdfsdf PRINCIPLESO F MATHEMATICAL MODELING SeconEdd ition CliLv.De y m HarveMyu ddC ollege ClaremoCnatl,if ornia Amsterdam • Boston • Heidelbe•rg London • New York• Oxford Paris • SanD iego• S anF rancisc•o S ingapo• reS ydney• Tokyo ELSEVIER ACADEMIC PRESS sdfsdf AcquisiEtdiiotno Bra:r baHroal land ProjeMcatn ageSra:r aHha jduk EditorAisaslis tTaonmtS :i nger M<lrketiMnagn ageLri:n dBae attie CoveDre sigMni:r iaDmy m CompositNieonw:g eInm agiSnygs te(mPs)L tdC.h,e nn<lIin,d ia PrintTehre:M <lple-VBaoiolkM anufacturing Group ElsevAiceard eimcP ress 200W heelReora dB,u rlingMtAo n0,1 803l,J SA 525B StreSeuti,t 1e9 00S,a nD iegCoa,l ifo9rn2i1a0 1-44U9S5A, 84T heobalRdo·asdL ,o ndoWnC IX8 RR,U K § Thibso oki sp rintoenda cid-fpraepee r. Copyrig©h2t0 04E,l sevIinecrA. l rlgi htrse served. PrinciopflM east hematical Mo1dsEetdl iitnigCo,ln i,vL e.D ym andE liz'lIbveetyh Copyri©g1h9t8 0A,c ademPirce sAsl.rl i ghrtess erved. Nop arotft hipsu bliicoanmt ayb er eproduocret dar nsmititnea dn yf oromr b ya nym eans, elercotniocrm echaniicnacll,u dpihnotgo copyr,e cordoirna gn,yi n fomratiosnt oraagned retriseyvsatle wmi,t hopuetr missiinwo rni tifnrgo mt hepu blsiher. Permissimoanybs e s ought difrreocmEt llseyv ieSrc'ise n8<c ]ee chnolRoiggyh ts DepartmeinnOt x ford,U K:p hone(:+ 441)8 658438f3a0x(,:+ 441)8 6585333e3-,m ail: permissions@else\'YiOelrli.I cloaamlys. I.:ouo km.p leytoeu rre queosntl ivnieta h eE lsevier homepag(eh ttp://elsevbiyse erl.eccot"miC)nu,gs tomSeurp poratn"dt hen "ObtainPienrgm issions." LibraorfCy o ngreCsast aloging-ill-PDuabtlai cation APPLICATISOUNB MITfED BritiLsihb raCrayt aloguiinPn ugb licaDtaitoan A catalorgeuwer fdo trh ibso oki sa vailafbrloemt heB ritiLsihb rary ISBN0:- 12-226551-3 Fora liln formatoinao lnAl c ademPirce spsu blicatviiosoniustr W ebs itaet www.acadclllicpressbooks.com Printientd h eV niteSdt atoefAs m erica 04 05 060 7 08 8 76 5 4 3 2 sdfsdf sdfsdf "���� � 'UU U U iUU J j UlJfJlI(� :U=tJ=]: - nnnn[c= .OOOn( ­ in ·n} ,l :'fUUl l(Ulc:l ::lU::'. l:: ::lM::l:[ J Contents Preface xiii .................................................................... . Acknowledgments . . xvii ................................................... . PARTA :F oundations . 1 ............................................... CHAPTER1 WhatI sM athematicMaold eling?. 3 ........ 11. Why Do We Do MathematicMaold eling? . 4 ................. 11.1. MathemaitcaMlo delinagn dt heS cientific Method ......................................4. ........... ., 11.2. MathematicMaold elign andt heP ractiocfe Engineering . . . . . 5 ............... .. ......................... 12. Princliepso fM athematicMaold eling 6 ........................ 13. Some Methodosf M athematicMaold eling . 8 ............... 13..1 DimensionHaolm ogeneiatnyd C onsistency 9 ...... 13..2 AbstractainodnS calin.g. . . . .. . 9 .. ... .. ..... ........... 13..3 ConservatainodnB alancPer inciples 1 0 .............. 13..4 ConstructLiinnge aMro dels . . . 11 ....................... 14. Summary . . . . . . 11 ........... ............................ ............ 15. Reference.s . . . 12 ................................................... vii viii Contents CHAPTER2 DimensionAanla lysi.s ... ..13 . ............. ... . . 2.1 Dimensioannsd U nits. ....................1.4. .................. 2.2 DimensionHaolm ogeneity. . . . 15 ... ............................ 2.3 Why Do We Do DimensionAanla lysis? .. 16 .................. 2.4 How Do We Do DimensionAanla lysis? . 19 ................... 2.4.1T heB asiMce thodo fD imensionAanla lys.i.s. 2.0. . 2.4.2T heB uckinghaPmiT heoremf or DimensionAanla lysi.s. . . . . 24 . .. . ....................... 2.5 SystemosfUnits . . . .. .. . 28 ................. .... ................... 2.6 Summary 30 ......................................................... 2.7 Reference.s . . . ... .. . . 31 ... ... ................ ...................... 2.8 Problem.s. . . . . . . .. 31 . .................... ................. .. .. . .... CHAPTER3 Scale .... ..... . .. ..3 3 .... .. ........ . ................... .. 3.1 AbstractainodnS cale . . . . 33 ............ .......................... 3.2 Sizea ndS hapeG:e ometric Sc.alin.g . . 35 ............ ......... 3.2.1G eometriScc alinagn dF ligMhuts cle FractioinnBs i rds . .. 36 .................................... 3.2.2L inearaintdyG eometriScc aling . 37 ................... 3.2.3" Log-logP"l otosf G eometriScc alinDga ta 38 ....... 3.3 Sizea ndF unctionB-iIr:d s and Fl.i..g.ht . .4 4 .. . .............. 3.3.1 TheP owerN eedefdo rH overi.n.g. .......4.5. ........ 3.3.2T heP owerA vailabfolreH overi.n.g.. ......4.6. ...... 3.3.3S oT hereI sa HoverinLgi mit..... .........4.7. ......... 3.4 Sizea ndF unctionH-eIaIr:i nagn dS peech. ........4.7. ..... 3.4.1H earinDge pendosn Siz.e. . . . 48 ............... ......... 3.4.2S peechD ependosn Si.ze .... 50 ..... ................... 3.5 Sizea ndL imitSsc:a lien E quations 51 ......................... 3.51. When a ModeIlsN o LongeArp plicable . 52 .......... 3.5.2S calinignE quation.s. .. . .. . 52 ... . ........ ............... 3.5.3C haracteriTismteisc. .........................5..4. ...... 3.6 ConsequencoefsC hoosinag S cale. ... .... 55 . .. . .... ......... 3.6.1S calinagn d DaAtcaq uisition. .. . 55 ........ . ........... 3.6.2S calinagn dt heD esigonf E xperimen.ts 59 .......... 3.6.3 Scalinagn dP erceptioofnP sr esentDeadt a 62 ....... 3.7 Summary . . . .. .. 65 ............................ ...... ................ 3.8 References .. .. . . 66 .............. . .................................. 3.9 Problems . . . 67 ........................... ........................... CHAPTER4 ApproximatainndgV aliadting Models ... . . 71 .......................................... 4.1 TaylorF'osr mul.a . .. . .. . . 71 ............. ......... ................. Contents ix 4..11 TaylorF'osr mulaan dS eries . . 72 ..................... .. 4..12 TayloSre rieosfT rigonometarnidc HyperbolFiucn ctions .. . . ..7 4 ...... ...................... 4..13 BinomiaElx pansio.ns. . . . 78 ..... ........................ 4.2 AlgebraAipcp roximatio.ns . . . ... ..82 .. ...... ... .. ....... .. .... . 4.3 NumericAaplp roximatiSoingsn:i ficFaingtu re.s .. 84 . .. . .. 4.4 ValidattihnegM odel-HIo:w Do We Know theM odelI sO K? . .. . 88 . ........... .......... ..................... 4.4.1C heckinDgi mensioannsd Units . .8 9 .................. 4.4.2C heckinQgu alitatainvdeL imit Beha.vior. 91 ....... . 4.5 ValidattihnegM odel-IHIo:w LargAer et heE rrors? 92 ..... 4.5.1E rror 93 ....................................................... 4.5.2A ccuracayn dP recision. . . . 94 .... ....... .............. .. 4.6 Fitting CutroDv aetsa . . . . 96 ........ .............................. 4.7 ElementaSrtya tis.t.i...c..s. ...................................9 9 4.7.1M ean,M ediana,n dS tandarDde viation . 100 ........ 4.7.2H istogram.s . . .1 02 . ... .................................... 4.8 Summary ... . .. . .. .. . . 106 ... . ........... . .... ........ ............ .. 4.9 AppendiExl:e mentaTrrya nscendenFtuanlc tions. 1 07 .... 4.10R eferences. . . . .. .. 110 ....... .. .............. ...................... 4.11P roblems. ... . . . .1 11 ... . . .. ........... ............................. PARTB :A pplicatio.ns. .. . . ... .1 15 .. ...... . . ................ . ....... CHAPTER5 ExponentGiraolw tha ndD ecay 1 17 ........ 5.1 How Do ThingGse tS o Outo fH and?. .. . .117 . . . ............ . 5.2 ExponentFiuanlc tioannsd T heiDri fferential Equations .. . . .. . .. .122 ..... ...................... ........... ....... 5.2.1C alclua tianngd D isplayiEnxgp onential Functio.n.s . . 122 . . ........................... .............. 5.2.2T heF irst-OrDdieffre rentEiqaualt ion dNI d t- AN 0 ............................1..2 .6. ...... = 5.3 RadioactDievcea y. . . . 127 ........................................ 5.4 Charginagn dD ischargian Cga pacito..r..". ............ 130 5.4.1A CapacitDoirs charges. .. .. . 131 ..... . ........... .. .... 5.4.2A CapacitIosCr h arged . 133 ............................. 5.5 ExponentMioadle lsi nM oneyM atters .1 36 .................. 5.5.1C om poundI ntere.s.t . . .. .. 136 . .............. ..... ...... 5.5.2I nflat.io.n . .. .. . . . . 138 ............... . .. ...... .............. 5.6 A NonlineMaord elo fP opulatiGorno wth. . 141 .. .. ......... 5.7 A CoupleMdo delo fF ightiAnrgm ies. . ..1 44 . ................ 5.8 Summar.y .. .. .. . . . ...1 47 . ................ ..... ... .... .... ........ .

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Science and engineering students depend heavily on concepts of mathematical modeling. In an age where almost everything is done on a computer, author Clive Dym believes that students need to understand and "own" the underlying mathematics that computers are doing on their behalf. His goal for Princi
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