On Fourth and Fifth Order Explicit Almost Runge Kutta Methods
|
|
|
- Merry Strickland
- 6 years ago
- Views:
Transcription
1 Itertiol Jourl of Sietifi d Iovtive Mthemtil Reerh (IJSIMR) Volume, Iue, Jur 6, PP ISSN 7-7X (Prit) & ISSN 7- (Olie) O Fourth d Fifth Order Expliit Almot Ruge Kutt Method Adulrhm Ndu Deprtmet of Mthemti Federl Uiverit of Teholog Mi, Nigeri.du@futmi.edu.g, du@hoo.om Khdeejh Jme Audu Deprtmet of Mthemti Federl Uiverit of Teholog Mi, Nigeri jmekhdeejh@hoo.om Atrt: Almot Ruge Kutt (ARK) method re peil l of geerl lier method tht hve loe ffiit to Ruge Kutt method for the umeril itegrtio of iitil vlue prolem of ordir differetil equtio, ut with ome dvtge over the lil Ruge Kutt method. I thi reerh work, we exploit the order, ihiltio d Ruge Kutt tilit oditio ormll oited with Ruge Kutt method to derive two ew expliit Almot Ruge Kutt method of order four (ARK) d five (ARK5)repetivel. The method were teted for oite d tilit d prove to tif the riteri for oth; hee their overgee i gurteed. I order to demotrte the effiie d reliilit of the propoed method, ome iitil vlue prolem were olved with the method. Numeril reult reveled tht the propoed method exhiited umeril error withi eptle limit whe ompred with the ext olutio. More o, the method performed etter th ome exitig Almot Ruge Kutt method of equl tdig. Keword: Iitil vlue prolem, Order oditio, Covergee, Stilit, Coite.. INTRODUCTION I the quet to hieve overll geerl formtio of umeril heme tht omie the multivlued feture of lier multitep method log with the multitge feture of Ruge-Kutt method, Buther [] itrodued the geerl lier method. I geerl lier method the multitge feture of Ruge-Kutt heme re liked with the pereptio of pig more vlue i- etwee tep. Almot Ruge-Kutt (ARK) method re peil tpe of geerl lier method whoe propertie loel reemle thoe of expliit Ruge-Kutt method. The were itrodued Buther [] for the purpoe of preervig the multi-tge hrter of Ruge-Kutt heme well pig m vlue etwee tep, there givig the method multi-vlue hrter []. The geerl ARK method tke the form: hf hf A U hf [ ] B V r r () where i the umer of iterl tge. For order method the three output vlue re ARC Pge 88
2 Adulrhm Ndu & Khdeejh Jme Audu Itertiol Jourl of Sietifi d Iovtive Mthemtil Reerh (IJSIMR) Pge 89 The oeffiiet of the method re hoe i reful w to eure the imple tilit propertie of Ruge Kutt method re retied. I thi reerh we will oetrte o method where i tritl lower trigulr, d hee the method i expliit. The geerl form of expliit ARK method i (),,, hf hf hf A Ae e where re the Iterl tge, re the Output pproximtio, re the Iput pproximtio d the tge derivtive re Jut like i the e of Ruge-Kutt method, i vetor of legth repreetig the weight d i vetor of legth repreetig the poitio t whih futio i evluted. The vetor i uit vetor of legth oitig of etirel. I thi pper, we eek to derive two expliit ARK method of order four d five with four d five tge repetivel (i.e.,method with d ).. METHODS. Derivtio of ARK The geerl fourth order four tge heme i give : () V B U A The i vetor,,. The otituet of the firt output pproximtio for order four with four tge re give elow:
3 O Fourth d Fifth Order Expliit Almot Ruge Kutt Method Comiig (7) d () we oti Either equtio () or the me oditio i () will e deigted ihiltio oditio. The vetor d, repreet the rd outgoig etimtio, whih i met to produe the reult. It follow tht, The Ruge-Kutt tilit oditio re, From (5), it follow tht From (5) - (8), we otied Thu, Ad, From () d () we evlute the remiig vlue of vetor. Evetull the followig ARK method i otied. Itertiol Jourl of Sietifi d Iovtive Mthemtil Reerh (IJSIMR) Pge 9
4 Adulrhm Ndu & Khdeejh Jme Audu ARK with, A B U V (6) 6 6. Derivtio of ARK The geerl fifth order, five tge heme i: A U B V (7) 5 where =. The Order oditiore: Itertiol Jourl of Sietifi d Iovtive Mthemtil Reerh (IJSIMR) Pge 9
5 O Fourth d Fifth Order Expliit Almot Ruge Kutt Method We hve free prmeter deigted, d. Thu, After otiig the remiig elemet of we oti the followig ARK5 heme. ARK5 with A B U V (5) Itertiol Jourl of Sietifi d Iovtive Mthemtil Reerh (IJSIMR) Pge
6 Adulrhm Ndu & Khdeejh Jme Audu. Covergee Ali of ARK The mtrix of the ARK heme of (6)mut hve ouded power for the method to e tle. Thu The hrteriti polomil of i Ad the eigevlue re foud to e. B Cle-Hmilto theorem, Thi implie tht Similrl, greter th ; it implie oitet ie it i of order. Covergee Ali of ARK5 The mtrix of ARK5 heme (5) i, for ever i ouded d thu the method i tle. The method i equll ; hee it overgee. The eigevlue of (59) lulted to e,, d. B Cle-Hmilto theorem the hrteriti equtio i Thi implie tht. Similrl, implie implie, for ever greter th ; it implie i ouded d thu the method i tle. The ARK5 method i lo oitet ie it i of order. Therefore it i overget.. RESULTS AND DISCUSSION The followig prolem re olved with the derived heme. Prolem : Thi prolem i foud i []. Prolem : Itertiol Jourl of Sietifi d Iovtive Mthemtil Reerh (IJSIMR) Pge 9
7 O Fourth d Fifth Order Expliit Almot Ruge Kutt Method Thi prolem i foud i [] d [5]. Prolem : Thi prolem i foud i [6]. Prolem, d olved uig the derived heme (ARK d ARK5). The reult re ompred with method of pproprite order derived [], [5] d [7]. The reult re preeted follow. Tle. Comprio of reult (ARK, Prolem )....E+..E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E- Tle. Comprio of reult (ARK, Prolem )....E+..E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E- Itertiol Jourl of Sietifi d Iovtive Mthemtil Reerh (IJSIMR) Pge 9
8 Adulrhm Ndu & Khdeejh Jme Audu Tle. Comprio of reult (ARK5, Prolem )....E+..E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E- Tle. Comprio of reult (ARK5, Prolem )....E+..E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E- I Tle d, the propoed ARK method dipled leer error th the method of [] d [5]. I Tle d 5, the propoed ARK5 method dipled leer error th the method of [7] d [5]. The reult of Tle reveled tht the propoed Almot Ruge-Kutt method of order d 5 exhiit egligile error i reltio to the ext olutio. The lo performed etter th the orrepodig exitig method the re ompred to exhiitig leer error.. CONCLUSION B the foregoig, it i itrutive tht the propoed Almot Ruge-Kutt method of order d 5 do ot ol exhiit effiie d reliilit, evidet their repetive ioequetil error i reltio to the ext olutio, ut lo perform etter th the exitig method. Itertiol Jourl of Sietifi d Iovtive Mthemtil Reerh (IJSIMR) Pge 95
![O Fourth d Fifth Order Expliit Almot Ruge Kutt Method REFERENCES [] Buther J. C., Geerl lier method, Computer Mthemti Applitio., 5- (996). [] Buther J. C., A itrodutio to Almot Ruge-Kutt method, Applied Numeril Mthemti.](/docs-images/77/75001613/images/9-0.jpg ", - (997). [] Rtteur N., Almot Ruge-kutt method for tiff d o-tiff prolem, Ph.D. Thei Uiverit of Aukld, New Zeld. Pp. 5, 57 (5). [] Ndu A.")
![, A optiml 6 tep impliit lier multitep method for iitil vlue prolem, Mter Thei, Federl Uiverit of Teholog Mi, Nigeri. Pp. 6 (). [5] Arhm O.](/docs-images/77/75001613/images/9-1.jpg ", Developmet of ome ew le of expliit Almot Ruge-Kutt method for o-tiff differetil equtio, Ph.D. Thei, Federl Uiverit of Teholog Mi, Nigeri. Pp. 5 (). [6] J. C.")
9 O Fourth d Fifth Order Expliit Almot Ruge Kutt Method REFERENCES [] Buther J. C., Geerl lier method, Computer Mthemti Applitio., 5- (996). [] Buther J. C., A itrodutio to Almot Ruge-Kutt method, Applied Numeril Mthemti., - (997). [] Rtteur N., Almot Ruge-kutt method for tiff d o-tiff prolem, Ph.D. Thei Uiverit of Aukld, New Zeld. Pp. 5, 57 (5). [] Ndu A., A optiml 6 tep impliit lier multitep method for iitil vlue prolem, Mter Thei, Federl Uiverit of Teholog Mi, Nigeri. Pp. 6 (). [5] Arhm O., Developmet of ome ew le of expliit Almot Ruge-Kutt method for o-tiff differetil equtio, Ph.D. Thei, Federl Uiverit of Teholog Mi, Nigeri. Pp. 5 (). [6] J. C. Buther, Numeril Method for Ordir Differetil Equtio, d ed. Chiheter, UK. Joh Wile & So, Ltd., 8, h., pp. 6. [7] Alimi O. K., O the performe of Rihrdo extrpoltio tehique i etimtig lol trutio error for expliit Almot Ruge-Kutt method, Mter Thei, Federl Uiverit of Teholog, Mi, Nigeri. Pp. (). AUTHORS BIOGRAPH Dr Adulrhm Ndu, otied PhD i Applied Mthemti from Federl Uiverit of Teholog, Mi, Nigeri i. He i Seior Leturer i the Deprtmet of Mthemti t Federl Uiverit of Teholog, Mi, Nigeri where he tehe vriou oure to udergrdute d potgrdute tudet. He i tive reerher i Numeril Ali d Biomthemti where he h everl pulitio i itertiol peer- reviewed jourl. Khdeejh Jme Audu, hold Mter degree i Mthemti from Federl Uiverit of Teholog, Mi, Nigeri (). Her reerh iteret i ill i Numeril Ali. She i urretl preprig to eroll for PhD progrmme i Applied Mthemti. Itertiol Jourl of Sietifi d Iovtive Mthemtil Reerh (IJSIMR) Pge 96
ROUTH-HURWITZ CRITERION
Automti Cotrol Sytem, Deprtmet of Mehtroi Egieerig, Germ Jordi Uiverity Routh-Hurwitz Criterio ite.google.om/ite/ziydmoud 7 ROUTH-HURWITZ CRITERION The Routh-Hurwitz riterio i lytil proedure for determiig
AUTOMATIC CONTROL SYSTEMS
9 HE UO ONROL SYSES OSVE SLE RELZONS OF ONNUOUS-E LNER SYSES deuz Kzore trt: he rolem for exitee d determitio of the et of oitive ymtotilly tle reliztio of roer trfer futio of lier otiuou-time ytem i formulted
Numerical Solutions of Fredholm Integral Equations Using Bernstein Polynomials
Numericl Solutios of Fredholm Itegrl Equtios Usig erstei Polyomils A. Shiri, M. S. Islm Istitute of Nturl Scieces, Uited Itertiol Uiversity, Dhk-, gldesh Deprtmet of Mthemtics, Uiversity of Dhk, Dhk-,
4 Application to the TA-40 Manipulator Introduction Description of the Manipulator
Applitio to the TA-0 Mipultor.. Itrodutio Util ow ol the theoretil priiple ued i thi thei hve ee overed. Thi hpter over how thi theor i pplied to the TA-0 mipultor. The TA-0 i rooti mipultor ued PETROBRAS
Introduction of Fourier Series to First Year Undergraduate Engineering Students
Itertiol Jourl of Adved Reserh i Computer Egieerig & Tehology (IJARCET) Volume 3 Issue 4, April 4 Itrodutio of Fourier Series to First Yer Udergrdute Egieerig Studets Pwr Tejkumr Dtttry, Hiremth Suresh
An Investigation of Continued Fractions
Ivetigtio o Cotiued rtio Kriti Ptto Ohio Norther Uiverity d Ohio 8 Emil: k-ptto@ou.edu dvior: Dr. dr hroeder -hroeder@ou.edu d Dr. Rihrd Dquil rdquil@mukigum.edu btrt: The tudy o otiued rtio h produed
Gauss Elimination, Gauss-Jordan Elimination and Gauss Seidel Iteration Method: Performance Comparison using MAT LAB
Itertiol Jourl of Egieerig Tehology Siee d Reserh www.ietsr.om ISSN 394 3386 Volume 4, Issue 0 Otober 07 Guss Elimitio, Guss-Jord Elimitio d Guss Seidel Itertio Method: Performe Compriso usig MAT LAB *Dr.
OBSERVATIONS ON THE NON- HOMOGENEOUS SEXTIC EQUATION WITH FOUR UNKNOWNS
ISSN: 1-87 Itertiol Jourl of Iovtive Reerch i Sciece, Egieerig d Techology Vol., Iue, My 01 OSERVTIONS ON THE NON- HOMOGENEOUS SEXTIC EQUTION WITH FOUR UNKNOWNS y ( k z S.Vidhylkhmi 1,M..Gopl,.Kvith Profeor,
Numerical Methods. Lecture 5. Numerical integration. dr hab. inż. Katarzyna Zakrzewska, prof. AGH. Numerical Methods lecture 5 1
Numeril Methods Leture 5. Numeril itegrtio dr h. iż. Ktrzy Zkrzewsk, pro. AGH Numeril Methods leture 5 Outlie Trpezoidl rule Multi-segmet trpezoidl rule Rihrdso etrpoltio Romerg's method Simpso's rule
ON n-fold FILTERS IN BL-ALGEBRAS
Jourl of Alger Numer Theor: Adves d Applitios Volume 2 Numer 29 Pges 27-42 ON -FOLD FILTERS IN BL-ALGEBRAS M. SHIRVANI-GHADIKOLAI A. MOUSSAVI A. KORDI 2 d A. AHMADI 2 Deprtmet of Mthemtis Trit Modres Uiversit
A New Estimator Using Auxiliary Information in Stratified Adaptive Cluster Sampling
Ope Jourl of ttitic, 03, 3, 78-8 ttp://d.doi.org/0.436/oj.03.3403 Publied Olie eptember 03 (ttp://www.cirp.org/jourl/oj) New Etimtor Uig uilir Iformtio i trtified dptive Cluter mplig Nippor Cutim *, Moc
Optimal Recovery of Holomorphic Functions from Inaccurate Information about Radial Integration
Ameri Jourl o Computtiol Mthemti,,, 58-68 http://ddoiorg/436/m435 Publihed Olie Deember (http://wwwsirporg/ourl/m) Optiml Reover o Holomorphi Futio rom Iurte Iormtio bout Rdil Itegrtio Arthur DeGrw Deprtmet
Thomas Whitham Sixth Form
Thoms Whithm ith Form Pure Mthemtis Uit lger Trigoometr Geometr lulus lger equees The ifiite sequee of umers U U U... U... is si to e () overget if U L fiite limit s () iverget to if U s Emple The sequee...
Addendum. Addendum. Vector Review. Department of Computer Science and Engineering 1-1
Addedum Addedum Vetor Review Deprtmet of Computer Siee d Egieerig - Coordite Systems Right hded oordite system Addedum y z Deprtmet of Computer Siee d Egieerig - -3 Deprtmet of Computer Siee d Egieerig
All the Laplace Transform you will encounter has the following form: Rational function X(s)
EE G Note: Chpter Itructor: Cheug Pge - - Iverio of Rtiol Fuctio All the Lplce Trform you will ecouter h the followig form: m m m m e τ 0...... Rtiol fuctio Dely Why? Rtiol fuctio come out turlly from
Introduction to Matrix Algebra
Itrodutio to Mtri Alger George H Olso, Ph D Dotorl Progrm i Edutiol Ledership Applhi Stte Uiversit Septemer Wht is mtri? Dimesios d order of mtri A p q dimesioed mtri is p (rows) q (olums) rr of umers,
The Real Numbers. RATIONAL FIELD Take rationals as given. is a field with addition and multiplication defined. BOUNDS. Addition: xy= yx, xy z=x yz,
MAT337H Itrodutio to Rel Alysis The Rel Numers RATIONAL FIELD Tke rtiols s give { m, m, R, } is field with dditio d multiplitio defied Additio: y= y, y z= yz, There eists Q suh tht = For eh Q there eists
AP Calculus AB AP Review
AP Clulus AB Chpters. Re limit vlues from grphsleft-h Limits Right H Limits Uerst tht f() vlues eist ut tht the limit t oes ot hve to.. Be le to ietify lel isotiuities from grphs. Do t forget out the 3-step
The Reimann Integral is a formal limit definition of a definite integral
MATH 136 The Reim Itegrl The Reim Itegrl is forml limit defiitio of defiite itegrl cotiuous fuctio f. The costructio is s follows: f ( x) dx for Reim Itegrl: Prtitio [, ] ito suitervls ech hvig the equl
Observations on the Non-homogeneous Quintic Equation with Four Unknowns
Itertiol Jourl of Mthemtic Reerch. ISSN 976-84 Volume, Number 1 (13), pp. 17-133 Itertiol Reerch Publictio Houe http://www.irphoue.com Obervtio o the No-homogeeou Quitic Equtio with Four Ukow S. Vidhylkhmi
Data Compression Techniques (Spring 2012) Model Solutions for Exercise 4
58487 Dt Compressio Tehiques (Sprig 0) Moel Solutios for Exerise 4 If you hve y fee or orretios, plese ott jro.lo t s.helsii.fi.. Prolem: Let T = Σ = {,,, }. Eoe T usig ptive Huffm oig. Solutio: R 4 U
Introduction to Modern Control Theory
Itroductio to Moder Cotrol Theory MM : Itroductio to Stte-Spce Method MM : Cotrol Deig for Full Stte Feedck MM 3: Etitor Deig MM 4: Itroductio of the Referece Iput MM 5: Itegrl Cotrol d Rout Trckig //4
Riemann Integral Oct 31, such that
Riem Itegrl Ot 31, 2007 Itegrtio of Step Futios A prtitio P of [, ] is olletio {x k } k=0 suh tht = x 0 < x 1 < < x 1 < x =. More suitly, prtitio is fiite suset of [, ] otiig d. It is helpful to thik of
ELEC 372 LECTURE NOTES, WEEK 6 Dr. Amir G. Aghdam Concordia University
ELEC 37 LECTURE NOTES, WEE 6 Dr mir G ghdm Cocordi Uiverity Prt of thee ote re dpted from the mteril i the followig referece: Moder Cotrol Sytem by Richrd C Dorf d Robert H Bihop, Pretice Hll Feedbck Cotrol
Definition Integral. over[ ab, ] the sum of the form. 2. Definite Integral
![Definition Integral. over[ ab, ] the sum of the form. 2. Definite Integral](/thumbs/83/87856510.jpg "Definition Integral. over[ ab, ] the sum of the form. 2. Definite Integral")
Defiite Itegrl Defiitio Itegrl. Riem Sum Let f e futio efie over the lose itervl with = < < < = e ritrr prtitio i suitervl. We lle the Riem Sum of the futio f over[, ] the sum of the form ( ξ ) S = f Δ
Neighborhoods of Certain Class of Analytic Functions of Complex Order with Negative Coefficients
Ge Mth Notes Vol 2 No Jury 20 pp 86-97 ISSN 229-784; Copyriht ICSRS Publitio 20 wwwi-srsor Avilble free olie t http://wwwemi Neihborhoods of Certi Clss of Alyti Futios of Complex Order with Netive Coeffiiets
Spherical refracting surface. Here, the outgoing rays are on the opposite side of the surface from the Incoming rays.
Sphericl refrctig urfce Here, the outgoig ry re o the oppoite ide of the urfce from the Icomig ry. The oject i t P. Icomig ry PB d PV form imge t P. All prxil ry from P which trike the phericl urfce will
Section IV.6: The Master Method and Applications
Sectio IV.6: The Mster Method d Applictios Defiitio IV.6.1: A fuctio f is symptoticlly positive if d oly if there exists rel umer such tht f(x) > for ll x >. A cosequece of this defiitio is tht fuctio
f ( x) ( ) dx =
Defiite Itegrls & Numeric Itegrtio Show ll work. Clcultor permitted o, 6,, d Multiple Choice. (Clcultor Permitted) If the midpoits of equl-width rectgles is used to pproximte the re eclosed etwee the x-xis
General properties of definite integrals
Roerto s Notes o Itegrl Clculus Chpter 4: Defiite itegrls d the FTC Sectio Geerl properties of defiite itegrls Wht you eed to kow lredy: Wht defiite Riem itegrl is. Wht you c ler here: Some key properties
Week 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right:
Week 1 Notes: 1) Riem Sum Aim: Compute Are Uder Grph Suppose we wt to fid out the re of grph, like the oe o the right: We wt to kow the re of the red re. Here re some wys to pproximte the re: We cut the
On AP-Henstock Integrals of Interval-Valued. Functions and Fuzzy-Number-Valued Functions.
Applied Mthemtis, 206, 7, 2285-2295 http://www.sirp.org/jourl/m ISSN Olie: 252-7393 ISSN Prit: 252-7385 O AP-Hestok Itegrls of Itervl-Vlued Futios d Fuzzy-Numer-Vlued Futios Muwy Elsheikh Hmid *, Alshikh
Chap8 - Freq 1. Frequency Response
Chp8 - Freq Frequecy Repoe Chp8 - Freq Aged Prelimirie Firt order ytem Frequecy repoe Low-p filter Secod order ytem Clicl olutio Frequecy repoe Higher order ytem Chp8 - Freq 3 Frequecy repoe Stedy-tte
Accuplacer Elementary Algebra Study Guide
Testig Ceter Studet Suess Ceter Aupler Elemetry Alger Study Guide The followig smple questios re similr to the formt d otet of questios o the Aupler Elemetry Alger test. Reviewig these smples will give
Fig. 1. I a. V ag I c. I n. V cg. Z n Z Y. I b. V bg
ymmetricl Compoets equece impedces Although the followig focuses o lods, the results pply eqully well to lies, or lies d lods. Red these otes together with sectios.6 d.9 of text. Cosider the -coected lced
Project 3: Using Identities to Rewrite Expressions
MAT 5 Projet 3: Usig Idetities to Rewrite Expressios Wldis I lger, equtios tht desrie properties or ptters re ofte lled idetities. Idetities desrie expressio e repled with equl or equivlet expressio tht
Numerical Integration by using Straight Line Interpolation Formula
Glol Jourl of Pure d Applied Mthemtics. ISSN 0973-1768 Volume 13, Numer 6 (2017), pp. 2123-2132 Reserch Idi Pulictios http://www.ripulictio.com Numericl Itegrtio y usig Stright Lie Iterpoltio Formul Mhesh
Convergence rates of approximate sums of Riemann integrals
Covergece rtes of pproximte sums of Riem itegrls Hiroyuki Tski Grdute School of Pure d Applied Sciece, Uiversity of Tsuku Tsuku Irki 5-857 Jp tski@mth.tsuku.c.jp Keywords : covergece rte; Riem sum; Riem
UNIT #5 SEQUENCES AND SERIES COMMON CORE ALGEBRA II
Awer Key Nme: Dte: UNIT # SEQUENCES AND SERIES COMMON CORE ALGEBRA II Prt I Quetio. For equece defied by f? () () 08 6 6 f d f f, which of the followig i the vlue of f f f f f f 0 6 6 08 (). I the viul
Motions of Infinite Mass-Spring Systems
Motio of Ifiite M-Sprig Syte KHIEM V. GO Deprtet of Aeroutic d Atroutic Stford Uiverity Stford, CA 9435-435 UITED STATES OF AMERICA Atrct: - I the tudy of phyicl, echicl, d electricl yte oe ofte ecouter
Name of the Student:
Egieerig Mthemtics 5 NAME OF THE SUBJECT : Mthemtics I SUBJECT CODE : MA65 MATERIAL NAME : Additiol Prolems MATERIAL CODE : HGAUM REGULATION : R UPDATED ON : M-Jue 5 (Sc the ove QR code for the direct
Chapter #5 EEE Control Systems
Sprig EEE Chpter #5 EEE Cotrol Sytem Deig Bed o Root Locu Chpter / Sprig EEE Deig Bed Root Locu Led Cotrol (equivlet to PD cotrol) Ued whe the tedy tte propertie of the ytem re ok but there i poor performce,
Last time: Completed solution to the optimum linear filter in real-time operation
6.3 tochatic Etimatio ad Cotrol, Fall 4 ecture at time: Completed olutio to the oimum liear filter i real-time operatio emi-free cofiguratio: t D( p) F( p) i( p) dte dp e π F( ) F( ) ( ) F( p) ( p) 4444443
Numerical Methods in Geophysics: High-Order Operators
: High-Order Opertors Why do we eed higher order opertors? Tylor Opertors Oe-sided opertors High-order Tylor Extrpoltio Ruge-Kutt Method Truted Fourier Opertors - Derivtive d Iterpoltio Aury o high-order
Section 2.2. Matrix Multiplication
Mtri Alger Mtri Multiplitio Setio.. Mtri Multiplitio Mtri multiplitio is little more omplite th mtri itio or slr multiplitio. If A is the prout A of A is the ompute s follow: m mtri, the is k mtri, 9 m
Asymptotic Properties of Solutions of Two Dimensional Neutral Difference Systems
Avilble t http://pvmuedu/m Appl Appl Mth ISSN: 192-9466 Vol 8, Iue 2 (December 21), pp 585 595 Applictio d Applied Mthemtic: A Itertiol Jourl (AAM) Aymptotic Propertie of Solutio of Two Dimeiol Neutrl
Area, Volume, Rotations, Newton s Method
Are, Volume, Rottio, Newto Method Are etwee curve d the i A ( ) d Are etwee curve d the y i A ( y) yd yc Are etwee curve A ( ) g( ) d where ( ) i the "top" d g( ) i the "ottom" yd Are etwee curve A ( y)
Hypergeometric Functions and Lucas Numbers
IOSR Jourl of Mthetis (IOSR-JM) ISSN: 78-78. Volue Issue (Sep-Ot. ) PP - Hypergeoetri utios d us Nuers P. Rjhow At Kur Bor Deprtet of Mthetis Guhti Uiversity Guwhti-78Idi Astrt: The i purpose of this pper
Note 7 Root-Locus Techniques
Lecture Note of Cotrol Syte I - ME 43/Alyi d Sythei of Lier Cotrol Syte - ME862 Note 7 Root-Locu Techique Deprtet of Mechicl Egieerig, Uiverity Of Sktchew, 57 Cpu Drive, Sktoo, S S7N 5A9, Cd Lecture Note
FREE Download Study Package from website: &
FREE Dolod Study Pkge from esite:.tekolsses.om &.MthsBySuhg.om Get Solutio of These Pkges & Ler y Video Tutorils o.mthsbysuhg.om SHORT REVISION. Defiitio : Retgulr rry of m umers. Ulike determits it hs
( ) 2 3 ( ) I. Order of operations II. Scientific Notation. Simplify. Write answers in scientific notation. III.
Assessmet Ceter Elemetry Alger Study Guide for the ACCUPLACER (CPT) The followig smple questios re similr to the formt d otet of questios o the Aupler Elemetry Alger test. Reviewig these smples will give
Chapter 2. LOGARITHMS
Chpter. LOGARITHMS Dte: - 009 A. INTRODUCTION At the lst hpter, you hve studied bout Idies d Surds. Now you re omig to Logrithms. Logrithm is ivers of idies form. So Logrithms, Idies, d Surds hve strog
Second Mean Value Theorem for Integrals By Ng Tze Beng. The Second Mean Value Theorem for Integrals (SMVT) Statement of the Theorem
Secod Me Vlue Theorem for Itegrls By Ng Tze Beg This rticle is out the Secod Me Vlue Theorem for Itegrls. This theorem, first proved y Hoso i its most geerlity d with extesio y ixo, is very useful d lmost
SOLUTION OF DIFFERENTIAL EQUATION FOR THE EULER-BERNOULLI BEAM
Jourl of Applied Mthemtics d Computtiol Mechics () 57-6 SOUION O DIERENIA EQUAION OR HE EUER-ERNOUI EAM Izbel Zmorsk Istitute of Mthemtics Czestochow Uiversit of echolog Częstochow Pold izbel.zmorsk@im.pcz.pl
The Elementary Arithmetic Operators of Continued Fraction
Americ-Eursi Jourl of Scietific Reserch 0 (5: 5-63, 05 ISSN 88-6785 IDOSI Pulictios, 05 DOI: 0.589/idosi.ejsr.05.0.5.697 The Elemetry Arithmetic Opertors of Cotiued Frctio S. Mugssi d F. Mistiri Deprtmet
Thomas J. Osler Mathematics Department Rowan University Glassboro NJ Introduction
Ot 0 006 Euler s little summtio formul d speil vlues of te zet futio Toms J Osler temtis Deprtmet Row Uiversity Glssboro J 0608 Osler@rowedu Itrodutio I tis ote we preset elemetry metod of determiig vlues
EXPONENTS AND LOGARITHMS
978--07-6- Mthemtis Stdrd Level for IB Diplom Eerpt EXPONENTS AND LOGARITHMS WHAT YOU NEED TO KNOW The rules of epoets: m = m+ m = m ( m ) = m m m = = () = The reltioship etwee epoets d rithms: = g where
SPH3UW Unit 7.5 Snell s Law Page 1 of Total Internal Reflection occurs when the incoming refraction angle is
SPH3UW Uit 7.5 Sell s Lw Pge 1 of 7 Notes Physis Tool ox Refrtio is the hge i diretio of wve due to hge i its speed. This is most ommoly see whe wve psses from oe medium to other. Idex of refrtio lso lled
BITSAT MATHEMATICS PAPER. If log 0.0( ) log 0.( ) the elogs to the itervl (, ] () (, ] [,+ ). The poit of itersectio of the lie joiig the poits i j k d i+ j+ k with the ple through the poits i+ j k, i
ELIMINATION OF FINITE EIGENVALUES OF STRONGLY SINGULAR SYSTEMS BY FEEDBACKS IN LINEAR SYSTEMS
73 M>D Tadeuz azore Waraw Uiverity of Tehology, Faulty of Eletrial Egieerig Ititute of Cotrol ad Idutrial Eletroi EIMINATION OF FINITE EIENVAUES OF STONY SINUA SYSTEMS BY FEEDBACS IN INEA SYSTEMS Tadeuz
On The Homogeneous Quintic Equation with Five Unknowns
IOSR Jourl of Mthemtics (IOSR-JM) e-issn: 78-78,p-ISSN: 319-76X, Volume 7, Issue 3 (Jul. - Aug. 013), PP 7-76 www.iosrjourls.org O The Homogeeous Quitic Equtio with Five Ukows y y 3 3 ( y ) 3(( y)( z w
Approximate Integration
Study Sheet (7.7) Approimte Itegrtio I this sectio, we will ler: How to fid pproimte vlues of defiite itegrls. There re two situtios i which it is impossile to fid the ect vlue of defiite itegrl. Situtio:
Ch. 12 Linear Bayesian Estimators
h. Lier Byesi stimtors Itrodutio I hpter we sw: the MMS estimtor tkes simple form whe d re joitly Gussi it is lier d used oly the st d d order momets (mes d ovries). Without the Gussi ssumptio, the Geerl
Fuzzy and interval finite element method for heat conduction problem
Fuzz itervl fiite elemet metho for het outio prolem Srgm Mjumr Sut N S. Chrvert Deprtmet of Mthemtis Ntiol Istitute of eholog Rourel Oish -769008 Ii Astrt ritiol fiite elemet metho is well-estlishe metho
the midpoint of the ith subinterval, and n is even for
Mth4 Project I (TI 89) Nme: Riem Sums d Defiite Itegrls The re uder the grph of positive fuctio is give y the defiite itegrl of the fuctio. The defiite itegrl c e pproimted y the followig sums: Left Riem
State space systems analysis
State pace ytem aalyi Repreetatio of a ytem i tate-pace (tate-pace model of a ytem To itroduce the tate pace formalim let u tart with a eample i which the ytem i dicuio i a imple electrical circuit with
lecture 16: Introduction to Least Squares Approximation
97 lecture 16: Itroductio to Lest Squres Approximtio.4 Lest squres pproximtio The miimx criterio is ituitive objective for pproximtig fuctio. However, i my cses it is more ppelig (for both computtio d
Numerical Solution of Fuzzy Fredholm Integral Equations of the Second Kind using Bernstein Polynomials
Jourl of Al-Nhri Uiversity Vol.5 (), Mrch,, pp.-9 Sciece Numericl Solutio of Fuzzy Fredholm Itegrl Equtios of the Secod Kid usig Berstei Polyomils Srmd A. Altie Deprtmet of Computer Egieerig d Iformtio
Transverse Electromagnetic Modes of Menger Sponge
Itertiol Jourl of Iformtio d Eletrois Egieerig, Vol. 5, No. 4, Jul 5 Trsverse Eletromgeti Modes of Meger Spoge M. J. Mughl, M. Omr, Sfiullh Kh, d A. Noor Astrt I this rtile, ltil solutio for the eletromgeti
Some Properties of Analytic Functions Defined. by a Linear Operator
It Jourl o Mth Alsis Vol 6 o 545-55 Some Proerties o Alti Futios Deied b ier Oertor S R Swm Dertmet o omuter Siee d ieeri R V ollee o ieeri Msore Rod Blore -56 59 Idi miltoswm@redimilom ABSTRAT The im
First assignment of MP-206
irt igmet of MP- er to quetio - 7- Norml tre log { : MP Priipl tree: I MP II MP III MP Priipl iretio: { I { II { III Iitill uppoe tht i tre tte eribe i the referee tem ' i the me tre tte but eribe i other
Limit of a function:
- Limit of fuctio: We sy tht f ( ) eists d is equl with (rel) umer L if f( ) gets s close s we wt to L if is close eough to (This defiitio c e geerlized for L y syig tht f( ) ecomes s lrge (or s lrge egtive
ECE 102 Engineering Computation
ECE Egieerig Computtio Phillip Wog Mth Review Vetor Bsis Mtri Bsis System of Lier Equtios Summtio Symol is the symol for summtio. Emple: N k N... 9 k k k k k the, If e e e f e f k Vetor Bsis A vetor is
Math1242 Project I (TI 84) Name:
Mth4 Project I (TI 84) Nme: Riem Sums d Defiite Itegrls The re uder the grph of positive fuctio is give y the defiite itegrl of the fuctio. The defiite itegrl c e pproimted y the followig sums: Left Riem
A New Three-Frequency, Geometry-Free, Technique for Ambiguity Resolution
A New Three-Frequey, Geometry-Free, Tehique or Amiguity Reolutio Rold R. Hth, NvCom Tehology, I. BIOGRAPHY Rold R. Hth w oe o the ouder o NvCom Tehology, Joh Deere Compy, d i urretly the Diretor o Nvigtio
CHAPTER 3 NETWORK ADMITTANCE AND IMPEDANCE MATRICES
CHAPTER NETWORK ADTTANCE AND PEDANCE ATRCES As we hve see i Chter tht ower system etwor c e coverted ito equivlet imedce digrm. This digrm forms the sis of ower flow (or lod flow) studies d short circuit
Vector Integration. Line integral: Let F ( x y,
Vetor Integrtion Thi hpter tret integrtion in vetor field. It i the mthemti tht engineer nd phiit ue to deribe fluid flow, deign underwter trnmiion ble, eplin the flow of het in tr, nd put tellite in orbit.
B. Examples 1. Finite Sums finite sums are an example of Riemann Sums in which each subinterval has the same length and the same x i
Mth 06 Clculus Sec. 5.: The Defiite Itegrl I. Riem Sums A. Def : Give y=f(x):. Let f e defied o closed itervl[,].. Prtitio [,] ito suitervls[x (i-),x i ] of legth Δx i = x i -x (i-). Let P deote the prtitio
AN DIGITAL FILTER DESIGN SCHEME BASED ON MATRIX TRANSFORMATION
Jourl of heoretil d Applied Iformtio ehology 5 th November. Vol. 5 No. 5 - JAI & LLS. All rights reserved. ISSN: 99-865 www.jtit.org E-ISSN: 87-95 AN DIGIAL FILER DESIGN SCHEME BASED ON MARIX RANSFORMAION
VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS Vector Integration
www.boopr.om VTU NOTES QUESTION PAPERS NEWS RESULTS FORUMS Vetor Integrtion Thi hpter tret integrtion in vetor field. It i the mthemti tht engineer nd phiit ue to deribe fluid flow, deign underwter trnmiion
20.2 Normal and Critical Slopes
Hdraulis Prof. B.. Thadaveswara Rao. Normal ad Critial lopes Whe disharge ad roughess are give, the Maig formula a e used for determiig the slope of the prismati hael i whih the flow is uiform at a give
Chapter #2 EEE Subsea Control and Communication Systems
EEE 87 Chpter # EEE 87 Sube Cotrol d Commuictio Sytem Trfer fuctio Pole loctio d -ple Time domi chrcteritic Extr pole d zero Chpter /8 EEE 87 Trfer fuctio Lplce Trform Ued oly o LTI ytem Differetil expreio
Discrete Time Sliding Mode Flow Controllers for Connection-Oriented Networks with Lossy Links
BULGARIAN ACADEMY OF SCIENCES CYBERNEICS AND INFORMAION ECHNOLOGIES Volume 3, No Sofi 3 Prit ISSN: 3-97; Olie ISSN: 34-48 DOI:.478/it-3-9 Disrete ime Slidig Mode Flow Cotrollers for Coetio-Orieted Networks
A 4-Step Implicit Collocation Method for Solution of First and Second Order Odes
Itertiol Jourl o Sciece d Tecolog Volume No. November A -Step Implicit Colloctio Metod or Solutio o First d Secod Order Odes Z A. Adegboe I. Aliu Deprtmet o Mts/Stts/Computer Sciece Kdu Poltecic Kdu.Nigeri.
Tranformations. Some slides adapted from Octavia Camps
Trformtio Some lide dpted from Octvi Cmp A m 3 3 m m 3m m Mtrice 5K C c m ij A ij m b ij B A d B mut hve the me dimeio m 3 Mtrice p m m p B A C H @K?J H @K?J m k kj ik ij b c A d B mut hve A d B mut hve
STRUNET CONCRETE DESIGN AIDS
Itrodutio to Corete Colum Deig Flow Chrt he Colum Deig Setio i Struet oti two mi prt: Chrt to develop tregth itertio digrm for give etio, d red -mde Colum Itertio Digrm, for quik deig of give olum. Corete
Kantorovich-Vlasov Method for the Flexural Analysis of Kirchhoff Plates with Opposite Edges Clamped, and Simply Supported (CSCS Plates)
ISSN (Prit) : 319-8613 ISSN (Olie) : 97- Oh H.N. et l. / Itertiol Jourl of Egieerig d Tehology (IJET) Ktorovih-Vlsov Method for the Flexurl Alysis of Kirhhoff Pltes with Opposite Edges Clmped, d Simply
Mathematical Notation Math Calculus & Analytic Geometry I
Mthemticl Nottio Mth - Clculus & Alytic Geometry I Nme : Use Wor or WorPerect to recrete the ollowig ocumets. Ech rticle is worth poits c e prite give to the istructor or emile to the istructor t jmes@richl.eu.
THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING
OLLSCOIL NA héireann, CORCAIGH THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING MATHEMATICS MA008 Clculus d Lier
Crushed Notes on MATH132: Calculus
Mth 13, Fll 011 Siyg Yg s Outlie Crushed Notes o MATH13: Clculus The otes elow re crushed d my ot e ect This is oly my ow cocise overview of the clss mterils The otes I put elow should ot e used to justify
Topic 4 Fourier Series. Today
Topic 4 Fourier Series Toy Wves with repetig uctios Sigl geertor Clssicl guitr Pio Ech istrumet is plyig sigle ote mile C 6Hz) st hrmoic hrmoic 3 r hrmoic 4 th hrmoic 6Hz 5Hz 783Hz 44Hz A sigle ote will
CHAPTER 7 SYMMETRICAL COMPONENTS AND REPRESENTATION OF FAULTED NETWORKS
HAPTER 7 SMMETRAL OMPOETS AD REPRESETATO OF FAULTED ETWORKS A uled three-phe yte e reolved ito three led yte i the iuoidl tedy tte. Thi ethod of reolvig uled yte ito three led phor yte h ee propoed y.
a f(x)dx is divergent.
Mth 250 Exm 2 review. Thursdy Mrh 5. Brig TI 30 lultor but NO NOTES. Emphsis o setios 5.5, 6., 6.2, 6.3, 3.7, 6.6, 8., 8.2, 8.3, prt of 8.4; HW- 2; Q-. Kow for trig futios tht 0.707 2/2 d 0.866 3/2. From
=> PARALLEL INTERCONNECTION. Basic Properties LTI Systems. The Commutative Property. Convolution. The Commutative Property. The Distributive Property
Lier Time-Ivrit Bsic Properties LTI The Commuttive Property The Distributive Property The Associtive Property Ti -6.4 / Chpter Covolutio y ] x ] ] x ]* ] x ] ] y] y ( t ) + x( τ ) h( t τ ) dτ x( t) * h(
INTEGRAL SOLUTIONS OF THE TERNARY CUBIC EQUATION
Itertiol Reserch Jourl of Egieerig d Techology IRJET) e-issn: 9-006 Volume: 04 Issue: Mr -017 www.irjet.et p-issn: 9-007 INTEGRL SOLUTIONS OF THE TERNRY CUBIC EQUTION y ) 4y y ) 97z G.Jki 1, C.Sry,* ssistt
International Journal of Mathematical Archive-6(10), 2015, Available online through ISSN
Itertiol Jourl of Mthemtil Arhive-(0 05 99-04 Aville olie through www.m.ifo ISSN 9 504 A FUZZY APPOACH FO SOLVING MIXED INTUITIONISTIC FUZZY TAVELLING SALESMAN POBLEM. N. JAT S. C. SHAMA SANJAY JAIN* ANCHAL
International Journal of Scientific & Engineering Research, Volume 6, Issue 12, December ISSN
Itertiol Jourl o Scietiic & Egieerig Reserc, Volume 6, Issue, December-0 0 ISSN -8 A Clss o New Bloc Geerlized Adms Implicit Ruge-Kutt Colloctio Metods Kumleg G.M., Collom J. P, Omgwu S. Abstrct I tis
Dynamics of Structures
UNION Dymis of Strutures Prt Zbigiew Wójii Je Grosel Projet o-fied by Europe Uio withi Europe Soil Fud UNION Mtries Defiitio of mtri mtri is set of umbers or lgebri epressios rrged i retgulr form with
King Fahd University of Petroleum & Minerals
Kig Fhd Uiversity of Petroleum & Mierls DEPARTMENT OF MATHEMATICAL CIENCE Techicl Report eries TR 434 April 04 A Direct Proof of the Joit Momet Geertig Fuctio of mple Me d Vrice Awr H. Jorder d A. Lrdji
GRAPHING LINEAR EQUATIONS. Linear Equations. x l ( 3,1 ) _x-axis. Origin ( 0, 0 ) Slope = change in y change in x. Equation for l 1.
GRAPHING LINEAR EQUATIONS Qudrt II Qudrt I ORDERED PAIR: The first umer i the ordered pir is the -coordite d the secod umer i the ordered pir is the y-coordite. (, ) Origi ( 0, 0 ) _-is Lier Equtios Qudrt