CO-LOCATED DIFFUSE APPROXIMATION METHOD FOR TWO DIMENSIONAL INCOMPRESSIBLE CHANNEL FLOWS

Size: px
Start display at page:

Download "CO-LOCATED DIFFUSE APPROXIMATION METHOD FOR TWO DIMENSIONAL INCOMPRESSIBLE CHANNEL FLOWS"

Transcription

1 CO-LOCATED DIFFUSE APPROXIMATION METHOD FOR TWO DIMENSIONAL INCOMPRESSIBLE CHANNEL FLOWS C.PRAX ad H.SADAT Laboratoire d'etudes Thermiques,URA CNRS , Aveue du Recteur Pieau Poitiers Cedex, Frace Itroductio The mai cotributio of this paper is the formulatio of a diffuse approximatio method(dam), for two-dimesioal chael flows. The proposed method is based o the vorticity-streamfuctio formulatio. The DAM which estimates derivates of a scalar field has the remarkable advatage to work o discretizatio poits (thus avoidig mesh geeratio). It has bee show that the DAM is much better tha the fiite elemet method for the computatio of gradiets [-2]. I a previous paper [3], we have show that it ca be used to solve lamiar atural covectio problems. I this work, we discuss the applicability of this method to chael flows with a particular emphasis o the form of the weightig fuctio. The Diffuse Approximatio Cosider a scalar field ϕ(x,y), i a two dimesioal domai, ad a set of poits Mi(x i,yi) i the viciity of a chose poit M(x,y). The diffuse approximatio provides estimates of ϕ ad its derivatives at M from the odal values ϕ i. The startig poit is to estimate the Taylor expasio of ϕ at M by a weighted least squares method which uses oly the values of ϕ at the earest poits Mi. By trocatig the series at order k, oe obtais the correspodig estimates of the derivatives at the same order. Let us the estimate the secod-order Taylor expasio of ϕ i at M as: ϕ i *(x i,y i ) = < p(m i,m)>. < α M > T () where <p(mi,m)> is the lie vector of polyomial basis ad <α M > T is the trasposed vector of the approximatio defied as : <p(m i,m)> = <,(x i -x),(y i -y),(x i -x) 2,(x i -x).(y i -y),(y i -y) 2 > (2)

2 2 <α M > T = < α 0, α, α 2, α 3, α 4, α 5 > T (3) The variables α M are determied by miimizig the quadratic expressio: I(α M ) = {ω(m,m i ). [ϕ i - < p(m i,m)>. < α M > T ] 2 } (4) i= where ω(m,m i ) is a cotiuous weightig fuctio, havig its maximum value at M ad decreasig rapidly to zero. Thus oly the earest poits to M are ivolved i (4). By writig the six coditios : α I(α M ) =0 j, j=0,5 (5) we obtai the (6x6) liear system : where [ A M ]. < α M > T = < B M > T (6) [ A M ]= ω(m,m i ).< p(m i,m)> T < p(m i,m)> (7) i= < B M > T = ω(m,m i ).< p(m i,m)> T.ϕ i (8) i= Oce the system (6) has bee solved, oe fially obtais the desired estimates of the derivatives at M: φ φ φ(x,y)=α 0; =α ; =α 2; x y φ φ φ =α 2 3; =α 4; =α 2 5; x y x y (9) The weightig fuctio ca be chose i may ways. Its radius must be large eough to overlap at least a umber of odes equal to the umber of terms αi. However, the situatio where the selected odes are aliged must be avoided i order to get a o sigular [A M ] matrix.

3 3 Implemetatio of the Diffuse Approximatio We cosider the Navier-Stokes equatios i the vorticitystreamfuctio formulatio : Ψ + ω = 0 (0) uω vω + = υ ω x y () Where ν is the kiematic viscosity. Our method for solvig the equatios (0) ad () by usig the diffuse approximatio has a remote similarity with the fiite differece method. The partial derivatives at a give poit are expressed as fuctios of the eighbourig odal values of Ψ or ω (by ivertig the matrix [A M ]): φ φ φ φ T φ M T = A 2 2 i i i x y x y x y i= φ,,,,,. ω (M,M ). p(m,m ).φ (2) By usig the relatios (2), the goverig equatios (0-) are ow replaced (at every ode M) by algebraic expressios i terms of the eighbourig odal values Ψ i or ω i. Two systems are the obtaied ad solved iteratively after the itroductio of the boudary coditios. I this work a relaxatio factor of 0.2 is used for each variable ad the covergece criteria iclude the relative chages betwee cosecutive iteratios: Ψ Ψ ω ω 0 ; 0 Ψ ω ew old 3 ew old 3 ew max ew max (3) The boudary coditio for the streamfuctio, at the outflow of the chael, is: Ψ x = 0 The vorticity values at the boudary are calculated i terms of the eighbourig streamfuctio values by usig the method of Kettleborough et al. [4].

4 4 Applicatios: Flow betwee Parallel Plates To evaluate the accuracy of the method, the developig lamiar flow betwee two parallel plates was computed. A uiform velocity is imposed at the ilet sectio, ad a parabolic profile is expected to form at about 0.04 Re [5], where Re is the Reyolds umber referred to the width of the chael. Calculatios were made for a chael with legth-to-width ratio L/D=0. We first cosidered a 0* grid ad the followig gaussia widow: ω (M,M i ) = Exp ( -2.( r σ ) 2 ) ω (M,M i ) = 0 if r 2 > σ 2 (4) where r 2 = (x i -x) 2 + (y i -y) 2 We the used a 5*2 grid where the ode desity ratio is equal to four ( x=4 y). I this case, the previous widow selects more poits i the trasverse directio tha i the streamwise directio ad the method fails to coverge. We have cosequetly modified the gaussia widow by settig: r 2 = (x i-x) (y i -y) 2 widows ode calculatio poit,5 U*,4,3,2, CVFEM (0*) DAM (0*) CVFEM (5*2) DAM (5*2) X* 0 FIG. FIG. 2 Weightig fuctios Ceterlie velocity These two widows are schematically represeted o Fig.. The calculated ceterlie velocity as a fuctio of the distace from the ilet for a Reyolds umber of 00 is show o Fig.2. The

5 5 secod mesh gives better results as expected. The umerical value i the fully developed regio agrees well with the aalytical value of.5. I order to compare the DAM with a classical method, we have also reported o FIG.2, the results obtaied by usig a cotrol volume fiite elemet method (CVFEM)[6]. It appears that the results obtaied by the DAM are comparable to those obtaied by the well established CVFEM. Lamiar Flow over a Backward Facig Step L=2 h= H=.5 x FIG. 3 Backward facig step Of cocer here is the lamiar flow over a facig step (FIG.3). A fully developed parabolic lamiar flow is prescribed at the iflow sectio. The goverig equatios are odimesioalized by defiig X = x H-h ; Y = y H-h ; U = u U max ; V = v U max ; Re = U max.(h-h) ν where U max is the maximum velocity at the iflow sectio. Calculatios were performed util X=24 for Re=50 ad Re=50 o two differet grids (62* ad 86*6) by usig a gaussia widow. I Table, the preset calculated reatachmet legth is compared with the results obtaied i [7] with a (62*50) mesh. The results obtaied by usig the CVFEM o a (62*) grid are also reported o Table for compariso. I FIG.4, the axial velocity profiles are preseted. We ca see a rather good agreemet with the referece ad with the CVFEM results.

6 6 TABLE. Reyolds umber SOU (62*50) H 2 2 Ref[7] DAM (62*) DAM(86*6) X=.6 DAM (62*) CVFEM(62*) DAM (86*6) Reattachemet legh,xr/(h-h) 0 X U FIG. 4 Axial velocity profiles X Coclusio The diffuse approximatio method has bee applied to fluid flow i chaels ad compared with a cotrol volume fiite elemet method. Its accuracy has bee show o two test cases. Fially, we have show that it is better to use a elogated weightig fuctio whe the ode desity is greater i oe directio. REFERENCES. B. Nayroles, G. Touzot ad P. Villo : L'approximatio diffuse, C.R.Acad. Sci. Paris, t. 33, Série II, p , Y.Marechal,J.L. Coulomb, G. Meuier ad G. Touzot : Use of the diffuse elemet method for electromagetic field computatio., IEEE Trasactios o magetics, vol 29, 2 March H.Sadat ad C.Prax : Applicatio of the diffuse approximatio method to the umerical solutio of fluid flow ad heat trasfer problems, (to be published i Iteratioal Joural of Heat ad Mass Trasfer). 4. C.F.Kettlebourough, S.R.Hussai ad C.Prakash, Solutio of Fluid Flow Problems with the Vorticity-Streamfuctio Formulatio ad the Cotrol-Volume-Based Fiite-Elemet Method, Numerical Heat Trasfer, Part B,Vol.6, pp.3-58, H.Schlichtig,Boudary-Layer theory, 7th ed., McGraw-Hill, New York, H.Sadat, P.Salagac, Further Results for Lamiar Natural Covectio i a two dimesioal trapezoidal eclosure,numerical Heat Trasfer, Part A, vol.27,pp , M.C.Melaae, Nostaggered calculatio of lamiar ad turbulet flows usig curviliear oorthogoal coordiates, Numerical Heat Trasfer, Part A, vol.24,pp , 993

Streamfunction-Vorticity Formulation

Streamfunction-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 information

A numerical Technique Finite Volume Method for Solving Diffusion 2D Problem

A 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 information

A NEW CLASS OF 2-STEP RATIONAL MULTISTEP METHODS

A 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 information

Lecture 8: Solving the Heat, Laplace and Wave equations using finite difference methods

Lecture 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 information

Taylor expansion: Show that the TE of f(x)= sin(x) around. sin(x) = x - + 3! 5! L 7 & 8: MHD/ZAH

Taylor 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 information

METHOD OF FUNDAMENTAL SOLUTIONS FOR HELMHOLTZ EIGENVALUE PROBLEMS IN ELLIPTICAL DOMAINS

METHOD 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 information

A collocation method for singular integral equations with cosecant kernel via Semi-trigonometric interpolation

A 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 information

L 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!

L 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 information

62. 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 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 information

THE NUMERICAL SOLUTION OF THE NEWTONIAN FLUIDS FLOW DUE TO A STRETCHING CYLINDER BY SOR ITERATIVE PROCEDURE ABSTRACT

THE 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 information

Numerical Conformal Mapping via a Fredholm Integral Equation using Fourier Method ABSTRACT INTRODUCTION

Numerical 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 information

THE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0.

THE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0. THE SOLUTION OF NONLINEAR EQUATIONS f( ) = 0. Noliear Equatio Solvers Bracketig. Graphical. Aalytical Ope Methods Bisectio False Positio (Regula-Falsi) Fied poit iteratio Newto Raphso Secat The root of

More information

The Method of Least Squares. To understand least squares fitting of data.

The 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 information

Section A assesses the Units Numerical Analysis 1 and 2 Section B assesses the Unit Mathematics for Applied Mathematics

Section A assesses the Units Numerical Analysis 1 and 2 Section B assesses the Unit Mathematics for Applied Mathematics X0/70 NATIONAL QUALIFICATIONS 005 MONDAY, MAY.00 PM 4.00 PM APPLIED MATHEMATICS ADVANCED HIGHER Numerical Aalysis Read carefully. Calculators may be used i this paper.. Cadidates should aswer all questios.

More information

TR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT

TR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT TR/46 OCTOBER 974 THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION by A. TALBOT .. Itroductio. A problem i approximatio theory o which I have recetly worked [] required for its solutio a proof that the

More information

The 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 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 information

3. 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 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 information

Boundary layer problem on conveyor belt. Gabriella Bognár University of Miskolc 3515 Miskolc-Egyetemváros, Hungary

Boundary layer problem on conveyor belt. Gabriella Bognár University of Miskolc 3515 Miskolc-Egyetemváros, Hungary Boudary layer problem o coveyor belt Gabriella Bogár Uiversity of Miskolc 355 Miskolc-Egyetemváros, Hugary e-mail: matvbg@ui-miskolc.hu Abstract: A techologically importat source of the boudary layer pheomeo

More information

Analytic Continuation

Analytic Continuation Aalytic Cotiuatio The stadard example of this is give by Example Let h (z) = 1 + z + z 2 + z 3 +... kow to coverge oly for z < 1. I fact h (z) = 1/ (1 z) for such z. Yet H (z) = 1/ (1 z) is defied for

More information

Taylor polynomial solution of difference equation with constant coefficients via time scales calculus

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 information

Ray Optics Theory and Mode Theory. Dr. Mohammad Faisal Dept. of EEE, BUET

Ray Optics Theory and Mode Theory. Dr. Mohammad Faisal Dept. of EEE, BUET Ray Optics Theory ad Mode Theory Dr. Mohammad Faisal Dept. of, BUT Optical Fiber WG For light to be trasmitted through fiber core, i.e., for total iteral reflectio i medium, > Ray Theory Trasmissio Ray

More information

Math 113, Calculus II Winter 2007 Final Exam Solutions

Math 113, Calculus II Winter 2007 Final Exam Solutions Math, Calculus II Witer 7 Fial Exam Solutios (5 poits) Use the limit defiitio of the defiite itegral ad the sum formulas to compute x x + dx The check your aswer usig the Evaluatio Theorem Solutio: I this

More information

Chapter 10: Power Series

Chapter 10: Power Series Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because

More information

Numerical Method for Blasius Equation on an infinite Interval

Numerical 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 information

Numerical solutions of 2-D steady incompressible flow in a driven skewed cavity.

Numerical solutions of 2-D steady incompressible flow in a driven skewed cavity. Published i : ZAMM - Joural of Applied Mathematics ad Mechaics (007 ZAMM - Z. Agew. Math. Mech. 007; Vol 87: pp 377-39 Numerical solutios of -D steady icompressible flow i a drive skewed cavity. Erca Erturk

More information

A STUDY ON MHD BOUNDARY LAYER FLOW OVER A NONLINEAR STRETCHING SHEET USING IMPLICIT FINITE DIFFERENCE METHOD

A 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 information

Math 257: Finite difference methods

Math 257: Finite difference methods Math 257: Fiite differece methods 1 Fiite Differeces Remember the defiitio of a derivative f f(x + ) f(x) (x) = lim 0 Also recall Taylor s formula: (1) f(x + ) = f(x) + f (x) + 2 f (x) + 3 f (3) (x) +...

More information

Infinite Sequences and Series

Infinite 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 information

The Advection-Diffusion equation!

The Advection-Diffusion equation! ttp://www.d.edu/~gtryggva/cf-course/! Te Advectio-iffusio equatio! Grétar Tryggvaso! Sprig 3! Navier-Stokes equatios! Summary! u t + u u x + v u y = P ρ x + µ u + u ρ y Hyperbolic part! u x + v y = Elliptic

More information

Higher-order iterative methods by using Householder's method for solving certain nonlinear equations

Higher-order iterative methods by using Householder's method for solving certain nonlinear equations Math Sci Lett, No, 7- ( 7 Mathematical Sciece Letters A Iteratioal Joural http://dxdoiorg/785/msl/5 Higher-order iterative methods by usig Householder's method for solvig certai oliear equatios Waseem

More information

Analysis of composites with multiple rigid-line reinforcements by the BEM

Analysis of composites with multiple rigid-line reinforcements by the BEM Aalysis of composites with multiple rigid-lie reiforcemets by the BEM Piotr Fedeliski* Departmet of Stregth of Materials ad Computatioal Mechaics, Silesia Uiversity of Techology ul. Koarskiego 18A, 44-100

More information

Salmon: Lectures on partial differential equations. 3. First-order linear equations as the limiting case of second-order equations

Salmon: Lectures on partial differential equations. 3. First-order linear equations as the limiting case of second-order equations 3. First-order liear equatios as the limitig case of secod-order equatios We cosider the advectio-diffusio equatio (1) v = 2 o a bouded domai, with boudary coditios of prescribed. The coefficiets ( ) (2)

More information

Information-based Feature Selection

Information-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 information

Chapter 7: The z-transform. Chih-Wei Liu

Chapter 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 information

Explicit Group Methods in the Solution of the 2-D Convection-Diffusion Equations

Explicit Group Methods in the Solution of the 2-D Convection-Diffusion Equations Proceedigs of the World Cogress o Egieerig 00 Vol III WCE 00 Jue 0 - July 00 Lodo U.K. Explicit Group Methods i the Solutio of the -D Covectio-Diffusio Equatios a Kah Bee orhashidah Hj. M. Ali ad Choi-Hog

More information

Chapter 2: Numerical Methods

Chapter 2: Numerical Methods Chapter : Numerical Methods. Some Numerical Methods for st Order ODEs I this sectio, a summar of essetial features of umerical methods related to solutios of ordiar differetial equatios is give. I geeral,

More information

Exact Solutions for a Class of Nonlinear Singular Two-Point Boundary Value Problems: The Decomposition Method

Exact 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 information

6.3 Testing Series With Positive Terms

6.3 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 information

PC5215 Numerical Recipes with Applications - Review Problems

PC5215 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 information

Some properties of Boubaker polynomials and applications

Some properties of Boubaker polynomials and applications Some properties of Boubaker polyomials ad applicatios Gradimir V. Milovaović ad Duša Joksimović Citatio: AIP Cof. Proc. 179, 1050 (2012); doi: 10.1063/1.756326 View olie: http://dx.doi.org/10.1063/1.756326

More information

Chapter 9: Numerical Differentiation

Chapter 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 information

Orthogonal Gaussian Filters for Signal Processing

Orthogonal Gaussian Filters for Signal Processing Orthogoal Gaussia Filters for Sigal Processig Mark Mackezie ad Kiet Tieu Mechaical Egieerig Uiversity of Wollogog.S.W. Australia Abstract A Gaussia filter usig the Hermite orthoormal series of fuctios

More information

TMA4205 Numerical Linear Algebra. The Poisson problem in R 2 : diagonalization methods

TMA4205 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 information

Boundary Element Method (BEM)

Boundary Element Method (BEM) Boudary Elemet Method BEM Zora Ilievski Wedesday 8 th Jue 006 HG 6.96 TU/e Talk Overview The idea of BEM ad its advatages The D potetial problem Numerical implemetatio Idea of BEM 3 Idea of BEM 4 Advatages

More information

Sequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018

Sequences, 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 information

Finite Difference Derivations for Spreadsheet Modeling John C. Walton Modified: November 15, 2007 jcw

Finite Difference Derivations for Spreadsheet Modeling John C. Walton Modified: November 15, 2007 jcw Fiite Differece Derivatios for Spreadsheet Modelig Joh C. Walto Modified: November 15, 2007 jcw Figure 1. Suset with 11 swas o Little Platte Lake, Michiga. Page 1 Modificatio Date: November 15, 2007 Review

More information

The Phi Power Series

The Phi Power Series The Phi Power Series I did this work i about 0 years while poderig the relatioship betwee the golde mea ad the Madelbrot set. I have fially decided to make it available from my blog at http://semresearch.wordpress.com/.

More information

1 1 2 = show that: over variables x and y. [2 marks] Write down necessary conditions involving first and second-order partial derivatives for ( x0, y

1 1 2 = show that: over variables x and y. [2 marks] Write down necessary conditions involving first and second-order partial derivatives for ( x0, y Questio (a) A square matrix A= A is called positive defiite if the quadratic form waw > 0 for every o-zero vector w [Note: Here (.) deotes the traspose of a matrix or a vector]. Let 0 A = 0 = show that:

More information

For example suppose we divide the interval [0,2] into 5 equal subintervals of length

For example suppose we divide the interval [0,2] into 5 equal subintervals of length Math 1206 Calculus Sec 1: Estimatig with Fiite Sums Abbreviatios: wrt with respect to! for all! there exists! therefore Def defiitio Th m Theorem sol solutio! perpedicular iff or! if ad oly if pt poit

More information

The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled

The 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 information

ENGI Series Page 6-01

ENGI Series Page 6-01 ENGI 3425 6 Series Page 6-01 6. Series Cotets: 6.01 Sequeces; geeral term, limits, covergece 6.02 Series; summatio otatio, covergece, divergece test 6.03 Stadard Series; telescopig series, geometric series,

More information

Lecture 2: Finite Difference Methods in Heat Transfer

Lecture 2: Finite Difference Methods in Heat Transfer Lecture 2: Fiite Differece Methods i Heat Trasfer V.Vuorie Aalto Uiversity School of Egieerig Heat ad Mass Trasfer Course, Autum 2016 November 1 st 2017, Otaiemi ville.vuorie@aalto.fi Overview Part 1 (

More information

Monte Carlo Optimization to Solve a Two-Dimensional Inverse Heat Conduction Problem

Monte Carlo Optimization to Solve a Two-Dimensional Inverse Heat Conduction Problem Australia Joural of Basic Applied Scieces, 5(): 097-05, 0 ISSN 99-878 Mote Carlo Optimizatio to Solve a Two-Dimesioal Iverse Heat Coductio Problem M Ebrahimi Departmet of Mathematics, Karaj Brach, Islamic

More information

Discrete Orthogonal Moment Features Using Chebyshev Polynomials

Discrete Orthogonal Moment Features Using Chebyshev Polynomials Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity 75450 Malacca, Malaysia. Istitute of Mathematical

More information

Calculus with Analytic Geometry 2

Calculus with Analytic Geometry 2 Calculus with Aalytic Geometry Fial Eam Study Guide ad Sample Problems Solutios The date for the fial eam is December, 7, 4-6:3p.m. BU Note. The fial eam will cosist of eercises, ad some theoretical questios,

More information

Free Surface Hydrodynamics

Free Surface Hydrodynamics Water Sciece ad Egieerig Free Surface Hydrodyamics y A part of Module : Hydraulics ad Hydrology Water Sciece ad Egieerig Dr. Shreedhar Maskey Seior Lecturer UNESCO-IHE Istitute for Water Educatio S. Maskey

More information

Numerical Solutions of Second Order Boundary Value Problems by Galerkin Residual Method on Using Legendre Polynomials

Numerical 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 information

FFTs in Graphics and Vision. The Fast Fourier Transform

FFTs in Graphics and Vision. The Fast Fourier Transform FFTs i Graphics ad Visio The Fast Fourier Trasform 1 Outlie The FFT Algorithm Applicatios i 1D Multi-Dimesioal FFTs More Applicatios Real FFTs 2 Computatioal Complexity To compute the movig dot-product

More information

AE/ME 339 Computational Fluid Dynamics (CFD)

AE/ME 339 Computational Fluid Dynamics (CFD) AE/ME 339 Computatioal Fluid Dyamics (CFD 0//004 Topic0_PresCorr_ Computatioal Fluid Dyamics (AE/ME 339 Pressure Correctio Method The pressure correctio formula (6.8.4 Calculatio of p. Coservatio form

More information

NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE

NEW 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 information

MIDTERM 3 CALCULUS 2. Monday, December 3, :15 PM to 6:45 PM. Name PRACTICE EXAM SOLUTIONS

MIDTERM 3 CALCULUS 2. Monday, December 3, :15 PM to 6:45 PM. Name PRACTICE EXAM SOLUTIONS MIDTERM 3 CALCULUS MATH 300 FALL 08 Moday, December 3, 08 5:5 PM to 6:45 PM Name PRACTICE EXAM S Please aswer all of the questios, ad show your work. You must explai your aswers to get credit. You will

More information

Journal of Computational Physics 149, (1999) Article ID jcph , available online at

Journal of Computational Physics 149, (1999) Article ID jcph , available online at Joural of Computatioal Physics 149, 418 422 (1999) Article ID jcph.1998.6131, available olie at http://www.idealibrary.com o NOTE Defiig Wave Amplitude i Characteristic Boudary Coditios Key Words: Euler

More information

An efficient time integration method for extra-large eddy simulations

An efficient time integration method for extra-large eddy simulations A efficiet time itegratio method for extra-large eddy simulatios M.A. Scheibeler Departmet of Mathematics Master s Thesis A efficiet time itegratio method for extra-large eddy simulatios M.A. Scheibeler

More information

Singular Continuous Measures by Michael Pejic 5/14/10

Singular Continuous Measures by Michael Pejic 5/14/10 Sigular Cotiuous Measures by Michael Peic 5/4/0 Prelimiaries Give a set X, a σ-algebra o X is a collectio of subsets of X that cotais X ad ad is closed uder complemetatio ad coutable uios hece, coutable

More information

A widely used display of protein shapes is based on the coordinates of the alpha carbons - - C α

A widely used display of protein shapes is based on the coordinates of the alpha carbons - - C α Nice plottig of proteis: I A widely used display of protei shapes is based o the coordiates of the alpha carbos - - C α -s. The coordiates of the C α -s are coected by a cotiuous curve that roughly follows

More information

Improving the Localization of Eigenvalues for Complex Matrices

Improving 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 information

NUMERICAL METHOD FOR SINGULARLY PERTURBED DELAY PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

NUMERICAL METHOD FOR SINGULARLY PERTURBED DELAY PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS THERMAL SCIENCE, Year 07, Vol., No. 4, pp. 595-599 595 NUMERICAL METHOD FOR SINGULARLY PERTURBED DELAY PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS by Yula WANG *, Da TIAN, ad Zhiyua LI Departmet of Mathematics,

More information

Avoidance of numerical singularities in free vibration analysis of Euler-Bernoulli beams using Green functions

Avoidance of numerical singularities in free vibration analysis of Euler-Bernoulli beams using Green functions WSEAS TRASACTIOS o APPLIED ad THEORETICAL MECHAICS Goraka Štimac Ročević, Braimir Ročević, Ate Skoblar, Saji Braut Avoidace of umerical sigularities i free vibratio aalysis of Euler-Beroulli beams usig

More information

Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series

Comparison 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 information

MATH 31B: MIDTERM 2 REVIEW

MATH 31B: MIDTERM 2 REVIEW MATH 3B: MIDTERM REVIEW JOE HUGHES. Evaluate x (x ) (x 3).. Partial Fractios Solutio: The umerator has degree less tha the deomiator, so we ca use partial fractios. Write x (x ) (x 3) = A x + A (x ) +

More information

Math 113 Exam 3 Practice

Math 113 Exam 3 Practice Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you

More information

Numerical Study on MHD Flow And Heat Transfer With The Effect Of Microrotational Parameter In The Porous Medium

Numerical Study on MHD Flow And Heat Transfer With The Effect Of Microrotational Parameter In The Porous Medium IOSR Joural of Egieerig (IOSRJEN) ISSN (e): 5-3, ISSN (p): 78-879 Vol. 5, Issue 4 (April. 5), V PP 8-7 www.iosrje.org Numerical Study o MHD Flow Ad Heat rasfer With he Effect Of Microrotatioal Parameter

More information

Basics of Probability Theory (for Theory of Computation courses)

Basics 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 information

Chandrasekhar Type Algorithms. for the Riccati Equation of Lainiotis Filter

Chandrasekhar Type Algorithms. for the Riccati Equation of Lainiotis Filter Cotemporary Egieerig Scieces, Vol. 3, 00, o. 4, 9-00 Chadrasekhar ype Algorithms for the Riccati Equatio of Laiiotis Filter Nicholas Assimakis Departmet of Electroics echological Educatioal Istitute of

More information

Solutions to Homework 7

Solutions to Homework 7 Solutios to Homework 7 Due Wedesday, August 4, 004. Chapter 4.1) 3, 4, 9, 0, 7, 30. Chapter 4.) 4, 9, 10, 11, 1. Chapter 4.1. Solutio to problem 3. The sum has the form a 1 a + a 3 with a k = 1/k. Sice

More information

Stochastic Simulation

Stochastic Simulation Stochastic Simulatio 1 Itroductio Readig Assigmet: Read Chapter 1 of text. We shall itroduce may of the key issues to be discussed i this course via a couple of model problems. Model Problem 1 (Jackso

More information

Introduction to Optimization Techniques. How to Solve Equations

Introduction 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 information

A solid Foundation for q-appell Polynomials

A solid Foundation for q-appell Polynomials Advaces i Dyamical Systems ad Applicatios ISSN 0973-5321, Volume 10, Number 1, pp. 27 35 2015) http://campus.mst.edu/adsa A solid Foudatio for -Appell Polyomials Thomas Erst Uppsala Uiversity Departmet

More information

(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is

(c) Write, but do not evaluate, an integral expression for the volume of the solid generated when R is Calculus BC Fial Review Name: Revised 7 EXAM Date: Tuesday, May 9 Remiders:. Put ew batteries i your calculator. Make sure your calculator is i RADIAN mode.. Get a good ight s sleep. Eat breakfast. Brig:

More information

Numerical Solution of the First-Order Hyperbolic Partial Differential Equation with Point-Wise Advance

Numerical 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 information

On forward improvement iteration for stopping problems

On 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 information

l -State Solutions of a New Four-Parameter 1/r^2 Singular Radial Non-Conventional Potential via Asymptotic Iteration Method

l -State Solutions of a New Four-Parameter 1/r^2 Singular Radial Non-Conventional Potential via Asymptotic Iteration Method America Joural of Computatioal ad Applied Mathematics 8, 8(): 7-3 DOI:.593/j.ajcam.88. l -State Solutios of a New Four-Parameter /r^ Sigular Radial No-Covetioal Potetial via Asymptotic Iteratio Method

More information

SECOND-LAW ANALYSIS OF LAMINAR NON- NEWTONIAN GRAVITY-DRIVEN LIQUID FILM ALONG AN INCLINED HEATED PLATE WITH VISCOUS DISSIPATION EFFECT

SECOND-LAW ANALYSIS OF LAMINAR NON- NEWTONIAN GRAVITY-DRIVEN LIQUID FILM ALONG AN INCLINED HEATED PLATE WITH VISCOUS DISSIPATION EFFECT Brazilia Joural of Chemical Egieerig ISSN -663 Prited i Brazil www.abeq.org.br/bjche Vol. 6, No., pp. 7 -, April - Jue, 9 SECOND-LAW ANALSIS OF LAMINAR NON- NEWTONIAN GRAVIT-DRIVEN LIQUID FILM ALONG AN

More information

10-701/ Machine Learning Mid-term Exam Solution

10-701/ 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 information

NUMERICAL METHODS FOR SOLVING EQUATIONS

NUMERICAL METHODS FOR SOLVING EQUATIONS Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 1 of 11 M.K. HOME TUITION Mathematics Revisio Guides Level: GCSE Higher Tier NUMERICAL METHODS FOR SOLVING EQUATIONS Versio:. Date:

More information

PAPER : IIT-JAM 2010

PAPER : IIT-JAM 2010 MATHEMATICS-MA (CODE A) Q.-Q.5: Oly oe optio is correct for each questio. Each questio carries (+6) marks for correct aswer ad ( ) marks for icorrect aswer.. Which of the followig coditios does NOT esure

More information

17 Phonons and conduction electrons in solids (Hiroshi Matsuoka)

17 Phonons and conduction electrons in solids (Hiroshi Matsuoka) 7 Phoos ad coductio electros i solids Hiroshi Matsuoa I this chapter we will discuss a miimal microscopic model for phoos i a solid ad a miimal microscopic model for coductio electros i a simple metal.

More information

Solution of Differential Equation from the Transform Technique

Solution 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 information

The Graetz Problem. R. Shankar Subramanian. , which can be larger or smaller than T 0

The Graetz Problem. R. Shankar Subramanian. , which can be larger or smaller than T 0 The Graetz Problem R. Shakar Subramaia As a good model problem, we cosider steady state heat trasfer to fluid i steady flow through a tube. The fluid eters the tube at a temperature T ad ecouters a wall

More information

7 Sequences of real numbers

7 Sequences of real numbers 40 7 Sequeces of real umbers 7. Defiitios ad examples Defiitio 7... A sequece of real umbers is a real fuctio whose domai is the set N of atural umbers. Let s : N R be a sequece. The the values of s are

More information

Random Walks on Discrete and Continuous Circles. by Jeffrey S. Rosenthal School of Mathematics, University of Minnesota, Minneapolis, MN, U.S.A.

Random Walks on Discrete and Continuous Circles. by Jeffrey S. Rosenthal School of Mathematics, University of Minnesota, Minneapolis, MN, U.S.A. Radom Walks o Discrete ad Cotiuous Circles by Jeffrey S. Rosethal School of Mathematics, Uiversity of Miesota, Mieapolis, MN, U.S.A. 55455 (Appeared i Joural of Applied Probability 30 (1993), 780 789.)

More information

x x x 2x x N ( ) p NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS By Newton-Raphson formula

x x x 2x x N ( ) p NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS By Newton-Raphson formula NUMERICAL METHODS UNIT-I-SOLUTION OF EQUATIONS AND EIGENVALUE PROBLEMS. If g( is cotiuous i [a,b], te uder wat coditio te iterative (or iteratio metod = g( as a uique solutio i [a,b]? '( i [a,b].. Wat

More information

Some New Iterative Methods for Solving Nonlinear Equations

Some New Iterative Methods for Solving Nonlinear Equations World Applied Scieces Joural 0 (6): 870-874, 01 ISSN 1818-495 IDOSI Publicatios, 01 DOI: 10.589/idosi.wasj.01.0.06.830 Some New Iterative Methods for Solvig Noliear Equatios Muhammad Aslam Noor, Khalida

More information

Computational Fluid Dynamics. Lecture 3

Computational Fluid Dynamics. Lecture 3 Computatioal Fluid Dyamics Lecture 3 Discretizatio Cotiued. A fourth order approximatio to f x ca be foud usig Taylor Series. ( + ) + ( + ) + + ( ) + ( ) = a f x x b f x x c f x d f x x e f x x f x 0 0

More information

THE APPEARANCE OF FIBONACCI AND LUCAS NUMBERS IN THE SIMULATION OF ELECTRICAL POWER LINES SUPPLIED BY TWO SIDES

THE APPEARANCE OF FIBONACCI AND LUCAS NUMBERS IN THE SIMULATION OF ELECTRICAL POWER LINES SUPPLIED BY TWO SIDES THE APPEARANCE OF FIBONACCI AND LUCAS NUMBERS IN THE SIMULATION OF ELECTRICAL POWER LINES SUPPLIED BY TWO SIDES Giuseppe Ferri Dipartimeto di Ihgegeria Elettrica-FacoM di Igegeria, Uiversita di L'Aquila

More information

We are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n

We 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 information

k- modeling using Modified Nodal Integral Method

k- modeling using Modified Nodal Integral Method Proceedigs of ICAPP 8 Aaheim, CA USA, Jue 8-, 8 - modelig usig Modified Nodal Itegral Method Sueet Sigh, Rizwa-uddi Departmet of Nuclear Plasma ad Radiological Egieerig Uiversity of Illiois at Urbaa-Champaig

More information

FINALTERM EXAMINATION Fall 9 Calculus & Aalytical Geometry-I Questio No: ( Mars: ) - Please choose oe Let f ( x) is a fuctio such that as x approaches a real umber a, either from left or right-had-side,

More information

IN many scientific and engineering applications, one often

IN many scientific and engineering applications, one often INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS, VOL 3, NO, FEBRUARY 07 5 Secod Degree Refiemet Jacobi Iteratio Method for Solvig System of Liear Equatio Tesfaye Kebede Abstract Several

More information

( ) (( ) ) ANSWERS TO EXERCISES IN APPENDIX B. Section B.1 VECTORS AND SETS. Exercise B.1-1: Convex sets. are convex, , hence. and. (a) Let.

( ) (( ) ) ANSWERS TO EXERCISES IN APPENDIX B. Section B.1 VECTORS AND SETS. Exercise B.1-1: Convex sets. are convex, , hence. and. (a) Let. Joh Riley 8 Jue 03 ANSWERS TO EXERCISES IN APPENDIX B Sectio B VECTORS AND SETS Exercise B-: Covex sets (a) Let 0 x, x X, X, hece 0 x, x X ad 0 x, x X Sice X ad X are covex, x X ad x X The x X X, which

More information