4. Runge-Kutta Formula For Differential Equations
|
|
|
- Hilda Smith
- 5 years ago
- Views:
Transcription
1 NCTU Deprme o Elecrcl d Compuer Egeerg 5 Sprg Course <Dymc Sysem Smulo d Implemeo> by Pro. Yo-Pg Ce. Ruge-Ku Formul For Derel Equos To solve e derel equos umerclly e mos useul ormul s clled Ruge-Ku ormul wc s bee wdely used umercl lyss. For dymc sysem wou pu s geerlly epressed s e ollowg rs-order derel equo: ( ( ( & ( α (- were s e l me d ( α s e l codo. Te problem o solve ( (- or > s clled e l vlue problem or IVP bre. For emple e ollowg equo ( ( & ( (- s IVP d s soluo c be obed closed orm s below: ( e However (- s more complced suc s ( ( s( (- & ( (- e s mpossble o d e soluo ( closed orm. Hece s requred o solve (- umercl meod. Te smples umercl meod s clled e Euler ormul wc ws propsed by Euler 768. W e use o ed grd sze e grd pos log re deoed s order were (-5 d e vlue ( s deed s ( (-6 were e l vlue ( α s ow. Accordg o e deo o dervve we ve ( ( & ( lm (-7 wc mples e dervve o ( c be ppromely epressed s ( ( ( ( & ( (-8 Subsug (-8 o (- leds o -
2 NCTU Deprme o Elecrcl d Compuer Egeerg 5 Sprg Course <Dymc Sysem Smulo d Implemeo> by Pro. Yo-Pg Ce - ( α (-9 d e ( α (- wc s e mous Euler ormul. Clerly we ry o ceve soluo more precsely e grd sze sould be cose s smll s possble. However reducg e grd sze s ece sce e cos o clculo me my crese remedously. Now le s roduce Tylor s epso o epl e error cused by Euler ormul (-. Accordg o Tylor s epso uco ( couous c be epressed s ( ( ( ( (- wose ger order dervves re ( ( ( ( & (- ( ( ( ( ( && (- ( ( ( ( ( ( &&& 5 5 (- From e bove equos we ve ( d ( ( or. Tus (- c be rewre s ( ( ( ( ( ( ( ( ( && & (-5 e we ve ( ( ( ( ( ( ( ( ( && & (-6.e. ( ( ( ( ( ( O (-7 were
3 ( O NCTU Deprme o Elecrcl d Compuer Egeerg 5 Sprg Course <Dymc Sysem Smulo d Implemeo> by Pro. Yo-Pg Ce ( ( ( Comprg (-7 w (- we ow e erm ( ormul wc s proporol o. - (-8 O s e error o Euler I order o reduce e error o we urer mody Euler orm (- s below: ( ( (-9 were d re vrbles o be deermed. Sce ( ( d s Tylor s epso ( d c be epressed s depeds o ( ( y( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (- were ll e coeces re cos. Te prl dervves o ( ( re ( ( ( ( ( y( ( y( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (- (- (- (- (-5 Clerly ll e coeces c be derved rom e bove prl dervves d epressed s
4 NCTU Deprme o Elecrcl d Compuer Egeerg 5 Sprg Course <Dymc Sysem Smulo d Implemeo> by Pro. Yo-Pg Ce - ( ( ( ( ( (. Subsug em o (- yelds ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (-6 e d ( e ( ( ( ( ( ( ( ( O (-7 Hece (-9 c be rerrged d rewre s ( ( [ ] ( ( [ ] ( ( ( [ ] ( 6 O (-8 I we w o e e precso o rom (-7 d (-8 we ve ( ( ( ( ( [ ] ( ( [ ] ( O O (-9 Sce ( ( & e erm ( c be cged o e ollowg orm: ( ( ( ( ( ( ( & (- As resul rom (-9 d (- we ob ( ( ( ( ( ( ( [ ] ( ( [ ] (-
5 NCTU Deprme o Elecrcl d Compuer Egeerg 5 Sprg Course <Dymc Sysem Smulo d Implemeo> by Pro. Yo-Pg Ce wc leds o (- From (- weve ( ( ( ( ( (- (- ( (-5 Sce ere re our vrbles d ree equos (- (- d (-5 e coce o ese vrbles s o uque d commoly ey re seleced s ( (-6 T mes e umercl soluo (-8 s cose s ( ( ( (-7 wc s clled e secod-order Ruge-Ku ormul. For e coveece o progrmmg e ormul s oe rerrged s below: were ( (-8 ( ( (-9 I cse ger precso s requred we oem employ e ger order Ruge-Ku ormul. For emple precso o s eeded e e our-order Ruge-Ku ormul mus be used wc s oe gve s were ( (- 6 ( ( (- Ts ormul s bee wdely ppled o lo o pplcos egeerg due o s smplcy d ccepble ccurcy. Ne le s use emple o sow e progrmmg o e our-order -5
6 NCTU Deprme o Elecrcl d Compuer Egeerg 5 Sprg Course <Dymc Sysem Smulo d Implemeo> by Pro. Yo-Pg Ce Ruge-Ku ormul MATAB. Cosder e ollowg equo: ( ( ( ( & (.5 (- d d e soluo ( or wc c be solved s below: ( 5e. (- Now le s pply e our-order Ruge-Ku ormul o very (- w e sep sze.. Te precso s e e order o 8. From (- we ve were ( (- 6 ( ( ( ( ( ( ( Te progrmmg MATAB s gve s below: >> % Four order Ruge-Ku meod >>.5;.; % l codo (.5 d sep sze.sec >> -*(-; -*(/-; -*(/-; -*(-; >> ((**/6; >> e((-(-.5*ep(-; % umercl error >> (; >> or :599 % ol smulo me 6 sec >> -*((-; -*((/-; >> -*((/-; -*((-; >> (((**/6; >> ((*; % -s >> e((-(-.5*ep(-(; % umercl error >> ed >> plo(; lbel( ; ylbel( ( (
7 >> plo(e; lbel( ; ylbel( e( NCTU Deprme o Elecrcl d Compuer Egeerg 5 Sprg Course <Dymc Sysem Smulo d Implemeo> by Pro. Yo-Pg Ce e( From e bove umercl resuls o error uco e( s rue e precso s roud wc s deed less ( 8. I MATAB some srucos re provded o solve e derel equos suc s ode d ode5. I c ese srucos re lso mplemeed by e Ruge-Ku meod. Now le s use e sruco ode o solve e sysem (- d use e sruco ode5 o solve e sysem descrbed s below: ( ( ( (.5s( s & (.5 (-5 wose soluo c o be epressed closed orm. Te progrmmg MATAB s gve e ollowg wc cludes wo m-les. Te le rs.m s wre or (- d solved by ode. Te le rs.m s wre or (-5 d solved by ode5. Clerly e resul o (-5 c o be epressed closed orm sce s more complced e resul o (-. Cree m-le: rs.m uco drs( d-; Cree m-le: rs.m uco drs( d-.5*s(s(*; >> % ey e ollowg srucos >> []ode(@rs[:.:6].5 >> plo(; lbel( ; ylbel( ( -7
8 NCTU Deprme o Elecrcl d Compuer Egeerg 5 Sprg Course <Dymc Sysem Smulo d Implemeo> by Pro. Yo-Pg Ce ( >> % ey e ollowg srucos >> []ode5(@rs[:.:6].5 >> plo(; lbel( ; ylbel( ( (
4. Runge-Kutta Formula For Differential Equations. A. Euler Formula B. Runge-Kutta Formula C. An Example for Fourth-Order Runge-Kutta Formula
NCTU Deprme o Elecrcl d Compuer Egeerg Seor Course By Pro. Yo-Pg Ce. Ruge-Ku Formul For Derel Equos A. Euler Formul B. Ruge-Ku Formul C. A Emple or Four-Order Ruge-Ku Formul
Chapter Simpson s 1/3 Rule of Integration. ( x)
Cper 7. Smpso s / Rule o Iegro Aer redg s per, you sould e le o. derve e ormul or Smpso s / rule o egro,. use Smpso s / rule o solve egrls,. develop e ormul or mulple-segme Smpso s / rule o egro,. use
Chapter Trapezoidal Rule of Integration
Cper 7 Trpezodl Rule o Iegro Aer redg s per, you sould e le o: derve e rpezodl rule o egro, use e rpezodl rule o egro o solve prolems, derve e mulple-segme rpezodl rule o egro, 4 use e mulple-segme rpezodl
Integral Equations and their Relationship to Differential Equations with Initial Conditions
Scece Refleco SR Vol 6 wwwscecereflecocom Geerl Leers Mhemcs GLM 6 3-3 Geerl Leers Mhemcs GLM Wese: hp://wwwscecereflecocom/geerl-leers--mhemcs/ Geerl Leers Mhemcs Scece Refleco Iegrl Equos d her Reloshp
Calculation of Effective Resonance Integrals
Clculo of ffecve Resoce egrls S.B. Borzkov FLNP JNR Du Russ Clculo of e effecve oce egrl wc cludes e rel eerg deedece of euro flux des d correco o e euro cure e smle s eeded for ccure flux deermo d euro
Quantum Mechanics II Lecture 11 Time-dependent perturbation theory. Time-dependent perturbation theory (degenerate or non-degenerate starting state)
Pro. O. B. Wrgh, Auum Quaum Mechacs II Lecure Tme-depede perurbao heory Tme-depede perurbao heory (degeerae or o-degeerae sarg sae) Cosder a sgle parcle whch, s uperurbed codo wh Hamloa H, ca exs a superposo
Decompression diagram sampler_src (source files and makefiles) bin (binary files) --- sh (sample shells) --- input (sample input files)
. Iroduco Probblsc oe-moh forecs gudce s mde b 50 esemble members mproved b Model Oupu scs (MO). scl equo s mde b usg hdcs d d observo d. We selec some prmeers for modfg forecs o use mulple regresso formul.
Differential Entropy 吳家麟教授
Deretl Etropy 吳家麟教授 Deto Let be rdom vrble wt cumultve dstrbuto ucto I F s cotuous te r.v. s sd to be cotuous. Let = F we te dervtve s deed. I te s clled te pd or. Te set were > 0 s clled te support set
Laplace Transform. Definition of Laplace Transform: f(t) that satisfies The Laplace transform of f(t) is defined as.
Lplce Trfor The Lplce Trfor oe of he hecl ool for olvg ordry ler dfferel equo. - The hoogeeou equo d he prculr Iegrl re olved oe opero. - The Lplce rfor cover he ODE o lgerc eq. σ j ple do. I he pole o
P-Convexity Property in Musielak-Orlicz Function Space of Bohner Type
J N Sce & Mh Res Vol 3 No (7) -7 Alble ole h://orlwlsogocd/deh/sr P-Coey Proery Msel-Orlcz Fco Sce o Boher ye Yl Rodsr Mhecs Edco Deree Fcly o Ss d echology Uerss sl Neger Wlsogo Cerl Jdoes Absrcs Corresodg
STOCHASTIC CALCULUS I STOCHASTIC DIFFERENTIAL EQUATION
The Bk of Thld Fcl Isuos Polcy Group Que Models & Fcl Egeerg Tem Fcl Mhemcs Foudo Noe 8 STOCHASTIC CALCULUS I STOCHASTIC DIFFERENTIAL EQUATION. ก Through he use of ordry d/or prl deres, ODE/PDE c rele
INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY
[Mjuh, : Jury, 0] ISSN: -96 Scefc Jourl Impc Fcr: 9 ISRA, Impc Fcr: IJESRT INTERNATIONAL JOURNAL OF ENINEERIN SCIENCES & RESEARCH TECHNOLOY HAMILTONIAN LACEABILITY IN MIDDLE RAPHS Mjuh*, MurlR, B Shmukh
Numerical Methods using the Successive Approximations for the Solution of a Fredholm Integral Equation
ece Advce Appled d eorecl ec uercl eod u e Succeve Approo or e Soluo o Fredol Ierl Equo AIA OBIŢOIU epre o ec d opuer Scece Uvery o Peroş Uvery Sree 6 Peroş OAIA rdorou@yoo.co Arc: pper pree wo eod or
CURVE FITTING LEAST SQUARES METHOD
Nuercl Alss for Egeers Ger Jord Uverst CURVE FITTING Although, the for of fucto represetg phscl sste s kow, the fucto tself ot be kow. Therefore, t s frequetl desred to ft curve to set of dt pots the ssued
A Remark on Generalized Free Subgroups. of Generalized HNN Groups
Ieraoal Mahemacal Forum 5 200 o 503-509 A Remar o Geeralzed Free Subroup o Geeralzed HNN Group R M S Mahmood Al Ho Uvery Abu Dhab POBo 526 UAE raheedmm@yahoocom Abrac A roup ermed eeralzed ree roup a ree
Differential Equation of Eigenvalues for Sturm Liouville Boundary Value Problem with Neumann Boundary Conditions
Ierol Reserc Jorl o Aled d Bsc Sceces 3 Avlle ole www.rjs.co ISSN 5-838X / Vol 4 : 997-33 Scece Exlorer Plcos Derel Eqo o Eevles or Sr Lovlle Bodry Vle Prole w Ne Bodry Codos Al Kll Gold Dere o Mecs Azr
TEACHERS ASSESS STUDENT S MATHEMATICAL CREATIVITY COMPETENCE IN HIGH SCHOOL
Jourl o See d rs Yer 5, No., pp. 5-, 5 ORIGINL PPER TECHERS SSESS STUDENT S MTHEMTICL CRETIVITY COMPETENCE IN HIGH SCHOOL TRN TRUNG TINH Musrp reeved: 9..5; eped pper:..5; Pulsed ole:..5. sr. ssessme s
Some Unbiased Classes of Estimators of Finite Population Mean
Itertol Jourl O Mtemtcs Ad ttstcs Iveto (IJMI) E-IN: 3 4767 P-IN: 3-4759 Www.Ijms.Org Volume Issue 09 etember. 04 PP-3-37 ome Ubsed lsses o Estmtors o Fte Poulto Me Prvee Kumr Msr d s Bus. Dertmet o ttstcs,
An improved Bennett s inequality
COMMUNICATIONS IN STATISTICS THEORY AND METHODS 017,VOL.0,NO.0,1 8 hps://do.org/10.1080/0361096.017.1367818 A mproved Bee s equly Sogfeg Zheg Deprme of Mhemcs, Mssour Se Uversy, Sprgfeld, MO, USA ABSTRACT
Numerical differentiation (Finite difference)
MA umercl Alyss Week 4 To tk d wrte portle progrms we eed strct model of ter computtol evromet. Ftful models do est ut tey revel tt evromet to e too dverse forcg portle progrmmers to lot eve te smplest
12 Iterative Methods. Linear Systems: Gauss-Seidel Nonlinear Systems Case Study: Chemical Reactions
HK Km Slghtly moded //9 /8/6 Frstly wrtte t Mrch 5 Itertve Methods er Systems: Guss-Sedel Noler Systems Cse Study: Chemcl Rectos Itertve or ppromte methods or systems o equtos cosst o guessg vlue d the
Explicit Representation of Green s Function for Linear Fractional. Differential Operator with Variable Coefficients
KSU-MH--E-R-: Verso 3 Epc Represeo of Gree s uco for er rco ffere Operor w Vrbe Coeffces Mog-H K d Hog-Co O cu of Mecs K Sug Uvers Pogg P R Kore Correspodg uor e-: oogco@ooco bsrc We provde epc represeos
Continuous Indexed Variable Systems
Ieraoal Joural o Compuaoal cece ad Mahemacs. IN 0974-389 Volume 3, Number 4 (20), pp. 40-409 Ieraoal Research Publcao House hp://www.rphouse.com Couous Idexed Varable ysems. Pouhassa ad F. Mohammad ghjeh
In Calculus I you learned an approximation method using a Riemann sum. Recall that the Riemann sum is
Mth Sprg 08 L Approxmtg Dete Itegrls I Itroducto We hve studed severl methods tht llow us to d the exct vlues o dete tegrls However, there re some cses whch t s ot possle to evlute dete tegrl exctly I
Interval Estimation. Consider a random variable X with a mean of X. Let X be distributed as X X
ECON 37: Ecoomercs Hypohess Tesg Iervl Esmo Wh we hve doe so fr s o udersd how we c ob esmors of ecoomcs reloshp we wsh o sudy. The queso s how comforble re we wh our esmors? We frs exme how o produce
1. Consider an economy of identical individuals with preferences given by the utility function
CO 755 Problem Se e Cbrer. Cosder ecoomy o decl dduls wh reereces e by he uly uco U l l Pre- rces o ll hree oods re ormled o oe. Idduls suly ood lbor < d cosume oods d. The oerme c mose d lorem es o oods
The Poisson Process Properties of the Poisson Process
Posso Processes Summary The Posso Process Properes of he Posso Process Ierarrval mes Memoryless propery ad he resdual lfeme paradox Superposo of Posso processes Radom seleco of Posso Pos Bulk Arrvals ad
A L A BA M A L A W R E V IE W
A L A BA M A L A W R E V IE W Volume 52 Fall 2000 Number 1 B E F O R E D I S A B I L I T Y C I V I L R I G HT S : C I V I L W A R P E N S I O N S A N D TH E P O L I T I C S O F D I S A B I L I T Y I N
Parameter Estimation and Hypothesis Testing of Two Negative Binomial Distribution Population with Missing Data
Avlble ole wwwsceceeccom Physcs Poce 0 475 480 0 Ieol Cofeece o Mecl Physcs Bomecl ee Pmee smo Hyohess es of wo Neve Boml Dsbuo Poulo wh Mss D Zhwe Zho Collee of MhemcsJl Noml UvesyS Ch zhozhwe@6com Absc
Chapter Runge-Kutta 2nd Order Method for Ordinary Differential Equations
Cter. Runge-Kutt nd Order Metod or Ordnr Derentl Eutons Ater redng ts cter ou sould be ble to:. understnd te Runge-Kutt nd order metod or ordnr derentl eutons nd ow to use t to solve roblems. Wt s te Runge-Kutt
Least Squares Fitting (LSQF) with a complicated function Theexampleswehavelookedatsofarhavebeenlinearintheparameters
Leas Squares Fg LSQF wh a complcaed fuco Theeampleswehavelookedasofarhavebeelearheparameers ha we have bee rg o deerme e.g. slope, ercep. For he case where he fuco s lear he parameers we ca fd a aalc soluo
Observations on the transcendental Equation
IOSR Jourl o Mecs IOSR-JM e-issn: 78-78-ISSN: 9-7 Volue 7 Issue Jul. - u. -7 www.osrjourls.or Oservos o e rscedel Euo M..Gol S.Vds T.R.Us R Dere o Mecs Sr Idr Gd Collee Trucrll- src: Te rscedel euo w ve
CS321. Numerical Analysis
CS Numercl Alyss Lecture 4 Numercl Itegrto Proessor Ju Zg Deprtmet o Computer Scece Uversty o Ketucky Legto, KY 456 6 Octoer 6, 5 Dete Itegrl A dete tegrl s tervl or tegrto. For ed tegrto tervl, te result
ICS141: Discrete Mathematics for Computer Science I
Uversty o Hw ICS: Dscrete Mthemtcs or Computer Scece I Dept. Iormto & Computer Sc., Uversty o Hw J Stelovsy bsed o sldes by Dr. Be d Dr. Stll Orgls by Dr. M. P. Fr d Dr. J.L. Gross Provded by McGrw-Hll
Solution set Stat 471/Spring 06. Homework 2
oluo se a 47/prg 06 Homework a Whe he upper ragular elemes are suppressed due o smmer b Le Y Y Y Y A weep o he frs colum o oba: A ˆ b chagg he oao eg ad ec YY weep o he secod colum o oba: Aˆ YY weep o
Available online Journal of Scientific and Engineering Research, 2014, 1(1): Research Article
Avalable ole wwwjsaercom Joural o Scec ad Egeerg Research, 0, ():0-9 Research Arcle ISSN: 39-630 CODEN(USA): JSERBR NEW INFORMATION INEUALITIES ON DIFFERENCE OF GENERALIZED DIVERGENCES AND ITS APPLICATION
Outline. Finite Difference Grids. Numerical Analysis. Finite Difference Grids II. Finite Difference Grids III
Itrodcto to Nmercl Alyss Mrc, 9 Nmercl Metods or PDEs Lrry Cretto Meccl Egeerg 5B Semr Egeerg Alyss Mrc, 9 Otle Revew mdterm soltos Revew bsc mterl o mercl clcls Expressos or dervtves, error d error order
International Mathematical Forum, Vol. 9, 2014, no. 13, HIKARI Ltd,
Ieol Mhemcl oum Vol. 9 4 o. 3 65-6 HIKARI Ld www.m-h.com hp//d.do.o/.988/m.4.43 Some Recuece Relo ewee he Sle Doule d Tple Mome o Ode Sc om Iveed mm Duo d hceo S. M. Ame * ollee o Scece d Hume Quwh Shq
VARIATIONAL ITERATION METHOD FOR DELAY DIFFERENTIAL-ALGEBRAIC EQUATIONS. Hunan , China,
Mahemacal ad Compuaoal Applcaos Vol. 5 No. 5 pp. 834-839. Assocao for Scefc Research VARIATIONAL ITERATION METHOD FOR DELAY DIFFERENTIAL-ALGEBRAIC EQUATIONS Hoglag Lu Aguo Xao Yogxag Zhao School of Mahemacs
AML710 CAD LECTURE 12 CUBIC SPLINE CURVES. Cubic Splines Matrix formulation Normalised cubic splines Alternate end conditions Parabolic blending
CUIC SLINE CURVES Cubc Sples Marx formulao Normalsed cubc sples Alerae ed codos arabolc bledg AML7 CAD LECTURE CUIC SLINE The ame sple comes from he physcal srume sple drafsme use o produce curves A geeral
MTH 146 Class 7 Notes
7.7- Approxmte Itegrto Motvto: MTH 46 Clss 7 Notes I secto 7.5 we lered tht some defte tegrls, lke x e dx, cot e wrtte terms of elemetry fuctos. So, good questo to sk would e: How c oe clculte somethg
F l a s h-b a s e d S S D s i n E n t e r p r i s e F l a s h-b a s e d S S D s ( S o-s ltiad t e D r i v e s ) a r e b e c o m i n g a n a t t r a c
L i f e t i m e M a n a g e m e n t o f F l a-b s ah s e d S S D s U s i n g R e c o v e r-a y w a r e D y n a m i c T h r o t t l i n g S u n g j i n L e, e T a e j i n K i m, K y u n g h o, Kainmd J
Chapter 2. Review of Hydrodynamics and Vector Analysis
her. Ree o Hdrodmcs d Vecor Alss. Tlor seres L L L L ' ' L L " " " M L L! " ' L " ' I s o he c e romed he Tlor seres. O he oher hd ' " L . osero o mss -dreco: L L IN ] OUT [mss l [mss l] mss ccmled h me
NUMERICAL SOLUTIONS TO ORDINARY DIFFERENTIAL EQUATIONS
NUMERICAL SOLUTIONS TO ORDINARY DIFFERENTIAL EQUATIONS If e eqao coas dervaves of a - order s sad o be a - order dffereal eqao. For eample a secod-order eqao descrbg e oscllao of a weg aced po b a sprg
Expectation and Moments
Her Sr d Joh W. Woods robbl Sscs d Rdom Vrbles or geers 4h ed. erso duco Ic.. ISB: 978----6 Cher 4 eco d omes Secos 4. eced Vlue o Rdom Vrble 5 O he Vld o quo 4.-8 8 4. Codol ecos Codol eco s Rdom Vrble
Cyclone. Anti-cyclone
Adveco Cycloe A-cycloe Lorez (963) Low dmesoal aracors. Uclear f hey are a good aalogy o he rue clmae sysem, bu hey have some appealg characerscs. Dscusso Is he al codo balaced? Is here a al adjusme
Density estimation III.
Lecure 4 esy esmao III. Mlos Hauskrec mlos@cs..edu 539 Seo Square Oule Oule: esy esmao: Mamum lkelood ML Bayesa arameer esmaes MP Beroull dsrbuo. Bomal dsrbuo Mulomal dsrbuo Normal dsrbuo Eoeal famly Eoeal
Determination of Antoine Equation Parameters. December 4, 2012 PreFEED Corporation Yoshio Kumagae. Introduction
refeed Soluos for R&D o Desg Deermao of oe Equao arameers Soluos for R&D o Desg December 4, 0 refeed orporao Yosho Kumagae refeed Iroduco hyscal propery daa s exremely mpora for performg process desg ad
Isotropic Non-Heisenberg Magnet for Spin S=1
Ierol Jourl of Physcs d Applcos. IN 974- Volume, Number (, pp. 7-4 Ierol Reserch Publco House hp://www.rphouse.com Isoropc No-Heseberg Mge for p = Y. Yousef d Kh. Kh. Mumov.U. Umrov Physcl-Techcl Isue
Chapter Simpson s 1/3 Rule of Integration. ( x)
Cpter 7. Smpso s / Rule o Itegrto Ater redg ts pter, you sould e le to. derve te ormul or Smpso s / rule o tegrto,. use Smpso s / rule t to solve tegrls,. develop te ormul or multple-segmet Smpso s / rule
Introduction to Neural Networks Computing. CMSC491N/691N, Spring 2001
Iroduco o Neurl Neorks Compug CMSC49N/69N, Sprg 00 us: cvo/oupu: f eghs: X, Y j X Noos, j s pu u, for oher us, j pu sgl here f. s he cvo fuco for j from u o u j oher books use Y f _ j j j Y j X j Y j bs:
Midterm Exam. Thursday, April hour, 15 minutes
Economcs of Grow, ECO560 San Francsco Sae Unvers Mcael Bar Sprng 04 Mderm Exam Tursda, prl 0 our, 5 mnues ame: Insrucons. Ts s closed boo, closed noes exam.. o calculaors of an nd are allowed. 3. Sow all
Integration by Parts for D K
Itertol OPEN ACCESS Jourl Of Moder Egeerg Reserc IJMER Itegrto y Prts for D K Itegrl T K Gr, S Ry 2 Deprtmet of Mtemtcs, Rgutpur College, Rgutpur-72333, Purul, West Begl, Id 2 Deprtmet of Mtemtcs, Ss Bv,
Xidian University Liu Congfeng Page 1 of 49
dom Sgl Processg Cher4 dom Processes Cher 4 dom Processes Coes 4 dom Processes... 4. Deo o dom Process... 4. Chrcerzo o dom Process...4 4.. ol Chrcerzo o dom Process...4 4.. Frs-Order Deses o dom Process...5
14. Poisson Processes
4. Posso Processes I Lecure 4 we roduced Posso arrvals as he lmg behavor of Bomal radom varables. Refer o Posso approxmao of Bomal radom varables. From he dscusso here see 4-6-4-8 Lecure 4 " arrvals occur
Boyce/DiPrima 9 th ed, Ch 7.6: Complex Eigenvalues
BocDPm 9 h d Ch 7.6: Compl Egvlus Elm Dffl Equos d Boud Vlu Poblms 9 h do b Wllm E. Boc d Rchd C. DPm 9 b Joh Wl & Sos Ic. W cosd g homogous ssm of fs od l quos wh cos l coffcs d hus h ssm c b w s ' A
1 4 6 is symmetric 3 SPECIAL MATRICES 3.1 SYMMETRIC MATRICES. Defn: A matrix A is symmetric if and only if A = A, i.e., a ij =a ji i, j. Example 3.1.
SPECIAL MATRICES SYMMETRIC MATRICES Def: A mtr A s symmetr f d oly f A A, e,, Emple A s symmetr Def: A mtr A s skew symmetr f d oly f A A, e,, Emple A s skew symmetr Remrks: If A s symmetr or skew symmetr,
Simpson s 1/3 rd Rule of Integration
Simpso s 1/3 rd Rule o Itegrtio Mjor: All Egieerig Mjors Authors: Autr Kw, Chrlie Brker Trsormig Numericl Methods Eductio or STEM Udergrdutes 1/10/010 1 Simpso s 1/3 rd Rule o Itegrtio Wht is Itegrtio?
DATA FITTING. Intensive Computation 2013/2014. Annalisa Massini
DATA FITTING Itesve Computto 3/4 Als Mss Dt fttg Dt fttg cocers the problem of fttg dscrete dt to obt termedte estmtes. There re two geerl pproches two curve fttg: Iterpolto Dt s ver precse. The strteg
P a g e 5 1 of R e p o r t P B 4 / 0 9
P a g e 5 1 of R e p o r t P B 4 / 0 9 J A R T a l s o c o n c l u d e d t h a t a l t h o u g h t h e i n t e n t o f N e l s o n s r e h a b i l i t a t i o n p l a n i s t o e n h a n c e c o n n e
Modeling and Predicting Sequences: HMM and (may be) CRF. Amr Ahmed Feb 25
Modelg d redcg Sequeces: HMM d m be CRF Amr Ahmed 070 Feb 25 Bg cure redcg Sgle Lbel Ipu : A se of feures: - Bg of words docume - Oupu : Clss lbel - Topc of he docume - redcg Sequece of Lbels Noo Noe:
Nonlocal Boundary Value Problem for Nonlinear Impulsive q k Symmetric Integrodifference Equation
OSR ol o Mec OSR-M e-ssn: 78-578 -SSN: 9-765X Vole e Ve M - A 7 PP 95- wwwojolog Nolocl Bo Vle Poble o Nole lve - Sec egoeece Eo Log Ceg Ceg Ho * Yeg He ee o Mec Yb Uve Yj PR C Abc: A oe ole lve egoeece
Chapter 12-b Integral Calculus - Extra
C - Itegrl Clulus Cpter - Itegrl Clulus - Etr Is Newto Toms Smpso BONUS Itroduto to Numerl Itegrto C - Itegrl Clulus Numerl Itegrto Ide s to do tegrl smll prts, lke te wy we preseted tegrto: summto. Numerl
Application of Multiple Exp-Function Method to Obtain Multi-Soliton Solutions of (2 + 1)- and (3 + 1)-Dimensional Breaking Soliton Equations
Amerc Jourl of Compuol Appled Mhemcs: ; (: 4-47 DOI:.593/j.jcm..8 Applco of Mulple Exp-Fuco Mehod o Ob Mul-Solo Soluos of ( + - (3 + -Dmesol Breg Solo Equos M. T. Drvsh,*, Mlheh Njf, Mohmmd Njf Deprme
Supplement: Gauss-Jordan Reduction
Suppleme: Guss-Jord Reducio. Coefficie mri d ugmeed mri: The coefficie mri derived from sysem of lier equios m m m m is m m m A O d he ugmeed mri derived from he ove sysem of lier equios is [ ] m m m m
Available online through
Avlble ole through wwwmfo FIXED POINTS FOR NON-SELF MAPPINGS ON CONEX ECTOR METRIC SPACES Susht Kumr Moht* Deprtmet of Mthemtcs West Begl Stte Uverst Brst 4 PrgsNorth) Kolt 76 West Begl Id E-ml: smwbes@yhoo
Numerical Differentiation
College o Egeerg ad Computer Scece Mecacal Egeerg Departmet Numercal Aalyss Notes November 4, 7 Istructor: Larry Caretto Numercal Deretato Itroducto Tese otes provde a basc troducto to umercal deretato
Chapter 5 Transient Analysis
hpr 5 rs Alyss Jsug Jg ompl rspos rs rspos y-s rspos m os rs orr co orr Dffrl Equo. rs Alyss h ffrc of lyss of crcus wh rgy sorg lms (ucors or cpcors) & m-ryg sgls wh rss crcus s h h quos rsulg from r
Modified Taylor's Method and Nonlinear Mixed Integral Equation
Uversl Jourl of Iegrl quos 4 (6), 9 wwwpperscecescom Modfed Tylor's Mehod d oler Mxed Iegrl quo R T Moog Fculy of Appled Scece, Umm Al Qurh Uversy Mkh, Kgdom of Sud Ar rmoog_777@yhoocom Asrc I hs pper,
Heat kernel methods in finance: the SABR model
He erel eods ce: e SABR odel Crelo Vccro Te wor reseed s reor s bee crred ou w e suor o Reuers Fcl Sowre Pueu Frce d uder e dreco o Adre Bourere o wo e uor s rculrl reul For rers coes or roosos o collboro
Partial Molar Properties of solutions
Paral Molar Properes of soluos A soluo s a homogeeous mxure; ha s, a soluo s a oephase sysem wh more ha oe compoe. A homogeeous mxures of wo or more compoes he gas, lqud or sold phase The properes of a
GENERALIZED METHOD OF LIE-ALGEBRAIC DISCRETE APPROXIMATIONS FOR SOLVING CAUCHY PROBLEMS WITH EVOLUTION EQUATION
Joural of Appled Maemacs ad ompuaoal Mecacs 24 3(2 5-62 GENERALIZED METHOD OF LIE-ALGEBRAI DISRETE APPROXIMATIONS FOR SOLVING AUHY PROBLEMS WITH EVOLUTION EQUATION Arkad Kdybaluk Iva Frako Naoal Uversy
Chapter 5. Curve fitting
Chapter 5 Curve ttg Assgmet please use ecell Gve the data elow use least squares regresso to t a a straght le a power equato c a saturato-growthrate equato ad d a paraola. Fd the r value ad justy whch
Area and the Definite Integral. Area under Curve. The Partition. y f (x) We want to find the area under f (x) on [ a, b ]
![Area and the Definite Integral. Area under Curve. The Partition. y f (x) We want to find the area under f (x) on [ a, b ]](/thumbs/83/87614710.jpg "Area and the Definite Integral. Area under Curve. The Partition. y f (x) We want to find the area under f (x) on [ a, b ]")
Are d the Defte Itegrl 1 Are uder Curve We wt to fd the re uder f (x) o [, ] y f (x) x The Prtto We eg y prttog the tervl [, ] to smller su-tervls x 0 x 1 x x - x -1 x 1 The Bsc Ide We the crete rectgles
Preliminary Examinations: Upper V Mathematics Paper 1
relmr Emtos: Upper V Mthemtcs per Jul 03 Emer: G Evs Tme: 3 hrs Modertor: D Grgortos Mrks: 50 INSTRUCTIONS ND INFORMTION Ths questo pper sts of 0 pges, cludg swer Sheet pge 8 d Iformto Sheet pges 9 d 0
NEIGHBOURHOODS OF A CERTAIN SUBCLASS OF STARLIKE FUNCTIONS. P. Thirupathi Reddy. E. mail:
NEIGHOURHOOD OF CERTIN UCL OF TRLIKE FUNCTION P Tirupi Reddy E mil: reddyp@yooom sr: Te im o is pper is o rodue e lss ( sulss o ( sisyig e odio wi is ( ) p < 0< E We sudy eigouroods o is lss d lso prove
A New Iterative Method for Solving Initial Value Problems
A New Ierave Meod or Solvg Ial Value Problems Mgse Wu ad Weu Hog Dearme o Ma., Sascs, ad CS, Uvers o Wscos-Sou, Meomoe, WI 5475, USA Dearme o Maemacs, Clao Sae Uvers, Morrow, GA 36, USA Absrac - I s aer,
Introduction to Numerical Integration Part II
Introducton to umercl Integrton Prt II CS 75/Mth 75 Brn T. Smth, UM, CS Dept. Sprng, 998 4/9/998 qud_ Intro to Gussn Qudrture s eore, the generl tretment chnges the ntegrton prolem to ndng the ntegrl w
FORCED VIBRATION of MDOF SYSTEMS
FORCED VIBRAION of DOF SSES he respose of a N DOF sysem s govered by he marx equao of moo: ] u C] u K] u 1 h al codos u u0 ad u u 0. hs marx equao of moo represes a sysem of N smulaeous equaos u ad s me
Solution of Impulsive Differential Equations with Boundary Conditions in Terms of Integral Equations
Joural of aheacs ad copuer Scece (4 39-38 Soluo of Ipulsve Dffereal Equaos wh Boudary Codos Ters of Iegral Equaos Arcle hsory: Receved Ocober 3 Acceped February 4 Avalable ole July 4 ohse Rabba Depare
Mu Sequences/Series Solutions National Convention 2014
Mu Sequeces/Seres Solutos Natoal Coveto 04 C 6 E A 6C A 6 B B 7 A D 7 D C 7 A B 8 A B 8 A C 8 E 4 B 9 B 4 E 9 B 4 C 9 E C 0 A A 0 D B 0 C C Usg basc propertes of arthmetc sequeces, we fd a ad bm m We eed
Chapter Unary Matrix Operations
Chpter 04.04 Ury trx Opertos After redg ths chpter, you should be ble to:. kow wht ury opertos mes,. fd the trspose of squre mtrx d t s reltoshp to symmetrc mtrces,. fd the trce of mtrx, d 4. fd the ermt
Math Week 5 concepts and homework, due Friday February 10
Mt 2280-00 Week 5 concepts nd omework, due Fridy Februry 0 Recll tt ll problems re good for seeing if you cn work wit te underlying concepts; tt te underlined problems re to be nded in; nd tt te Fridy
CS321. Introduction to Numerical Methods
CS Itroducto to Numercl Metods Lecture Revew Proessor Ju Zg Deprtmet o Computer Scece Uversty o Ketucky Legto, KY 6 6 Mrc 7, Number Coverso A geerl umber sould be coverted teger prt d rctol prt seprtely
CONTROLLABILITY OF A CLASS OF SINGULAR SYSTEMS
44 Asa Joural o Corol Vol 8 No 4 pp 44-43 December 6 -re Paper- CONTROLLAILITY OF A CLASS OF SINGULAR SYSTEMS Guagmg Xe ad Log Wag ASTRACT I hs paper several dere coceps o corollably are vesgaed or a class
Chapter 5. Long Waves
ape 5. Lo Waes Wae e s o compaed ae dep: < < L π Fom ea ae eo o s s ; amos ozoa moo z p s ; dosac pesse Dep-aeaed coseao o mass
Complex Variables. Chapter 19 Series and Residues. March 26, 2013 Lecturer: Shih-Yuan Chen
omplex Vrble hpter 9 Sere d Redue Mrch 6, Lecturer: Shh-Yu he Except where otherwe oted, cotet lceed uder BY-N-SA. TW Lcee. otet Sequece & ere Tylor ere Luret ere Zero & pole Redue & redue theorem Evluto
i+1 by A and imposes Ax
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 09.9 NUMERICAL FLUID MECHANICS FALL 009 Mody, October 9, 009 QUIZ : SOLUTIONS Notes: ) Multple solutos
Representation of Solutions of Linear Homogeneous Caputo Fractional Differential Equations with Continuous Variable Coefficients
Repor Nuber: KSU MATH 3 E R 6 Represeo o Souos o Ler Hoogeeous puo Fro ere Equos w ouous Vrbe oees Su-Ae PAK Mog-H KM d Hog-o O * Fu o Mes K Sug Uvers Pogg P R Kore * orrespodg uor e: oogo@ooo Absr We
The Mathematical Appendix
The Mathematcal Appedx Defto A: If ( Λ, Ω, where ( λ λ λ whch the probablty dstrbutos,,..., Defto A. uppose that ( Λ,,..., s a expermet type, the σ-algebra o λ λ λ are defed s deoted by ( (,,...,, σ Ω.
G1-Renewal Process as Repairable System Model
G-Reewl Process s Reprble Sysem Model M.P. Kmsky d V.V. Krvsov Uversy of Mryld College Prk USA Ford Moor Compy Derbor USA Absrc Ths pper cosders po process model wh moooclly decresg or cresg ROCOF d he
8. Queueing systems lect08.ppt S Introduction to Teletraffic Theory - Fall
8. Queueg sysems lec8. S-38.45 - Iroduco o Teleraffc Theory - Fall 8. Queueg sysems Coes Refresher: Smle eleraffc model M/M/ server wag laces M/M/ servers wag laces 8. Queueg sysems Smle eleraffc model
MATRIX AND VECTOR NORMS
Numercl lyss for Egeers Germ Jord Uversty MTRIX ND VECTOR NORMS vector orm s mesure of the mgtude of vector. Smlrly, mtr orm s mesure of the mgtude of mtr. For sgle comoet etty such s ordry umers, the
Numerical Differentiation
College o Egeerg ad Computer Scece Mecacal Egeerg Departmet ME 9 Numercal Aalyss Marc 4, 4 Istructor: Larry Caretto Numercal Deretato Itroducto Tese otes provde a basc troducto to umercal deretato usg
Mathematically, integration is just finding the area under a curve from one point to another. It is b
Numerl Metods or Eg [ENGR 9] [Lyes KADEM 7] CHAPTER VI Numerl Itegrto Tops - Rem sums - Trpezodl rule - Smpso s rule - Rrdso s etrpolto - Guss qudrture rule Mtemtlly, tegrto s just dg te re uder urve rom
A NOTE ON THE APPLICATION OF THE GUERMOND-PASQUETTI MASS LUMPING CORRECTION TECHNIQUE FOR CONVECTION-DIFFUSION PROBLEMS ( ) SERGII V.
NTE N THE PPLICTIN F THE UERMND-PSQUETTI MSS LUMPIN CRRECTIN TECHNIQUE FR CNVECTIN-DIFFUSIN PRLEMS Prer submed o Elsever jourl 0 My 0 SERII V. SIRYK Nol Teccl Uversy of Ure "Igor Sorsy Kyv Polyecc Isue",
Assignment 7/MATH 247/Winter, 2010 Due: Friday, March 19. Powers of a square matrix
Assgmet 7/MATH 47/Wter, 00 Due: Frday, March 9 Powers o a square matrx Gve a square matrx A, ts powers A or large, or eve arbtrary, teger expoets ca be calculated by dagoalzg A -- that s possble (!) Namely,
BEST PATTERN OF MULTIPLE LINEAR REGRESSION
ERI COADA GERMAY GEERAL M.R. SEFAIK AIR FORCE ACADEMY ARMED FORCES ACADEMY ROMAIA SLOVAK REPUBLIC IERAIOAL COFERECE of SCIEIFIC PAPER AFASES Brov 6-8 M BES PAER OF MULIPLE LIEAR REGRESSIO Corel GABER PEROLEUM-GAS
THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 10, Number 2/2009, pp
THE PUBLIHING HOUE PROCEEDING OF THE ROMANIAN ACADEMY, eres A, OF THE ROMANIAN ACADEMY Volume 0, Number /009,. 000-000 ON ZALMAI EMIPARAMETRIC DUALITY MODEL FOR MULTIOBJECTIVE FRACTIONAL PROGRAMMING WITH
The ray paths and travel times for multiple layers can be computed using ray-tracing, as demonstrated in Lab 3.
C. Trael me cures for mulple reflecors The ray pahs ad rael mes for mulple layers ca be compued usg ray-racg, as demosraed Lab. MATLAB scrp reflec_layers_.m performs smple ray racg. (m) ref(ms) ref(ms)