The Application of hybrid BEM/FEM methods to solve Electrical Impedance Tomography s forward problem for the human head
|
|
- Miles Dickerson
- 6 years ago
- Views:
Transcription
1 The Applcato of hybrd / methods to solve Electrcal Impedace Tomography s forward problem for the huma head S.R. Arrdge, R.H. ayford, L. Horesh ad J. Skora March, Abstract The forward problem Electrcal Impedace Tomography (EIT) requres a accurate estmato of ts soluto. Although aalytcal solutos est for a lmted umber of cases, there s o sutable aalytcal soluto for the geometry of the huma head. A commoly adopted method s the use of Fte Elemet Methods (). However, the huma head cossts of layers, whch clude the scalp ad CSF, whch are appromately mm thck. Meshg these layers s techcally dffcult ad troduces sgfcat errors to the soluto. A terestg alteratve s the use of hybrd methods combg both ad. The oudary Elemet Method () s a alteratve to. Ths method essetally uses Gree s theorem to map boudary dstrbutos to boudary dstrbutos, usg the eplct form of the Gree s fuctos. Although has the dsadvatage of dese matrces, ad assumes the rego s homogeous, t would be of great advatage for the th layer of the huma such as the scalp ad CSF, whch are homogeous. The larger regos are better suted to, whch are composed of o-uform coductvty dstrbuto. A logcal approach for ths forward problem s to combe the ad to produce hybrd code, whch assumes some rego, have cotact values. We preset results for D spheres ad dscuss the requred modfcatos of order to mprove relatvely hgh error at the, we also show for spheres that wth certa dscretzato soluto error for a mm th layer could be reduced. Itroducto There s a lot of dscusso about the advatages ad dsadvatages of the E method whe compared to the FE oe. Clearly there are certa applcatos where oe techque s more sutable tha the other. ut for electrcal mpedace or optcal tomography problems, combg both techques the same computer program, would be the most effcet way of modellg th layers lke scull or CSF a huma head. I order to take advatage of ad, ther couplg has bee vestgated etesvely several egeerg felds, such as geomechacs [, 3], sold mechacs [], fracture mechacs [] ad electromagetcs [,,,, ] There are several dfferet method of couplg ad [9,, ]. - couplg Ths problem s closely related to the mult rego problem of the E method such as preseted Fg.. The mult rego aalyss has to fulfl cotuty ad equlbrum codtos alog the le betwee Ω ad Ω regos. Ths results the
2 Ω Ω Ω Ω approach we have assume that o the we have addtoal ukow flu epressed by Neuma boudary codtos. Normally to solve FE system the boudary codtos have to be mposed allow us to solve the system of equatos. So ow the FE system of equatos ts matr form could be epressed as follows A (FE) Φ (FE) (FE) Φ(FE) = + F Ω FE () Fg. : The mult rego boudary elemet aalyss (left) ad hybrd dscretzato (rght) followg two relatoshps Φ () = Φ () () Φ () = Φ() Let the sub rego Ω be dscretzed by the Fte Elemets ad the Ω by the oudary Elemets. Alog the commo has tobesatsfedeq.(). Cotuty of the state fucto Φ ca be mataed by usg the same order of bass fuctos both FE ad E formulatos. Thus, f a three ode soparametrc quadratc boudary elemet s used a equvalet fte elemet such as for eample eght ode quadrlateral quadratc elemet or s odes soparametrc tragle has to be used for the fte elemet appromato. The essetalty of the problem les the fact that the terpolato for the dervatves of the potetal for the les oe order lower tha the order of the potetal tself, whereas for the formulato developed here, the terpolato fuctos has the same order ot oly for the potetal but also for ts dervatves. Such uequal terpolato of the ormal dervatves o the mplats a error to the resultg system of equatos. ecause alog the the cotuty ad equlbrum codtos have to be fulflled for the FE where Φ (FE) ad Φ(FE) are colum matrces cotag the odal values for the potetal (electrc potetal or photo desty) ad ts ormal dervatves (electrc curret or curret photo). The correspodg boudary tegral equato for the sub doma s gve by A (E) (E) Φ(E) Φ = + q o E (3) where Φ (E) ad Φ(E) are the odal potetals ad thers ormal dervatve vectors respectvely.. Goverg equatos The boudary value problem for the FE sub rego s defed by the secod order dfferetal equato [7] D(r) Φ(r)+k Φ(r) = () cojucto wth the boudary codtos Φ = Φ o Φ = ψ o () where D(r) s the coductvty (EIT) or dffuso coeffcet (OT), k teds to zero case of Laplace s equato (EIT). The equvalet varatoal problem for the boudary value problem defed above s gve by δf(φ) = ()
3 cojucto wth the boudary codtos Eq.(), form where F (Φ) = [ D Φ + k Φ ] dω+ Φψd (7) Ω (FE) To dscretze the fuctoal Eq.(7), the FE sub rego s dvded to elemets ME wth M odes ad boudary (see Fg. ) s broke to MS segmets wth M umber of odes. Usually, M s much larger tha M. Wth each area elemet, for eample s ode tragle, the feld s epressed as Φ(, y) = N e (, y)φ e = {N e } T {Φ e } = = = {Φ e } T {N e } () ad o each le segmet o the the feld s epressed as Φ (, y) = 3 N (, y)φ = {N } T {Φ } = = = {Φ } T {N } (9) Assumg that the boudary s a smooth cotour, the ormal dervatve, whch s ψ, s well defed at each ode ad therefore ca also be epressed as ψ (, y) = 3 N (, y)ψ = {N } T {ψ } = = = {ψ } T {N } () Ths s a weak pot of ths approach because the Φ ad ts ormal dervatve ψ are appromated by the same shape fuctos. Results of such approach wll be demostrated later. Substtutg Eq.( ) to Eq.(7), we obta F = M e= MS E{Φ e } T [A e ]{Φ e } + {Φ } T [ ]{ψ } () = where case of D space matr [A e ] wll take the ad [A e ] = + [ ]= [ { N }{ } e N e T D + Ω e { }{ } ] N e N e T ddy y y Ω e k {N e }{N e } T ddy () {N }{N } T d (3) Provded that the elemet legth of the s small, the Jacoba of trasformato to local coordate system may be assumed costat ad take out of the tegral sg Eq.(3) wthout causg sgfcat errors. Therefore, by substtutg the eplct epressos for the shape fuctos, t s easy to perform the dcated tegratos aalytcally. So the etres of matr [ ] case of the quadrlateral three odes soparametrc elemets of local coordate system are defed by [ ] = = N N NN NN 3 N N N N N N 3 J(ξ) = N3 N N3 N N3 N 3 J(ξ) () Tha, performg the assembly Eq.() ca be wrtte as F = {Φ}T [ A (FE)] {Φ} + {Φ} T [ (FE)] {ψ} () where A (FE) s a M M square matr, (FE) s a M M rectagular matr, Φ s a colum vector represetg the odal values of electrc potetal or photo testy ad ψ s a colum vector represetg the odal values of Φ o the M odes of the. Dfferetatg F wth respect to each 3
4 odal value of Φ ad equatg the resultg epresso to zero yelds a system of lear equatos A (FE) Φ (FE) (FE) Φ(FE) + = () As a result we wll get the matr wth the followg structure (see Fg. ). l u Fg. : The matr structure (left) ad a eemplary mage for the dscretzato show Fg. 3 Matr form of the E sub rego ca be wrtte as A (E) Φ (E) (E) Φ(E) = q (7) Now the system of boudary equatos could be corporated to the system of fte elemets. A (FE) (FE) A (FE) (FE) A (E) (E) A (E) (E) Φ (FE) Φ (FE) Φ (E) Φ (E) = = q q Note that the superscrpts (E) ad (FE) dcate to whch subrego partcular matr s prescrbed. The matr s usymmetrcal wth much bgger badwdth wth two addtoal group of o zero elemets caused by the betwee FE ad E sub domas. Rearragemet of resultg matr s very smlar to that requred mult rego problems. The smplest approach to solvg Eq.(7) ad Eq.() s to solve them smultaeously, that s, to solve a (M +3M +M ) (M +3M +M ) matr system, where M s the umber of ukows o the boudary Fg.. As prevous secto, the matrces wll eed some rearragemet to accommodate the codtos Eq.() ad stadard boudary codtos. 3 Numercal eamples Let us cosder several eamples D space, startg from the smplest oe ad tha move to the stadard EIT or optcal tomography bechmark. 3. echmark problem (?) As a bechmark problem let us cosder the steady state for a dffuso equato a square rego (see Fg. 3) Φ() μ a D Φ() = Φ() k Φ() = () where k s so called a wave umber ad for the EIT k become zero. The followg boudary codtos are mposed: Drchlet boudary codtos Φ() = = Φ() =a = (9) ad Neuma boudary codtos Φ() = Φ() = y = Φ() = y = () =a Ths problem, due to the geometry ad boudary codtos s oe dmesoal fact, but t would be solved D space ad results wll compare wth the aalytcal soluto whch s Φ() = ( Φ(a) D D e k(a ) J(a) k e k(a ) + + Φ() D e k + J() ) k e k () where for Optcal Tomography, usually dffuso coeffcet D =.3cm ad μ a =.cm so k =
5 μa /D.7cm,ad J() = Φ() = kd (cosh(kδ)φ() Φ(a)) sh(kδ) J(a) = kdsh(kδ)φ() + () + cosh(kδ)kd (cosh(kδ)φ() Φ(a)) sh(kδ) I order to acheve the followg results wth the ad of, rego has to be dscretzed by elemets wth 33 odes (badwdth equal 3) ad wth the ad of, rego has to be dscretszed by elemets ad odes. Results of calculatos are preseted Fg. 3. We ca observe how the error crease o the le. A A couplg Aalytc soluto couplg 3 Aalytc soluto relatve error [%] Fg. : Dscretzato ad soluto alog the le A A for (left) ad alog the le (rght) for dese mesh relatve error [%] 9 couplg Aalytc soluto relatve error [%] relatve error [%] Fg. 3: Dscretzato ad soluto for the coarse mesh (left) ad dese (rght) for the hybrd soluto 3. Two cocetrc squares Now let us cosder slghtly more advaced eample whe the FE rego s mmersed the E rego as t s show Fg.. Ifluece of the gap dmeso o the error soluto s show Fg.??. What s terestg ad a very promsg feature of the hybrd approach, s the fact that the error of the soluto for such cofgurato as Fg. s ot sestve o dscretzato. Cocluso???? Fg. : Dscretzato ad soluto alog the le for dese mesh ad the small gap betwee ad sub regos Refereces [] M. H. Alabad. The oudary Elemet Method; Volume ; Applcatos Solds ad Structures. Joh Wley & Sos, LTD,. [] G. eer. Programmg the oudary Elemet Method. A Itroducto for Egeers. Joh Wley & Sos,. [3] G. eer ad J.O. Watso. Itroducto to Fte ad oudary Elemet Methods for Egeers. Joh Wley & Sos, 99. [] C.P. radley, G.M. Harrs, ad A.J. Pulla. The computatoal performace of a hgh order coupled fem/bem procedure elektropotetal
6 problems. IEEE Trasactos o omedcal Egeerg, ():3, November. [] M.V.K. Char ad S.J. Salo. Numercal Methods Electromagetsm. Academc Press,. [] I. Guve ad E. Madec. Traset heat coductg aalyss a pecewse homogeeous doma by a coupled boudary ad fte elemet method. It. Jour. for Numercal Meth. Egeerg, :3 3, 3. [7] Jamg J. The Fte Elemet Method Electromagetcs. Joh Wley & Sos, 993. [] S. Kurz ad S. Russeschuck. Accurate calculato of magetc felds the ed regos of supercoductg accelerator magets usg the bem fem couplg method, 999. Proceedgs of the 999 Partcle Accelerator Coferece, New York. [9] O.K. Paagoul ad P.D. Paagotopulos. The fem ad bem for fractal boudares ad s. applcatos to ulateral problems. Computers ad Structures, ( ):39 339, 997. [] M. Trlep, L. Skerget,. Kreča, ad. Hrberk. Hybrd fte elemet boudary elemet method for olear electromagetc problems. IEEE Trasactos o Magetcs, 3(3):3 33, May 99. [] O.C. Zekewcz, D.W. Kelly, ad P. ettess. The couplg of the fte elemets method ad boudary soluto procedures. It. Jour. for Numercal Meth. Egeerg, pages 3 37, 977.
Functions of Random Variables
Fuctos of Radom Varables Chapter Fve Fuctos of Radom Varables 5. Itroducto A geeral egeerg aalyss model s show Fg. 5.. The model output (respose) cotas the performaces of a system or product, such as weght,
More informationPGE 310: Formulation and Solution in Geosystems Engineering. Dr. Balhoff. Interpolation
PGE 30: Formulato ad Soluto Geosystems Egeerg Dr. Balhoff Iterpolato Numercal Methods wth MATLAB, Recktewald, Chapter 0 ad Numercal Methods for Egeers, Chapra ad Caale, 5 th Ed., Part Fve, Chapter 8 ad
More informationCubic Nonpolynomial Spline Approach to the Solution of a Second Order Two-Point Boundary Value Problem
Joural of Amerca Scece ;6( Cubc Nopolyomal Sple Approach to the Soluto of a Secod Order Two-Pot Boudary Value Problem W.K. Zahra, F.A. Abd El-Salam, A.A. El-Sabbagh ad Z.A. ZAk * Departmet of Egeerg athematcs
More informationC-1: Aerodynamics of Airfoils 1 C-2: Aerodynamics of Airfoils 2 C-3: Panel Methods C-4: Thin Airfoil Theory
ROAD MAP... AE301 Aerodyamcs I UNIT C: 2-D Arfols C-1: Aerodyamcs of Arfols 1 C-2: Aerodyamcs of Arfols 2 C-3: Pael Methods C-4: Th Arfol Theory AE301 Aerodyamcs I Ut C-3: Lst of Subects Problem Solutos?
More informationLecture 12 APPROXIMATION OF FIRST ORDER DERIVATIVES
FDM: Appromato of Frst Order Dervatves Lecture APPROXIMATION OF FIRST ORDER DERIVATIVES. INTRODUCTION Covectve term coservato equatos volve frst order dervatves. The smplest possble approach for dscretzato
More informationESS Line Fitting
ESS 5 014 17. Le Fttg A very commo problem data aalyss s lookg for relatoshpetwee dfferet parameters ad fttg les or surfaces to data. The smplest example s fttg a straght le ad we wll dscuss that here
More informationBeam Warming Second-Order Upwind Method
Beam Warmg Secod-Order Upwd Method Petr Valeta Jauary 6, 015 Ths documet s a part of the assessmet work for the subject 1DRP Dfferetal Equatos o Computer lectured o FNSPE CTU Prague. Abstract Ths documet
More informationECE 595, Section 10 Numerical Simulations Lecture 19: FEM for Electronic Transport. Prof. Peter Bermel February 22, 2013
ECE 595, Secto 0 Numercal Smulatos Lecture 9: FEM for Electroc Trasport Prof. Peter Bermel February, 03 Outle Recap from Wedesday Physcs-based devce modelg Electroc trasport theory FEM electroc trasport
More informationAnalysis of Lagrange Interpolation Formula
P IJISET - Iteratoal Joural of Iovatve Scece, Egeerg & Techology, Vol. Issue, December 4. www.jset.com ISS 348 7968 Aalyss of Lagrage Iterpolato Formula Vjay Dahya PDepartmet of MathematcsMaharaja Surajmal
More informationChapter 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
More informationFractional Order Finite Difference Scheme For Soil Moisture Diffusion Equation And Its Applications
IOS Joural of Mathematcs (IOS-JM e-iss: 78-578. Volume 5, Issue 4 (Ja. - Feb. 3, PP -8 www.osrourals.org Fractoal Order Fte Dfferece Scheme For Sol Mosture Dffuso quato Ad Its Applcatos S.M.Jogdad, K.C.Takale,
More informationIntroduction to local (nonparametric) density estimation. methods
Itroducto to local (oparametrc) desty estmato methods A slecture by Yu Lu for ECE 66 Sprg 014 1. Itroducto Ths slecture troduces two local desty estmato methods whch are Parze desty estmato ad k-earest
More informationCS5620 Intro to Computer Graphics
CS56 Itro to Computer Graphcs Geometrc Modelg art II Geometrc Modelg II hyscal Sples Curve desg pre-computers Cubc Sples Stadard sple put set of pots { } =, No dervatves specfed as put Iterpolate by cubc
More informationBayesian Inferences for Two Parameter Weibull Distribution Kipkoech W. Cheruiyot 1, Abel Ouko 2, Emily Kirimi 3
IOSR Joural of Mathematcs IOSR-JM e-issn: 78-578, p-issn: 9-765X. Volume, Issue Ver. II Ja - Feb. 05, PP 4- www.osrjourals.org Bayesa Ifereces for Two Parameter Webull Dstrbuto Kpkoech W. Cheruyot, Abel
More informationChapter 4 (Part 1): Non-Parametric Classification (Sections ) Pattern Classification 4.3) Announcements
Aoucemets No-Parametrc Desty Estmato Techques HW assged Most of ths lecture was o the blacboard. These sldes cover the same materal as preseted DHS Bometrcs CSE 90-a Lecture 7 CSE90a Fall 06 CSE90a Fall
More informationComparison of Analytical and Numerical Results in Modal Analysis of Multispan Continuous Beams with LS-DYNA
th Iteratoal S-N Users oferece Smulato Techology omparso of alytcal ad Numercal Results Modal alyss of Multspa otuous eams wth S-N bht Mahapatra ad vk hatteree etral Mechacal Egeerg Research Isttute, urgapur
More informationKernel-based Methods and Support Vector Machines
Kerel-based Methods ad Support Vector Maches Larr Holder CptS 570 Mache Learg School of Electrcal Egeerg ad Computer Scece Washgto State Uverst Refereces Muller et al. A Itroducto to Kerel-Based Learg
More informationQuantization in Dynamic Smarandache Multi-Space
Quatzato Dyamc Smaradache Mult-Space Fu Yuhua Cha Offshore Ol Research Ceter, Beg, 7, Cha (E-mal: fuyh@cooc.com.c ) Abstract: Dscussg the applcatos of Dyamc Smaradache Mult-Space (DSMS) Theory. Supposg
More informationECE606: Solid State Devices Lecture 13 Solutions of the Continuity Eqs. Analytical & Numerical
ECE66: Sold State Devces Lecture 13 Solutos of the Cotuty Eqs. Aalytcal & Numercal Gerhard Klmeck gekco@purdue.edu Outle Aalytcal Solutos to the Cotuty Equatos 1) Example problems ) Summary Numercal Solutos
More informationThe solution of Euler-Bernoulli beams using variational derivative method
Scetfc Research ad Essays Vol. 5(9), pp. 9-4, 4 May Avalable ole at http://www.academcjourals.org/sre ISSN 99-48 Academc Jourals Full egth Research Paper The soluto of Euler-Beroull beams usg varatoal
More informationMOLECULAR VIBRATIONS
MOLECULAR VIBRATIONS Here we wsh to vestgate molecular vbratos ad draw a smlarty betwee the theory of molecular vbratos ad Hückel theory. 1. Smple Harmoc Oscllator Recall that the eergy of a oe-dmesoal
More information1 Lyapunov Stability Theory
Lyapuov Stablty heory I ths secto we cosder proofs of stablty of equlbra of autoomous systems. hs s stadard theory for olear systems, ad oe of the most mportat tools the aalyss of olear systems. It may
More informationIntegral Equation Methods. Jacob White. Thanks to Deepak Ramaswamy, Michal Rewienski, Xin Wang and Karen Veroy
Itroducto to Smulato - Lecture 22 Itegral Equato ethods Jacob Whte Thaks to Deepak Ramaswamy, chal Rewesk, X Wag ad Kare Veroy Outle Itegral Equato ethods Exteror versus teror problems Start wth usg pot
More informationSimple Linear Regression
Statstcal Methods I (EST 75) Page 139 Smple Lear Regresso Smple regresso applcatos are used to ft a model descrbg a lear relatoshp betwee two varables. The aspects of least squares regresso ad correlato
More informationL5 Polynomial / Spline Curves
L5 Polyomal / Sple Curves Cotets Coc sectos Polyomal Curves Hermte Curves Bezer Curves B-Sples No-Uform Ratoal B-Sples (NURBS) Mapulato ad Represetato of Curves Types of Curve Equatos Implct: Descrbe a
More informationBAYESIAN INFERENCES FOR TWO PARAMETER WEIBULL DISTRIBUTION
Iteratoal Joural of Mathematcs ad Statstcs Studes Vol.4, No.3, pp.5-39, Jue 06 Publshed by Europea Cetre for Research Trag ad Developmet UK (www.eajourals.org BAYESIAN INFERENCES FOR TWO PARAMETER WEIBULL
More informationTransforms that are commonly used are separable
Trasforms s Trasforms that are commoly used are separable Eamples: Two-dmesoal DFT DCT DST adamard We ca the use -D trasforms computg the D separable trasforms: Take -D trasform of the rows > rows ( )
More informationPart 4b Asymptotic Results for MRR2 using PRESS. Recall that the PRESS statistic is a special type of cross validation procedure (see Allen (1971))
art 4b Asymptotc Results for MRR usg RESS Recall that the RESS statstc s a specal type of cross valdato procedure (see Alle (97)) partcular to the regresso problem ad volves fdg Y $,, the estmate at the
More informationNewton s Power Flow algorithm
Power Egeerg - Egll Beedt Hresso ewto s Power Flow algorthm Power Egeerg - Egll Beedt Hresso The ewto s Method of Power Flow 2 Calculatos. For the referece bus #, we set : V = p.u. ad δ = 0 For all other
More informationFourth Order Four-Stage Diagonally Implicit Runge-Kutta Method for Linear Ordinary Differential Equations ABSTRACT INTRODUCTION
Malasa Joural of Mathematcal Sceces (): 95-05 (00) Fourth Order Four-Stage Dagoall Implct Ruge-Kutta Method for Lear Ordar Dfferetal Equatos Nur Izzat Che Jawas, Fudzah Ismal, Mohamed Sulema, 3 Azm Jaafar
More informationBlock-Based Compact Thermal Modeling of Semiconductor Integrated Circuits
Block-Based Compact hermal Modelg of Semcoductor Itegrated Crcuts Master s hess Defese Caddate: Jg Ba Commttee Members: Dr. Mg-Cheg Cheg Dr. Daqg Hou Dr. Robert Schllg July 27, 2009 Outle Itroducto Backgroud
More informationFREQUENCY ANALYSIS OF A DOUBLE-WALLED NANOTUBES SYSTEM
Joural of Appled Matematcs ad Computatoal Mecacs 04, 3(4), 7-34 FREQUENCY ANALYSIS OF A DOUBLE-WALLED NANOTUBES SYSTEM Ata Cekot, Stasław Kukla Isttute of Matematcs, Czestocowa Uversty of Tecology Częstocowa,
More informationDYNAMIC ANALYSIS OF CONCRETE RECTANGULAR LIQUID STORAGE TANKS
The 4 th World Coferece o Earthquake Egeerg October 2-7, 28, Bejg, Cha DYNAMIC ANAYSIS OF CONCRETE RECTANGUAR IQUID STORAGE TANKS J.Z. Che, A.R. Ghaemmagham 2 ad M.R. Kaoush 3 Structural Egeer, C2M I Caada,
More informationx y exp λ'. x exp λ 2. x exp 1.
egecosmcd Egevalue-egevector of the secod dervatve operator d /d hs leads to Fourer seres (se, cose, Legedre, Bessel, Chebyshev, etc hs s a eample of a systematc way of geeratg a set of mutually orthogoal
More informationA Collocation Method for Solving Abel s Integral Equations of First and Second Kinds
A Collocato Method for Solvg Abel s Itegral Equatos of Frst ad Secod Kds Abbas Saadatmad a ad Mehd Dehgha b a Departmet of Mathematcs, Uversty of Kasha, Kasha, Ira b Departmet of Appled Mathematcs, Faculty
More informationEstimation of Stress- Strength Reliability model using finite mixture of exponential distributions
Iteratoal Joural of Computatoal Egeerg Research Vol, 0 Issue, Estmato of Stress- Stregth Relablty model usg fte mxture of expoetal dstrbutos K.Sadhya, T.S.Umamaheswar Departmet of Mathematcs, Lal Bhadur
More informationChapter 9 Jordan Block Matrices
Chapter 9 Jorda Block atrces I ths chapter we wll solve the followg problem. Gve a lear operator T fd a bass R of F such that the matrx R (T) s as smple as possble. f course smple s a matter of taste.
More informationA Conventional Approach for the Solution of the Fifth Order Boundary Value Problems Using Sixth Degree Spline Functions
Appled Matheatcs, 1, 4, 8-88 http://d.do.org/1.4/a.1.448 Publshed Ole Aprl 1 (http://www.scrp.org/joural/a) A Covetoal Approach for the Soluto of the Ffth Order Boudary Value Probles Usg Sth Degree Sple
More informationBERNSTEIN COLLOCATION METHOD FOR SOLVING NONLINEAR DIFFERENTIAL EQUATIONS. Aysegul Akyuz Dascioglu and Nese Isler
Mathematcal ad Computatoal Applcatos, Vol. 8, No. 3, pp. 293-300, 203 BERNSTEIN COLLOCATION METHOD FOR SOLVING NONLINEAR DIFFERENTIAL EQUATIONS Aysegul Ayuz Dascoglu ad Nese Isler Departmet of Mathematcs,
More informationDIFFERENTIAL GEOMETRIC APPROACH TO HAMILTONIAN MECHANICS
DIFFERENTIAL GEOMETRIC APPROACH TO HAMILTONIAN MECHANICS Course Project: Classcal Mechacs (PHY 40) Suja Dabholkar (Y430) Sul Yeshwath (Y444). Itroducto Hamltoa mechacs s geometry phase space. It deals
More informationModule 1 : The equation of continuity. Lecture 5: Conservation of Mass for each species. & Fick s Law
Module : The equato of cotuty Lecture 5: Coservato of Mass for each speces & Fck s Law NPTEL, IIT Kharagpur, Prof. Sakat Chakraborty, Departmet of Chemcal Egeerg 2 Basc Deftos I Mass Trasfer, we usually
More informationMMJ 1113 FINITE ELEMENT METHOD Introduction to PART I
MMJ FINITE EEMENT METHOD Cotut requremets Assume that the fuctos appearg uder the tegral the elemet equatos cota up to (r) th order To esure covergece N must satsf Compatblt requremet the fuctos must have
More informationEvolution Operators and Boundary Conditions for Propagation and Reflection Methods
voluto Operators ad for Propagato ad Reflecto Methods Davd Yevck Departmet of Physcs Uversty of Waterloo Physcs 5/3/9 Collaborators Frak Schmdt ZIB Tlma Frese ZIB Uversty of Waterloo] atem l-refae Nortel
More information2006 Jamie Trahan, Autar Kaw, Kevin Martin University of South Florida United States of America
SOLUTION OF SYSTEMS OF SIMULTANEOUS LINEAR EQUATIONS Gauss-Sedel Method 006 Jame Traha, Autar Kaw, Kev Mart Uversty of South Florda Uted States of Amerca kaw@eg.usf.edu Itroducto Ths worksheet demostrates
More informationChapter 8. Inferences about More Than Two Population Central Values
Chapter 8. Ifereces about More Tha Two Populato Cetral Values Case tudy: Effect of Tmg of the Treatmet of Port-We tas wth Lasers ) To vestgate whether treatmet at a youg age would yeld better results tha
More informationLecture 3 Probability review (cont d)
STATS 00: Itroducto to Statstcal Iferece Autum 06 Lecture 3 Probablty revew (cot d) 3. Jot dstrbutos If radom varables X,..., X k are depedet, the ther dstrbuto may be specfed by specfyg the dvdual dstrbuto
More informationComparison of Dual to Ratio-Cum-Product Estimators of Population Mean
Research Joural of Mathematcal ad Statstcal Sceces ISS 30 6047 Vol. 1(), 5-1, ovember (013) Res. J. Mathematcal ad Statstcal Sc. Comparso of Dual to Rato-Cum-Product Estmators of Populato Mea Abstract
More informationNumerical Simulations of the Complex Modied Korteweg-de Vries Equation. Thiab R. Taha. The University of Georgia. Abstract
Numercal Smulatos of the Complex Moded Korteweg-de Vres Equato Thab R. Taha Computer Scece Departmet The Uversty of Georga Athes, GA 002 USA Tel 0-542-2911 e-mal thab@cs.uga.edu Abstract I ths paper mplemetatos
More information( ) = ( ) ( ) Chapter 13 Asymptotic Theory and Stochastic Regressors. Stochastic regressors model
Chapter 3 Asmptotc Theor ad Stochastc Regressors The ature of eplaator varable s assumed to be o-stochastc or fed repeated samples a regresso aalss Such a assumpto s approprate for those epermets whch
More informationDKA method for single variable holomorphic functions
DKA method for sgle varable holomorphc fuctos TOSHIAKI ITOH Itegrated Arts ad Natural Sceces The Uversty of Toushma -, Mamhosama, Toushma, 770-8502 JAPAN Abstract: - Durad-Kerer-Aberth (DKA method for
More information5 Short Proofs of Simplified Stirling s Approximation
5 Short Proofs of Smplfed Strlg s Approxmato Ofr Gorodetsky, drtymaths.wordpress.com Jue, 20 0 Itroducto Strlg s approxmato s the followg (somewhat surprsg) approxmato of the factoral,, usg elemetary fuctos:
More informationPoint Estimation: definition of estimators
Pot Estmato: defto of estmators Pot estmator: ay fucto W (X,..., X ) of a data sample. The exercse of pot estmato s to use partcular fuctos of the data order to estmate certa ukow populato parameters.
More informationBayes (Naïve or not) Classifiers: Generative Approach
Logstc regresso Bayes (Naïve or ot) Classfers: Geeratve Approach What do we mea by Geeratve approach: Lear p(y), p(x y) ad the apply bayes rule to compute p(y x) for makg predctos Ths s essetally makg
More informationAnalysis of von Kármán plates using a BEM formulation
Boudary Elemets ad Other Mesh Reducto Methods XXIX 213 Aalyss of vo Kármá plates usg a BEM formulato L. Wademam & W. S. Vetur São Carlos School of Egeerg, Uversty of São Paulo, Brazl Abstract Ths work
More informationOrdinary Least Squares Regression. Simple Regression. Algebra and Assumptions.
Ordary Least Squares egresso. Smple egresso. Algebra ad Assumptos. I ths part of the course we are gog to study a techque for aalysg the lear relatoshp betwee two varables Y ad X. We have pars of observatos
More informationUNIT 2 SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS
Numercal Computg -I UNIT SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS Structure Page Nos..0 Itroducto 6. Objectves 7. Ital Approxmato to a Root 7. Bsecto Method 8.. Error Aalyss 9.4 Regula Fals Method
More informationBoundary Elements and Other Mesh Reduction Methods XXIX 13
Boudary Elemets ad Other Mesh Reducto Methods XXIX 3 No-overlappg doma decomposto scheme for the symmetrc radal bass fucto meshless approach wth double collocato at the sub-doma terfaces H. Power, A. Heradez
More informationDerivation of 3-Point Block Method Formula for Solving First Order Stiff Ordinary Differential Equations
Dervato of -Pot Block Method Formula for Solvg Frst Order Stff Ordary Dfferetal Equatos Kharul Hamd Kharul Auar, Kharl Iskadar Othma, Zara Bb Ibrahm Abstract Dervato of pot block method formula wth costat
More informationLecture 5: Interpolation. Polynomial interpolation Rational approximation
Lecture 5: Iterpolato olyomal terpolato Ratoal appromato Coeffcets of the polyomal Iterpolato: Sometme we kow the values of a fucto f for a fte set of pots. Yet we wat to evaluate f for other values perhaps
More informationEVALUATION OF FUNCTIONAL INTEGRALS BY MEANS OF A SERIES AND THE METHOD OF BOREL TRANSFORM
EVALUATION OF FUNCTIONAL INTEGRALS BY MEANS OF A SERIES AND THE METHOD OF BOREL TRANSFORM Jose Javer Garca Moreta Ph. D research studet at the UPV/EHU (Uversty of Basque coutry) Departmet of Theoretcal
More informationResearch Article A New Derivation and Recursive Algorithm Based on Wronskian Matrix for Vandermonde Inverse Matrix
Mathematcal Problems Egeerg Volume 05 Artcle ID 94757 7 pages http://ddoorg/055/05/94757 Research Artcle A New Dervato ad Recursve Algorthm Based o Wroska Matr for Vadermode Iverse Matr Qu Zhou Xja Zhag
More information1 0, x? x x. 1 Root finding. 1.1 Introduction. Solve[x^2-1 0,x] {{x -1},{x 1}} Plot[x^2-1,{x,-2,2}] 3
Adrew Powuk - http://www.powuk.com- Math 49 (Numercal Aalyss) Root fdg. Itroducto f ( ),?,? Solve[^-,] {{-},{}} Plot[^-,{,-,}] Cubc equato https://e.wkpeda.org/wk/cubc_fucto Quartc equato https://e.wkpeda.org/wk/quartc_fucto
More informationCS475 Parallel Programming
CS475 Parallel Programmg Deretato ad Itegrato Wm Bohm Colorado State Uversty Ecept as otherwse oted, the cotet o ths presetato s lcesed uder the Creatve Commos Attrbuto.5 lcese. Pheomea Physcs: heat, low,
More informationhp calculators HP 30S Statistics Averages and Standard Deviations Average and Standard Deviation Practice Finding Averages and Standard Deviations
HP 30S Statstcs Averages ad Stadard Devatos Average ad Stadard Devato Practce Fdg Averages ad Stadard Devatos HP 30S Statstcs Averages ad Stadard Devatos Average ad stadard devato The HP 30S provdes several
More informationANALYTICAL SOLUTION FOR NONLINEAR BENDING OF FG PLATES BY A LAYERWISE THEORY
6 TH ITERATIOAL COFERECE O COMPOSITE MATERIALS AALYTICAL SOLUTIO FOR OLIEAR BEDIG OF FG PLATES BY A LAYERWISE THEORY M. Taha*, S.M. Mrababaee** *Departmet of Mechacal Egeerg, Ferdows Uversty of Mashhad,
More informationModule 7: Probability and Statistics
Lecture 4: Goodess of ft tests. Itroducto Module 7: Probablty ad Statstcs I the prevous two lectures, the cocepts, steps ad applcatos of Hypotheses testg were dscussed. Hypotheses testg may be used to
More informationObjectives of Multiple Regression
Obectves of Multple Regresso Establsh the lear equato that best predcts values of a depedet varable Y usg more tha oe eplaator varable from a large set of potetal predctors {,,... k }. Fd that subset of
More informationApplied Fitting Theory VII. Building Virtual Particles
Appled Fttg heory II Paul Avery CBX 98 38 Jue 8, 998 Apr. 7, 999 (rev.) Buldg rtual Partcles I Statemet of the problem I may physcs aalyses we ecouter the problem of mergg a set of partcles to a sgle partcle
More informationu 1 Figure 1 3D Solid Finite Elements
Sold Elemets he Fte Elemet Lbrary of the MIDAS Famly Programs cludes the follog Sold Elemets: - ode tetrahedro, -ode petahedro, ad -ode hexahedro sho Fg.. he fte elemet formulato of all elemet types s
More informationLecture 3. Sampling, sampling distributions, and parameter estimation
Lecture 3 Samplg, samplg dstrbutos, ad parameter estmato Samplg Defto Populato s defed as the collecto of all the possble observatos of terest. The collecto of observatos we take from the populato s called
More informationCentroids & Moments of Inertia of Beam Sections
RCH 614 Note Set 8 S017ab Cetrods & Momets of erta of Beam Sectos Notato: b C d d d Fz h c Jo L O Q Q = ame for area = ame for a (base) wdth = desgato for chael secto = ame for cetrod = calculus smbol
More informationLecture 07: Poles and Zeros
Lecture 07: Poles ad Zeros Defto of poles ad zeros The trasfer fucto provdes a bass for determg mportat system respose characterstcs wthout solvg the complete dfferetal equato. As defed, the trasfer fucto
More informationThird handout: On the Gini Index
Thrd hadout: O the dex Corrado, a tala statstca, proposed (, 9, 96) to measure absolute equalt va the mea dfferece whch s defed as ( / ) where refers to the total umber of dvduals socet. Assume that. The
More informationA New Family of Transformations for Lifetime Data
Proceedgs of the World Cogress o Egeerg 4 Vol I, WCE 4, July - 4, 4, Lodo, U.K. A New Famly of Trasformatos for Lfetme Data Lakhaa Watthaacheewakul Abstract A famly of trasformatos s the oe of several
More informationNumerical Solution of Linear Second Order Ordinary Differential Equations with Mixed Boundary Conditions by Galerkin Method
Mathematcs ad Computer Scece 7; (5: 66-78 http://www.scecepublshggroup.com//mcs do:.648/.mcs.75. Numercal Soluto of Lear Secod Order Ordary Dfferetal Equatos wth Mxed Boudary Codtos by Galer Method Aalu
More informationd dt d d dt dt Also recall that by Taylor series, / 2 (enables use of sin instead of cos-see p.27 of A&F) dsin
Learzato of the Swg Equato We wll cover sectos.5.-.6 ad begg of Secto 3.3 these otes. 1. Sgle mache-fte bus case Cosder a sgle mache coected to a fte bus, as show Fg. 1 below. E y1 V=1./_ Fg. 1 The admttace
More informationSTA 108 Applied Linear Models: Regression Analysis Spring Solution for Homework #1
STA 08 Appled Lear Models: Regresso Aalyss Sprg 0 Soluto for Homework #. Let Y the dollar cost per year, X the umber of vsts per year. The the mathematcal relato betwee X ad Y s: Y 300 + X. Ths s a fuctoal
More informationNumerical Analysis Formulae Booklet
Numercal Aalyss Formulae Booklet. Iteratve Scemes for Systems of Lear Algebrac Equatos:.... Taylor Seres... 3. Fte Dfferece Approxmatos... 3 4. Egevalues ad Egevectors of Matrces.... 3 5. Vector ad Matrx
More informationCHAPTER VI Statistical Analysis of Experimental Data
Chapter VI Statstcal Aalyss of Expermetal Data CHAPTER VI Statstcal Aalyss of Expermetal Data Measuremets do ot lead to a uque value. Ths s a result of the multtude of errors (maly radom errors) that ca
More informationENGI 4430 Numerical Integration Page 5-01
ENGI 443 Numercal Itegrato Page 5-5. Numercal Itegrato I some o our prevous work, (most otaly the evaluato o arc legth), t has ee dcult or mpossle to d the dete tegral. Varous symolc algera ad calculus
More informationSemiconductor Device Physics
1 Semcoductor evce Physcs Lecture 7 htt://ztomul.wordress.com 0 1 3 Semcoductor evce Physcs Chater 6 Jucto odes: I-V Characterstcs 3 Chater 6 Jucto odes: I-V Characterstcs Qualtatve ervato Majorty carrers
More informationAnalysis of Variance with Weibull Data
Aalyss of Varace wth Webull Data Lahaa Watthaacheewaul Abstract I statstcal data aalyss by aalyss of varace, the usual basc assumptos are that the model s addtve ad the errors are radomly, depedetly, ad
More informationModule 7. Lecture 7: Statistical parameter estimation
Lecture 7: Statstcal parameter estmato Parameter Estmato Methods of Parameter Estmato 1) Method of Matchg Pots ) Method of Momets 3) Mamum Lkelhood method Populato Parameter Sample Parameter Ubased estmato
More informationDynamic Analysis of Axially Beam on Visco - Elastic Foundation with Elastic Supports under Moving Load
Dyamc Aalyss of Axally Beam o Vsco - Elastc Foudato wth Elastc Supports uder Movg oad Saeed Mohammadzadeh, Seyed Al Mosayeb * Abstract: For dyamc aalyses of ralway track structures, the algorthm of soluto
More informationCH E 374 Computational Methods in Engineering Fall 2007
CH E 7 Computatoal Methods Egeerg Fall 007 Sample Soluto 5. The data o the varato of the rato of stagato pressure to statc pressure (r ) wth Mach umber ( M ) for the flow through a duct are as follows:
More informationFinite Difference Approximations for Fractional Reaction-Diffusion Equations and the Application In PM2.5
Iteratoal Symposum o Eergy Scece ad Chemcal Egeerg (ISESCE 5) Fte Dfferece Appromatos for Fractoal Reacto-Dffuso Equatos ad the Applcato I PM5 Chagpg Xe, a, Lag L,b, Zhogzha Huag,c, Jya L,d, PegLag L,e
More informationThe Necessarily Efficient Point Method for Interval Molp Problems
ISS 6-69 Eglad K Joural of Iformato ad omputg Scece Vol. o. 9 pp. - The ecessarly Effcet Pot Method for Iterval Molp Problems Hassa Mshmast eh ad Marzeh Alezhad + Mathematcs Departmet versty of Ssta ad
More information0/1 INTEGER PROGRAMMING AND SEMIDEFINTE PROGRAMMING
CONVEX OPIMIZAION AND INERIOR POIN MEHODS FINAL PROJEC / INEGER PROGRAMMING AND SEMIDEFINE PROGRAMMING b Luca Buch ad Natala Vktorova CONENS:.Itroducto.Formulato.Applcato to Kapsack Problem 4.Cuttg Plaes
More informationAnalyzing Fuzzy System Reliability Using Vague Set Theory
Iteratoal Joural of Appled Scece ad Egeerg 2003., : 82-88 Aalyzg Fuzzy System Relablty sg Vague Set Theory Shy-Mg Che Departmet of Computer Scece ad Iformato Egeerg, Natoal Tawa versty of Scece ad Techology,
More informationA MLPG Meshless Method for Numerical Simulaton of Unsteady Incompressible Flows
Joural of Appled Flud Mechacs, Vol. x, No. x, pp. x-x, 00x. Avalable ole at www.afmole.et, ISSN 735-3645. A MLPG Meshless Method for Numercal Smulato of Usteady Icompressble Flows Ira Saeedpaah Assstat
More informationPROJECTION PROBLEM FOR REGULAR POLYGONS
Joural of Mathematcal Sceces: Advaces ad Applcatos Volume, Number, 008, Pages 95-50 PROJECTION PROBLEM FOR REGULAR POLYGONS College of Scece Bejg Forestry Uversty Bejg 0008 P. R. Cha e-mal: sl@bjfu.edu.c
More information4 Inner Product Spaces
11.MH1 LINEAR ALGEBRA Summary Notes 4 Ier Product Spaces Ier product s the abstracto to geeral vector spaces of the famlar dea of the scalar product of two vectors or 3. I what follows, keep these key
More information1. A real number x is represented approximately by , and we are told that the relative error is 0.1 %. What is x? Note: There are two answers.
PROBLEMS A real umber s represeted appromately by 63, ad we are told that the relatve error s % What s? Note: There are two aswers Ht : Recall that % relatve error s What s the relatve error volved roudg
More informationBounds on the expected entropy and KL-divergence of sampled multinomial distributions. Brandon C. Roy
Bouds o the expected etropy ad KL-dvergece of sampled multomal dstrbutos Brado C. Roy bcroy@meda.mt.edu Orgal: May 18, 2011 Revsed: Jue 6, 2011 Abstract Iformato theoretc quattes calculated from a sampled
More informationDescriptive Statistics
Page Techcal Math II Descrptve Statstcs Descrptve Statstcs Descrptve statstcs s the body of methods used to represet ad summarze sets of data. A descrpto of how a set of measuremets (for eample, people
More informationA NEW NUMERICAL APPROACH FOR SOLVING HIGH-ORDER LINEAR AND NON-LINEAR DIFFERANTIAL EQUATIONS
Secer, A., et al.: A New Numerıcal Approach for Solvıg Hıgh-Order Lıear ad No-Lıear... HERMAL SCIENCE: Year 8, Vol., Suppl., pp. S67-S77 S67 A NEW NUMERICAL APPROACH FOR SOLVING HIGH-ORDER LINEAR AND NON-LINEAR
More informationAssignment 5/MATH 247/Winter Due: Friday, February 19 in class (!) (answers will be posted right after class)
Assgmet 5/MATH 7/Wter 00 Due: Frday, February 9 class (!) (aswers wll be posted rght after class) As usual, there are peces of text, before the questos [], [], themselves. Recall: For the quadratc form
More informationMulti-Step Methods Applied to Nonlinear Equations of Power Networks
Electrcal ad Electroc Egeerg 03, 3(5): 8-3 DOI: 0.593/j.eee.030305.0 Mult-Step s Appled to olear Equatos of Power etworks Rubé llafuerte D.,*, Rubé A. llafuerte S., Jesús Meda C. 3, Edgar Meja S. 3 Departmet
More information{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution:
Chapter 4 Exercses Samplg Theory Exercse (Smple radom samplg: Let there be two correlated radom varables X ad A sample of sze s draw from a populato by smple radom samplg wthout replacemet The observed
More informationENGI 4421 Joint Probability Distributions Page Joint Probability Distributions [Navidi sections 2.5 and 2.6; Devore sections
ENGI 441 Jot Probablty Dstrbutos Page 7-01 Jot Probablty Dstrbutos [Navd sectos.5 ad.6; Devore sectos 5.1-5.] The jot probablty mass fucto of two dscrete radom quattes, s, P ad p x y x y The margal probablty
More information