Monte Carlo Optimization to Solve a Two-Dimensional Inverse Heat Conduction Problem
|
|
- Justina Campbell
- 5 years ago
- Views:
Transcription
1 Australia Joural of Basic Applied Scieces, 5(): , 0 ISSN Mote Carlo Optimizatio to Solve a Two-Dimesioal Iverse Heat Coductio Problem M Ebrahimi Departmet of Mathematics, Karaj Brach, Islamic Azad Uiversity, Karaj, Ira Abstract: This paper establishes a umerical-probabilistic algorithm based o a discretizatio scheme a simulatio method for solvig a two-dimesioal iverse heat coductio problem with two uow boudary coditios At the begiig of the algorithm, alterative directio implicit scheme is used as a umerical method to discretize the problem domai the a approach of Mote Carlo method is employed as a probabilistic simulatio techique to solve the obtaied large sparse systems of liear algebraic equatios The rom search algorithm i Mote Carlo method for global optimizatio is adopted to fid the solutio of our iterest iverse problem Numerical results show that a excellet estimatio o the uow boudary coditios ca be obtaied withi a couple of miutes CPU time at petium IV-4 GHz PC Key words: Iverse heat coductio problem, Mote Carlo Optimizatio, Fiite differeces method, System of liear algebraic equatios, Mote Carlo simulatio AMS Subject Classificatio: Primary 35R30, 65C05; Secodary 65M06, 78M50 INTRODUCTION Iverse heat coductio problems (IHCP's) have a variety of applicatios i may idustries such as spacecraft, uclear reactors, supercoductive geerators, etc There have bee umerous applicatios of IHCP i various braches of sciece egieerig, such as the predictio of the ier wall temperature of a reactor, the determiatio of the heat trasfer coefficiets the outer surface coditios of a space vehicle, the predictio of temperature or heat flux at the tool-worpiece iterface of machie cuttig The estimatio for the boudary coditios i IHCP's has received a great deal of attetio i recet years To date various methods have bee developed for the aalysis of the IHCP's ivolvig the estimatio of boudary coditio or diffusio coefficiet from measured temperature iside the material (Shidfar, A et al, 006; Ebrahimi, M, 0; Che, HT et al, 00; Lai, CH, 00) Shidfar et al, have applied a umerical algorithm based o fiite differeces method least-squares scheme for solvig a oliear diffusio problem Recetly Ebrahimi (Ebrahimi, M, 0) has bee ivestigated a stochastic algorithm based o Feyma-Kac formula for a iverse heat coductio problem with uow diffusio coefficiet However, most aalytical umerical methods were oly employed to deal with oe-dimesioal iverse problems Few wors were preseted for two-dimesioal parabolic iverse problems because the difficulty of these problem was more proouced The literature reviews showed that Che et al, (00) have applied a umerical method for solvig a two-dimesioal parabolic iverse problem CH Lai et al, (00) have studied a two dimesioal oliear, parabolic IHCP for weldig of metals alloys I the preset study, a two-dimesioal liear parabolic iverse heat coductio problem is solved usig a umerical-probabilistic algorithm ivolvig the combied use of the fiite differeces method Mote Carlo simulatio based o rom samplig The fuctioal forms of the boudary coditios are uow priori The uow boudary coditios are approximated by the polyomial forms Mote Carlo optimizatio method is used for estimatio uow coefficiets of the polyomials Numerical experimets cofirm the accuracy efficiecy of the preset umerical-probabilistic algorithm for a parabolic iverse problem i a fiite regio Accordig to latest iformatio from the research wors it is believed that the solutio of the preset two dimesio IHCP with two uow boudary coditio based o umerical-probabilistic algorithm icluded the Mote Carlo optimizatio has bee ivestigated for the first time i the preset study Statemet of the Problem: Cosider a ifiitely log bar with costat thermal properties with a square cross sectio of uit sidethe adiabatic coditios are applied at the side of 0 The coditio o the side of 0 is isothermal the temperature is oe uit It is iitially at a uiform temperature the suddely two temperature fuctios,, are applied to the sides 0, respectively The mathematical formulatio of the two dimesioal liear parabolic problem cocered to the above metioed physical model ca be give as: Correspodig Author: M Ebrahimi, Departmet of Mathematics, Karaj Brach, Islamic Azad Uiversity, Karaj, Ira moebrahimi@iauacir; Tel: (098)735406; Fax: (+98)
2 Aust J Basic & Appl Sci, 5(): , 0,0,0,0, (),,,, 0, 0, () 0,, 0, 0, 0, (3), 0,,, 0, 0, (4),, 0, 0, 0, (5),,00,0,0 (6) The direct problem cosidered here is cocered with the determiatio of the medium temperatures whe the surface temperatures,,, the iitial boudary coditios o the boudaries of square plae,:0,0 are ow For the iverse problem, the surface temperatures,, are regarded as beig uow I additio, temperature measuremets tae at some grid locatios time o the square plae are also cosidered available I fact, to estimate the uow fuctios,, the additioal iformatio of discrete temperature measuremets is required Therefore, for the iverse problem of fidig,,, measuremets of,, are assumed to be available o a grid of size Let the temperature measuremets tae at these grid poits over the time period be deoted by:,,,,,,, (7) where is the fial time for temperature measuremets represets the umber of grid i directios The measured data, could be obtaied as,,,, where is average rom error that may be is cosidered withi -05 to 05, obtaied from the solutio of the direct problem ()-(6) We ote that the measured temperature, should cotais measuremets errors Therefore the iverse problem ca be stated as follows: by utilizig the above-metioed measured temperature data,, estimate the uow fuctios,, over the etire space time domai Overview of the Numerical-Probabilistic Algorithm: At first, we will cosider a umerical problem as a mappig :, where is some Baach space We will call is a solutio operator The elemets of are the data, for which the problem has to be solved for Υ, Υ is the exact solutio For a give Υ we wat to compute (or approximate) Υ Now, the applicatio of the preset umerical-probabilistic algorithm to fid the solutio of the iverse problem ()-(7) ca be divided ito the followig steps Step : Fiite Differeces Method: Let us cosider the direct problem ()-(6) The discrete problem ca be cosidered as a mappig of fuctio Υ, oto I fact, the purpose of this sectio is to describe the implemetatio of the solutio operator (or discrete operator) that maps,,, as a iput, oto a output,, For a give Υ, we wat to compute (or approximate), Sice the problem ()-(6) is a well-posed problem, the liear operator, mappig the data,,, 0, oto,,, is well defied Now, solvig the iverse problem ()-(7), is thus equivalet to solvig the operator equatio, Υ By discretizig the direct problem ()-(6) we ca compute a approximatio of the operator I the preset wor we use fiite differeces method a equidistat grid o Therefore, defie the co-ordiates,, of the mesh poits of the solutio domai by,, where, are positive itegers Also, 098
3 Aust J Basic & Appl Sci, 5(): , 0 deote the values of,, at these mesh poits by,,,, Now, We use Alterate Directio Implicit (ADI) method for discretize equatio () cetral-differeces approximatio for equatios (3) (5) We assume the solutio is ow for time the replacig oly oe of the secod-order derivatives, say, by a implicit fiite differeces approximatio i terms of uow pivotal values of from the time level The other secod-order derivative,, beig replaced by a explicit fiite differeces approximatio Therefore the equatios,,,,,,,,,, (8),,,,,,,,,,,, (9),,,,, 0, (0),,,, 0, (),,,,,, () U p,q,0 = 0, r=0, (3) where 0, are used to advace the solutio from the to the time step Now, we cosider the equatios (8)-(3) may be writte i the followig matrix form:,, (4) where B ( ) ( ) ( ), ( ),,,,,,,,,,,,,,,,,,,, 0 0,,,, 099
4 Aust J Basic & Appl Sci, 5(): , 0 We ote that i equatio (4), U 0 U U By the Gerschgoris circle theorem, matrix method [5], we cocluded that the fiite differeces scheme (8) is ucoditioally stable Now, we assume the solutio is ow for time the advacemet of the solutio to the time level is obtaied by replacig by a implicit fiite differeces approximatio, by a explicit oe Hece the equatio,,,,,,,,,, (5),,,,,,, I cojuctio with the equatios (9)-(3), while 0, are used for advacemet the solutio from the to the time step I this case, if we cosider the may by we use the followig matrix form:, 0, (6) where B ( ) ( ) ( ), ( ),,,,,,,,,,,,, Ad, 0 0 We ote that i equatio (6), U,,,,, U U Note that based o Gerschgoris circle theorem, ucoditioally stability of the fiite differeces scheme (5) will be obtaied, easily Step Mote Carlo Simulatio: we assume, 00
5 Aust J Basic & Appl Sci, 5(): , 0, Therefore from equatio (4) we obtai, (7) where B B To solve the liear system (7), we cosider the followig iterative method, (8) where i,, N (0,] This is called the Jacobi overrelaxatio iterative method with relaxatio parameter (0,] Equatio (8) may be writte i the followig matrix form U ( ) LU ( ) f,,, (9) where D diag( B ( ) ( ) ( ) t ( u,, un is the -th iterative solutio of (8), L I _ DA, f DY U ),, B N, N ),, is a diagoal matrix I fact, we covert the system (7) ito a equivalet system of the followig form U LU f (0) Now, we cosider the Marov chai X x x x x () 0 trasitio matrix P pi, j, i, j,, N Let P( x0 i) pi, P( x N j x N i) pij, where p i p ij are the iitial distributio the trasitio probabilities of the Marov chai, respectively The weight fuctio W, for Marov chai () with m N states, is defied by usig the recursio formula with state space,,, N l xm, m W0, Wm Wm, m,, p xm, m Now the followig rom variable is defied H [ H ] p x0 x0 m0 W m c x m, x 0 which is associated with the sample path x x x t where is a give iteger umber H h,, h ) is a give vector We also cosider the problem of fidig the ier product ( N 0
6 Aust J Basic & Appl Sci, 5(): , 0 H, U hu h N u N, where U is the solutio of problem ()-(6) The solutio operator for the above formulated problem ca be writte i the followig form: K ( F, G) H, U * * Now, we defie a adjoit operator K try to fid the umerical solutio of K ( F, G) U Theorem : The mathematical expectatio value of the rom variable [H ] is equal to the ier product E H ( ), U hu h u ( ) [ H ] H, U H, U N N,, ie, Proof see referece (Rubistei, RY, 98) To estimate h u h u ( ) ( ) ( ) N N, We simulate N rom paths ( s) ( s) ( s) ( s) x0 x x x, s () M, Each with the legth of, evaluate the sample mea [ H ] N [ H ] E ( ) [ H ] H, U ( ) ( ) I fact, from Theorem we coclude tha [H ] is a ubiased estimator of the ier product H, U t ( ) ( ) It is readily see that by settig H (0,,0,,0,,0 ) we obtai H, U u j, j,, N Hece j [H ] is a ubiased estimator of the ( ) u j Step 3 Mote Carlo Optimizatio Techique: I this wor the polyomial form proposed for the uow fuctios F ( y, G ( x, before performig the iverse calculatios Therefore F ( y, G ( x, are approximated as F app G app ( y, ( a ( x, ( b a ( a 3 a4 y ( a 5 a6 y ( a a b ( b3 b4 x ( b5 b6 x ( b b where ( a,, a ) ( b,, b ) are costats which remai to be determied simultaeously These uow coefficiets ca be determied i such a way that the followig fuctioal is miimized: x y 0
7 Aust J Basic & Appl Sci, 5(): , 0 tf N N cal mea (,,,,, ) [ pq, ( p, q, ) pq, ( p, q, )], t 0 p q cal Here, p q ( x p, yq J a a b b U x y t U x y t dt U, are the calculated temperatures o the pla at the grid locatios ) These quatities are determied from the solutio of the direct problem give previously by usig a approximated F app ( y, G app ( x, for the exact F ( y, G ( x,, respectively The estimated values of a j, j,, b i, i,, are determied util the value of J ( a,, a, b,, b ) is miimum The computatioal procedure for estimatig uow coefficiets a j b i are described as follows: Cosider the followig determiistic optimizatio problem * * mi J( A) J( A ) J, ( ) AR () where J (A) is real-valued bouded fuctio defied o A,,,,, It is assumed that achieves its maximum value at a uique poit A The fuctio J (A) may have may local maximum i but oly oe global maximum Rom Search Algorithm: For solvig problem () we cosider the followig rom search algorithm: Geerate dimesioal rom variables Z, from a -dimesioal ormal, Z distributio with zero mea covariace matrix C that is Z ~ N (0, C) Z Z, Select a iitial poit J( A ) 3 Compute 4 Set 5 If, go to step 8 6 Set A A 7 Go to step 0 8 J ( A ) Compute i Z i 9 If J ( Ai Z i ) J ( Ai ) the (where 0) set A A Z else set A A 0 If the stoppig criterio is met, stop; otherwise, set Go to step 5, Z Numerical Experimet: I this sectio, we are goig to demostrate some umerical results for determie ( U ( x, y,, F(, y,, G( x,0, ) i the iverse problem ()-(7) All the computatios are performed o the PC However, to further demostratig the accuracy efficiecy of this method, the preset problem is ivestigated the followig example is cosidered Example: Cosider ()-(7) with F app ( y, 0 04t 03y 07 yt 0y 0 0y t G app ( x, 05 03t 0x 04xt 05x 0 x t Table shows the values of U i x px, x, t rt, whe x y 00 I Table, f 03
8 Aust J Basic & Appl Sci, 5(): , 0 U (0,08, t ), U (0, 0, t ), U ( 0 5,08, t ), U ( 09,0, U (09,0 8, betwee the exact results the preset umerical results Tables 3 shows the exact umerical values of F (, y, G( x,0, respectively Figures shows the graphical results of Tables 3, respectively Table : Results for U with t f 04 t Table : Results for F (, y, with x y 00 t=005 y Numerical Exact Table 3: Results for G(x,0, with x y 00 t=005 y Numerical Exact t=05 Numerical Exact t=05 Numerical Exat t=05 Numerical Exat t= 05 Numerical Exat Fig : Results for F (, y, with x y 00
9 Aust J Basic & Appl Sci, 5(): , 0 Fig : Results for G ( x,0, with x y 00 Coclusio: I this paper a umerical-probabilistic algorithm ivolvig the fiite differeces method i cojuctio with the Mote Carlo optimizatio is employed to solve a two-dimesioal parabolic iverse problem, successfully the followig results are obtaied: Mote Carlo methods are preferable for solvig large sparse systems, such as those arisig from approximatios of partial differetial equatios Oe of the most importat advatages of Mote carlo methods is that they ca be used to evaluate oly oe compoet of the solutio or some liear form of the solutio This advatage is of great practical iterest 3 From the illustrated example it ca be see that the proposed method is efficiet accurate to estimate the uow boudary coditios i a two-dimesioal IHCP REFERENCES Che, HT, SY Li LC Fag, 00 Estimatio of surface temperature i two-dimesioal iverse heat coductio problems Iteratioal Joural of Heat Mass Trasfer, 44: Ebrahimi, M, 0 Estimatio of diffusio coefficiet i gas exchage process withi huma respiratio via a iverse problem Australia Joural of Basic Applied Scieces, (to appear) Faroosh, R M Ebrahimi, 007 Mote Carlo method via a umerical algorithm to solve a parabolic problem Applied Mathematics Computatio, 90(): Lai, CH, CS Ierotheou, CJ Palasuriya KA Pericleous, 00, A domai decompositio algorithm for iverse weldig problems Computig Visualizatio i Sciece, 4: Rubistei, RY, 98 Simulatio the Mote Carlo method, Wiley, New Yor Shidfar, A, R Pourgholi M Ebrahimi, 006 A umerical method for solvig of a oliear iverse diffusio problem Computers Mathematics with Applicatios, 5:
Taylor polynomial solution of difference equation with constant coefficients via time scales calculus
TMSCI 3, o 3, 129-135 (2015) 129 ew Treds i Mathematical Scieces http://wwwtmscicom Taylor polyomial solutio of differece equatio with costat coefficiets via time scales calculus Veysel Fuat Hatipoglu
More informationDECOMPOSITION METHOD FOR SOLVING A SYSTEM OF THIRD-ORDER BOUNDARY VALUE PROBLEMS. Park Road, Islamabad, Pakistan
Mathematical ad Computatioal Applicatios, Vol. 9, No. 3, pp. 30-40, 04 DECOMPOSITION METHOD FOR SOLVING A SYSTEM OF THIRD-ORDER BOUNDARY VALUE PROBLEMS Muhammad Aslam Noor, Khalida Iayat Noor ad Asif Waheed
More informationNumerical Solution of the Two Point Boundary Value Problems By Using Wavelet Bases of Hermite Cubic Spline Wavelets
Australia Joural of Basic ad Applied Scieces, 5(): 98-5, ISSN 99-878 Numerical Solutio of the Two Poit Boudary Value Problems By Usig Wavelet Bases of Hermite Cubic Splie Wavelets Mehdi Yousefi, Hesam-Aldie
More informationA NEW CLASS OF 2-STEP RATIONAL MULTISTEP METHODS
Jural Karya Asli Loreka Ahli Matematik Vol. No. (010) page 6-9. Jural Karya Asli Loreka Ahli Matematik A NEW CLASS OF -STEP RATIONAL MULTISTEP METHODS 1 Nazeeruddi Yaacob Teh Yua Yig Norma Alias 1 Departmet
More informationThe Method of Least Squares. To understand least squares fitting of data.
The Method of Least Squares KEY WORDS Curve fittig, least square GOAL To uderstad least squares fittig of data To uderstad the least squares solutio of icosistet systems of liear equatios 1 Motivatio Curve
More informationA collocation method for singular integral equations with cosecant kernel via Semi-trigonometric interpolation
Iteratioal Joural of Mathematics Research. ISSN 0976-5840 Volume 9 Number 1 (017) pp. 45-51 Iteratioal Research Publicatio House http://www.irphouse.com A collocatio method for sigular itegral equatios
More informationLainiotis filter implementation. via Chandrasekhar type algorithm
Joural of Computatios & Modellig, vol.1, o.1, 2011, 115-130 ISSN: 1792-7625 prit, 1792-8850 olie Iteratioal Scietific Press, 2011 Laiiotis filter implemetatio via Chadrasehar type algorithm Nicholas Assimais
More informationTEACHER CERTIFICATION STUDY GUIDE
COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra
More informationSolution of Differential Equation from the Transform Technique
Iteratioal Joural of Computatioal Sciece ad Mathematics ISSN 0974-3189 Volume 3, Number 1 (2011), pp 121-125 Iteratioal Research Publicatio House http://wwwirphousecom Solutio of Differetial Equatio from
More information10-701/ Machine Learning Mid-term Exam Solution
0-70/5-78 Machie Learig Mid-term Exam Solutio Your Name: Your Adrew ID: True or False (Give oe setece explaatio) (20%). (F) For a cotiuous radom variable x ad its probability distributio fuctio p(x), it
More informationNumerical Conformal Mapping via a Fredholm Integral Equation using Fourier Method ABSTRACT INTRODUCTION
alaysia Joural of athematical Scieces 3(1): 83-93 (9) umerical Coformal appig via a Fredholm Itegral Equatio usig Fourier ethod 1 Ali Hassa ohamed urid ad Teh Yua Yig 1, Departmet of athematics, Faculty
More informationChapter 7: The z-transform. Chih-Wei Liu
Chapter 7: The -Trasform Chih-Wei Liu Outlie Itroductio The -Trasform Properties of the Regio of Covergece Properties of the -Trasform Iversio of the -Trasform The Trasfer Fuctio Causality ad Stability
More information3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,
3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [
More informationMETHOD OF FUNDAMENTAL SOLUTIONS FOR HELMHOLTZ EIGENVALUE PROBLEMS IN ELLIPTICAL DOMAINS
Please cite this article as: Staisław Kula, Method of fudametal solutios for Helmholtz eigevalue problems i elliptical domais, Scietific Research of the Istitute of Mathematics ad Computer Sciece, 009,
More informationDETERMINATION OF MECHANICAL PROPERTIES OF A NON- UNIFORM BEAM USING THE MEASUREMENT OF THE EXCITED LONGITUDINAL ELASTIC VIBRATIONS.
ICSV4 Cairs Australia 9- July 7 DTRMINATION OF MCHANICAL PROPRTIS OF A NON- UNIFORM BAM USING TH MASURMNT OF TH XCITD LONGITUDINAL LASTIC VIBRATIONS Pavel Aokhi ad Vladimir Gordo Departmet of the mathematics
More informationStreamfunction-Vorticity Formulation
Streamfuctio-Vorticity Formulatio A. Salih Departmet of Aerospace Egieerig Idia Istitute of Space Sciece ad Techology, Thiruvaathapuram March 2013 The streamfuctio-vorticity formulatio was amog the first
More informationTHE KALMAN FILTER RAUL ROJAS
THE KALMAN FILTER RAUL ROJAS Abstract. This paper provides a getle itroductio to the Kalma filter, a umerical method that ca be used for sesor fusio or for calculatio of trajectories. First, we cosider
More informationLecture 8: Solving the Heat, Laplace and Wave equations using finite difference methods
Itroductory lecture otes o Partial Differetial Equatios - c Athoy Peirce. Not to be copied, used, or revised without explicit writte permissio from the copyright ower. 1 Lecture 8: Solvig the Heat, Laplace
More informationChapter 4 : Laplace Transform
4. Itroductio Laplace trasform is a alterative to solve the differetial equatio by the complex frequecy domai ( s = σ + jω), istead of the usual time domai. The DE ca be easily trasformed ito a algebraic
More informationL 5 & 6: RelHydro/Basel. f(x)= ( ) f( ) ( ) ( ) ( ) n! 1! 2! 3! If the TE of f(x)= sin(x) around x 0 is: sin(x) = x - 3! 5!
aylor epasio: Let ƒ() be a ifiitely differetiable real fuctio. At ay poit i the eighbourhood of =0, the fuctio ca be represeted as a power series of the followig form: X 0 f(a) f() ƒ() f()= ( ) f( ) (
More informationNEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE
UPB Sci Bull, Series A, Vol 79, Iss, 207 ISSN 22-7027 NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE Gabriel Bercu We itroduce two ew sequeces of Euler-Mascheroi type which have fast covergece
More informationPAijpam.eu ON TENSOR PRODUCT DECOMPOSITION
Iteratioal Joural of Pure ad Applied Mathematics Volume 103 No 3 2015, 537-545 ISSN: 1311-8080 (prited versio); ISSN: 1314-3395 (o-lie versio) url: http://wwwijpameu doi: http://dxdoiorg/1012732/ijpamv103i314
More informationOn forward improvement iteration for stopping problems
O forward improvemet iteratio for stoppig problems Mathematical Istitute, Uiversity of Kiel, Ludewig-Mey-Str. 4, D-24098 Kiel, Germay irle@math.ui-iel.de Albrecht Irle Abstract. We cosider the optimal
More informationGenerating Functions for Laguerre Type Polynomials. Group Theoretic method
It. Joural of Math. Aalysis, Vol. 4, 2010, o. 48, 257-266 Geeratig Fuctios for Laguerre Type Polyomials α of Two Variables L ( xy, ) by Usig Group Theoretic method Ajay K. Shula* ad Sriata K. Meher** *Departmet
More informationProperties and Hypothesis Testing
Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Cross-sectioal data. 2. Time series data.
More informationNumerical Solutions of Second Order Boundary Value Problems by Galerkin Residual Method on Using Legendre Polynomials
IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578, p-issn: 319-765X. Volume 11, Issue 6 Ver. IV (Nov. - Dec. 15), PP 1-11 www.iosrjourals.org Numerical Solutios of Secod Order Boudary Value Problems
More informationSequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018
CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical
More informationIntroduction to Optimization Techniques. How to Solve Equations
Itroductio to Optimizatio Techiques How to Solve Equatios Iterative Methods of Optimizatio Iterative methods of optimizatio Solutio of the oliear equatios resultig form a optimizatio problem is usually
More informationA numerical Technique Finite Volume Method for Solving Diffusion 2D Problem
The Iteratioal Joural Of Egieerig d Sciece (IJES) Volume 4 Issue 10 Pages PP -35-41 2015 ISSN (e): 2319 1813 ISSN (p): 2319 1805 umerical Techique Fiite Volume Method for Solvig Diffusio 2D Problem 1 Mohammed
More informationFUZZY ALTERNATING DIRECTION IMPLICIT METHOD FOR SOLVING PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS IN THREE DIMENSIONS
FUZZY ALTERNATING DIRECTION IMPLICIT METHOD FOR SOLVING PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS IN THREE DIMENSIONS N.Mugutha *1, B.Jessaili Jeba #2 *1 Assistat Professor, Departmet of Mathematics, M.V.Muthiah
More informationExact Solutions for a Class of Nonlinear Singular Two-Point Boundary Value Problems: The Decomposition Method
Exact Solutios for a Class of Noliear Sigular Two-Poit Boudary Value Problems: The Decompositio Method Abd Elhalim Ebaid Departmet of Mathematics, Faculty of Sciece, Tabuk Uiversity, P O Box 741, Tabuki
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationSimilarity Solutions to Unsteady Pseudoplastic. Flow Near a Moving Wall
Iteratioal Mathematical Forum, Vol. 9, 04, o. 3, 465-475 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/imf.04.48 Similarity Solutios to Usteady Pseudoplastic Flow Near a Movig Wall W. Robi Egieerig
More informationChapter 9: Numerical Differentiation
178 Chapter 9: Numerical Differetiatio Numerical Differetiatio Formulatio of equatios for physical problems ofte ivolve derivatives (rate-of-chage quatities, such as velocity ad acceleratio). Numerical
More informationTaylor expansion: Show that the TE of f(x)= sin(x) around. sin(x) = x - + 3! 5! L 7 & 8: MHD/ZAH
Taylor epasio: Let ƒ() be a ifiitely differetiable real fuctio. A ay poit i the eighbourhood of 0, the fuctio ƒ() ca be represeted by a power series of the followig form: X 0 f(a) f() f() ( ) f( ) ( )
More informationTMA4205 Numerical Linear Algebra. The Poisson problem in R 2 : diagonalization methods
TMA4205 Numerical Liear Algebra The Poisso problem i R 2 : diagoalizatio methods September 3, 2007 c Eiar M Røquist Departmet of Mathematical Scieces NTNU, N-749 Trodheim, Norway All rights reserved A
More informationμ are complex parameters. Other
A New Numerical Itegrator for the Solutio of Iitial Value Problems i Ordiary Differetial Equatios. J. Suday * ad M.R. Odekule Departmet of Mathematical Scieces, Adamawa State Uiversity, Mubi, Nigeria.
More informationInverse Nodal Problems for Differential Equation on the Half-line
Australia Joural of Basic ad Applied Scieces, 3(4): 4498-4502, 2009 ISSN 1991-8178 Iverse Nodal Problems for Differetial Equatio o the Half-lie 1 2 3 A. Dabbaghia, A. Nematy ad Sh. Akbarpoor 1 Islamic
More informationChapter 2 The Monte Carlo Method
Chapter 2 The Mote Carlo Method The Mote Carlo Method stads for a broad class of computatioal algorithms that rely o radom sampligs. It is ofte used i physical ad mathematical problems ad is most useful
More informationThe picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled
1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how
More informationCS537. Numerical Analysis and Computing
CS57 Numerical Aalysis ad Computig Lecture Locatig Roots o Equatios Proessor Ju Zhag Departmet o Computer Sciece Uiversity o Ketucky Leigto KY 456-6 Jauary 9 9 What is the Root May physical system ca be
More informationCO-LOCATED DIFFUSE APPROXIMATION METHOD FOR TWO DIMENSIONAL INCOMPRESSIBLE CHANNEL FLOWS
CO-LOCATED DIFFUSE APPROXIMATION METHOD FOR TWO DIMENSIONAL INCOMPRESSIBLE CHANNEL FLOWS C.PRAX ad H.SADAT Laboratoire d'etudes Thermiques,URA CNRS 403 40, Aveue du Recteur Pieau 86022 Poitiers Cedex,
More informationR. van Zyl 1, A.J. van der Merwe 2. Quintiles International, University of the Free State
Bayesia Cotrol Charts for the Two-parameter Expoetial Distributio if the Locatio Parameter Ca Take o Ay Value Betwee Mius Iity ad Plus Iity R. va Zyl, A.J. va der Merwe 2 Quitiles Iteratioal, ruaavz@gmail.com
More informationBasics of Probability Theory (for Theory of Computation courses)
Basics of Probability Theory (for Theory of Computatio courses) Oded Goldreich Departmet of Computer Sciece Weizma Istitute of Sciece Rehovot, Israel. oded.goldreich@weizma.ac.il November 24, 2008 Preface.
More informationNumerical Methods in Geophysics: Implicit Methods
Numerical Methods i Geophysics: What is a implicit scheme? Explicit vs. implicit scheme for Newtoia oolig rak-nicholso Scheme (mixed explicit-implicit Explicit vs. implicit for the diffusio equatio Relaxatio
More informationRandom Variables, Sampling and Estimation
Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig
More informationPoincaré Problem for Nonlinear Elliptic Equations of Second Order in Unbounded Domains
Advaces i Pure Mathematics 23 3 72-77 http://dxdoiorg/4236/apm233a24 Published Olie Jauary 23 (http://wwwscirporg/oural/apm) Poicaré Problem for Noliear Elliptic Equatios of Secod Order i Ubouded Domais
More informationLinear Regression Demystified
Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to
More informationON POINTWISE BINOMIAL APPROXIMATION
Iteratioal Joural of Pure ad Applied Mathematics Volume 71 No. 1 2011, 57-66 ON POINTWISE BINOMIAL APPROXIMATION BY w-functions K. Teerapabolar 1, P. Wogkasem 2 Departmet of Mathematics Faculty of Sciece
More informationEstimation of Backward Perturbation Bounds For Linear Least Squares Problem
dvaced Sciece ad Techology Letters Vol.53 (ITS 4), pp.47-476 http://dx.doi.org/.457/astl.4.53.96 Estimatio of Bacward Perturbatio Bouds For Liear Least Squares Problem Xixiu Li School of Natural Scieces,
More informationFinite Difference Method for the Estimation of a Heat Source Dependent on Time Variable ABSTRACT
Malaysia Joural of Matematical Scieces 6(S): 39-5 () Special Editio of Iteratioal Worsop o Matematical Aalysis (IWOMA) Fiite Differece Metod for te Estimatio of a Heat Source Depedet o Time Variable, Allabere
More informationLinear Elliptic PDE s Elliptic partial differential equations frequently arise out of conservation statements of the form
Liear Elliptic PDE s Elliptic partial differetial equatios frequetly arise out of coservatio statemets of the form B F d B Sdx B cotaied i bouded ope set U R. Here F, S deote respectively, the flux desity
More informationInformation-based Feature Selection
Iformatio-based Feature Selectio Farza Faria, Abbas Kazeroui, Afshi Babveyh Email: {faria,abbask,afshib}@staford.edu 1 Itroductio Feature selectio is a topic of great iterest i applicatios dealig with
More informationROLL CUTTING PROBLEMS UNDER STOCHASTIC DEMAND
Pacific-Asia Joural of Mathematics, Volume 5, No., Jauary-Jue 20 ROLL CUTTING PROBLEMS UNDER STOCHASTIC DEMAND SHAKEEL JAVAID, Z. H. BAKHSHI & M. M. KHALID ABSTRACT: I this paper, the roll cuttig problem
More informationPRELIM PROBLEM SOLUTIONS
PRELIM PROBLEM SOLUTIONS THE GRAD STUDENTS + KEN Cotets. Complex Aalysis Practice Problems 2. 2. Real Aalysis Practice Problems 2. 4 3. Algebra Practice Problems 2. 8. Complex Aalysis Practice Problems
More informationThe z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j
The -Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discrete-time LTI systems. 7.
More informationPAijpam.eu ON DERIVATION OF RATIONAL SOLUTIONS OF BABBAGE S FUNCTIONAL EQUATION
Iteratioal Joural of Pure ad Applied Mathematics Volume 94 No. 204, 9-20 ISSN: 3-8080 (prited versio); ISSN: 34-3395 (o-lie versio) url: http://www.ijpam.eu doi: http://dx.doi.org/0.2732/ijpam.v94i.2 PAijpam.eu
More informationLecture 2: Monte Carlo Simulation
STAT/Q SCI 43: Itroductio to Resamplig ethods Sprig 27 Istructor: Ye-Chi Che Lecture 2: ote Carlo Simulatio 2 ote Carlo Itegratio Assume we wat to evaluate the followig itegratio: e x3 dx What ca we do?
More informationResearch Article A New Second-Order Iteration Method for Solving Nonlinear Equations
Abstract ad Applied Aalysis Volume 2013, Article ID 487062, 4 pages http://dx.doi.org/10.1155/2013/487062 Research Article A New Secod-Order Iteratio Method for Solvig Noliear Equatios Shi Mi Kag, 1 Arif
More information8. Applications To Linear Differential Equations
8. Applicatios To Liear Differetial Equatios 8.. Itroductio 8.. Review Of Results Cocerig Liear Differetial Equatios Of First Ad Secod Orders 8.3. Eercises 8.4. Liear Differetial Equatios Of Order N 8.5.
More informationA Taylor Series Based Method for Solving a Two-dimensional Second-order Equation
Applied Mathematical Scieces, Vol. 8, 2014, o. 66, 3255-3261 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.45347 A Taylor Series Based Method for Solvig a Two-dimesioal Secod-order Equatio
More informationDefinition 4.2. (a) A sequence {x n } in a Banach space X is a basis for X if. unique scalars a n (x) such that x = n. a n (x) x n. (4.
4. BASES I BAACH SPACES 39 4. BASES I BAACH SPACES Sice a Baach space X is a vector space, it must possess a Hamel, or vector space, basis, i.e., a subset {x γ } γ Γ whose fiite liear spa is all of X ad
More informationTHE NUMERICAL SOLUTION OF THE NEWTONIAN FLUIDS FLOW DUE TO A STRETCHING CYLINDER BY SOR ITERATIVE PROCEDURE ABSTRACT
Europea Joural of Egieerig ad Techology Vol. 3 No., 5 ISSN 56-586 THE NUMERICAL SOLUTION OF THE NEWTONIAN FLUIDS FLOW DUE TO A STRETCHING CYLINDER BY SOR ITERATIVE PROCEDURE Atif Nazir, Tahir Mahmood ad
More informationNumerical Solution of the First-Order Hyperbolic Partial Differential Equation with Point-Wise Advance
Iteratioal oural of Sciece ad Research (ISR) ISSN (Olie): 39-74 Ide Copericus Value (3): 4 Impact Factor (3): 4438 Numerical Solutio of the First-Order Hyperbolic Partial Differetial Equatio with Poit-Wise
More informationA statistical method to determine sample size to estimate characteristic value of soil parameters
A statistical method to determie sample size to estimate characteristic value of soil parameters Y. Hojo, B. Setiawa 2 ad M. Suzuki 3 Abstract Sample size is a importat factor to be cosidered i determiig
More informationThe Adomian Polynomials and the New Modified Decomposition Method for BVPs of nonlinear ODEs
Mathematical Computatio March 015, Volume, Issue 1, PP.1 6 The Adomia Polyomials ad the New Modified Decompositio Method for BVPs of oliear ODEs Jusheg Dua # School of Scieces, Shaghai Istitute of Techology,
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationTime-Domain Representations of LTI Systems
2.1 Itroductio Objectives: 1. Impulse resposes of LTI systems 2. Liear costat-coefficiets differetial or differece equatios of LTI systems 3. Bloc diagram represetatios of LTI systems 4. State-variable
More informationAbstract Vector Spaces. Abstract Vector Spaces
Astract Vector Spaces The process of astractio is critical i egieerig! Physical Device Data Storage Vector Space MRI machie Optical receiver 0 0 1 0 1 0 0 1 Icreasig astractio 6.1 Astract Vector Spaces
More informationComparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series
Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series
More informationMachine Learning for Data Science (CS 4786)
Machie Learig for Data Sciece CS 4786) Lecture & 3: Pricipal Compoet Aalysis The text i black outlies high level ideas. The text i blue provides simple mathematical details to derive or get to the algorithm
More informationStatistical Inference Based on Extremum Estimators
T. Rotheberg Fall, 2007 Statistical Iferece Based o Extremum Estimators Itroductio Suppose 0, the true value of a p-dimesioal parameter, is kow to lie i some subset S R p : Ofte we choose to estimate 0
More informationCS321. Numerical Analysis and Computing
CS Numerical Aalysis ad Computig Lecture Locatig Roots o Equatios Proessor Ju Zhag Departmet o Computer Sciece Uiversity o Ketucky Leigto KY 456-6 September 8 5 What is the Root May physical system ca
More informationThe standard deviation of the mean
Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider
More informationEstimation of Population Mean Using Co-Efficient of Variation and Median of an Auxiliary Variable
Iteratioal Joural of Probability ad Statistics 01, 1(4: 111-118 DOI: 10.593/j.ijps.010104.04 Estimatio of Populatio Mea Usig Co-Efficiet of Variatio ad Media of a Auxiliary Variable J. Subramai *, G. Kumarapadiya
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationOptimally Sparse SVMs
A. Proof of Lemma 3. We here prove a lower boud o the umber of support vectors to achieve geeralizatio bouds of the form which we cosider. Importatly, this result holds ot oly for liear classifiers, but
More informationACO Comprehensive Exam 9 October 2007 Student code A. 1. Graph Theory
1. Graph Theory Prove that there exist o simple plaar triagulatio T ad two distict adjacet vertices x, y V (T ) such that x ad y are the oly vertices of T of odd degree. Do ot use the Four-Color Theorem.
More informationPC5215 Numerical Recipes with Applications - Review Problems
PC55 Numerical Recipes with Applicatios - Review Problems Give the IEEE 754 sigle precisio bit patter (biary or he format) of the followig umbers: 0 0 05 00 0 00 Note that it has 8 bits for the epoet,
More informationA STUDY ON MHD BOUNDARY LAYER FLOW OVER A NONLINEAR STRETCHING SHEET USING IMPLICIT FINITE DIFFERENCE METHOD
IRET: Iteratioal oural of Research i Egieerig ad Techology eissn: 39-63 pissn: 3-7308 A STUDY ON MHD BOUNDARY LAYER FLOW OVER A NONLINEAR STRETCHING SHEET USING IMPLICIT FINITE DIFFERENCE METHOD Satish
More informationIntroduction to Machine Learning DIS10
CS 189 Fall 017 Itroductio to Machie Learig DIS10 1 Fu with Lagrage Multipliers (a) Miimize the fuctio such that f (x,y) = x + y x + y = 3. Solutio: The Lagragia is: L(x,y,λ) = x + y + λ(x + y 3) Takig
More informationJ. Stat. Appl. Pro. Lett. 2, No. 1, (2015) 15
J. Stat. Appl. Pro. Lett. 2, No. 1, 15-22 2015 15 Joural of Statistics Applicatios & Probability Letters A Iteratioal Joural http://dx.doi.org/10.12785/jsapl/020102 Martigale Method for Rui Probabilityi
More informationMost text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t
Itroductio to Differetial Equatios Defiitios ad Termiolog Differetial Equatio: A equatio cotaiig the derivatives of oe or more depedet variables, with respect to oe or more idepedet variables, is said
More informationSeed and Sieve of Odd Composite Numbers with Applications in Factorization of Integers
IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578, p-issn: 319-75X. Volume 1, Issue 5 Ver. VIII (Sep. - Oct.01), PP 01-07 www.iosrjourals.org Seed ad Sieve of Odd Composite Numbers with Applicatios i
More informationPrinciple Of Superposition
ecture 5: PREIMINRY CONCEP O RUCUR NYI Priciple Of uperpositio Mathematically, the priciple of superpositio is stated as ( a ) G( a ) G( ) G a a or for a liear structural system, the respose at a give
More informationImproving the Localization of Eigenvalues for Complex Matrices
Applied Mathematical Scieces, Vol. 5, 011, o. 8, 1857-1864 Improvig the Localizatio of Eigevalues for Complex Matrices P. Sargolzaei 1, R. Rakhshaipur Departmet of Mathematics, Uiversity of Sista ad Baluchesta
More informationTHE TRANSFORMATION MATRIX OF CHEBYSHEV IV BERNSTEIN POLYNOMIAL BASES
Joural of Mathematical Aalysis ISSN: 17-341, URL: http://iliriascom/ma Volume 7 Issue 4(16, Pages 13-19 THE TRANSFORMATION MATRIX OF CHEBYSHEV IV BERNSTEIN POLYNOMIAL BASES ABEDALLAH RABABAH, AYMAN AL
More informationIP Reference guide for integer programming formulations.
IP Referece guide for iteger programmig formulatios. by James B. Orli for 15.053 ad 15.058 This documet is iteded as a compact (or relatively compact) guide to the formulatio of iteger programs. For more
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationMarkov Decision Processes
Markov Decisio Processes Defiitios; Statioary policies; Value improvemet algorithm, Policy improvemet algorithm, ad liear programmig for discouted cost ad average cost criteria. Markov Decisio Processes
More informationOptimization Methods MIT 2.098/6.255/ Final exam
Optimizatio Methods MIT 2.098/6.255/15.093 Fial exam Date Give: December 19th, 2006 P1. [30 pts] Classify the followig statemets as true or false. All aswers must be well-justified, either through a short
More informationNumerical Method for Blasius Equation on an infinite Interval
Numerical Method for Blasius Equatio o a ifiite Iterval Alexader I. Zadori Omsk departmet of Sobolev Mathematics Istitute of Siberia Brach of Russia Academy of Scieces, Russia zadori@iitam.omsk.et.ru 1
More informationComputation of Error Bounds for P-matrix Linear Complementarity Problems
Mathematical Programmig mauscript No. (will be iserted by the editor) Xiaoju Che Shuhuag Xiag Computatio of Error Bouds for P-matrix Liear Complemetarity Problems Received: date / Accepted: date Abstract
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationWe are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n
Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at
More informationDirection: This test is worth 250 points. You are required to complete this test within 50 minutes.
Term Test October 3, 003 Name Math 56 Studet Number Directio: This test is worth 50 poits. You are required to complete this test withi 50 miutes. I order to receive full credit, aswer each problem completely
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationZeros of Polynomials
Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree
More informationThe Random Walk For Dummies
The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli
More information