Topic02_PDE. 8/29/2006 topic02_pde 1. Computational Fluid Dynamics (AE/ME 339) MAE Dept., UMR
|
|
- Felicity Wright
- 5 years ago
- Views:
Transcription
1 MEAE 9 Computational Fluid Dnamics Topic0_ topic0_ Partial Dierential Equations () (CLW: 7., 7., 7.4) s can be linear or nonlinear Order : Determined b the order o the highest derivative. Linear, nd order s are classiied as the elliptic, hperbolic Parabolic tpe. Example: u u u u A A A B x z x u u + B + B + C u + D = z topic0_
2 Coeicients A, A, A Ma be +, - or zero. u is the dependent variable and x,, z are the independent variables. Note that we do not have an cross-derivative terms. Classiication: Elliptic: A, A, A are non-zero and have the same sign, then is o the elliptic tpe topic0_ Hperbolic: A, A, A are non-zero and o mixed sign, the is hperbolic Parabolic: I one o A, A, A, sa A is zero and the rest are o the same sign, and i B is non-zero, the is parabolic. Since A i, B i, C and D ma be unctions o x, and z, the classiication ma depend on position in space. In man CFD problems, one o the independent variables will be time and the rest will be space coordinates such as x,, z or or transormed variables such as ξ, η, ζ topic0_ 4
3 Two-Dimensional Examples Elliptic u x u + = 0 Hperbolic Parabolic u u u = + t x = + t x u u u topic0_ 5 Numerical solution o requires a inite number o points to Descretize the equations. Examples: See Figure in the next slide topic0_ 6
4 89006 topic0_ 7 Solution requires initial and boundar conditions depending on the Problem. Indices i, j, n can be used to label the nodes in x,, t directions(see ig.) I the origin has i=0, j=0 and n = 0, then the node i, j, n has coordinates i x, j, n t, where x,, t are the uniorm intervals between nodes along x,, t coordinate directions. Let uxt (,,) u be the exact solution o the and ijn,, be the approximations to be determined at each grid point. v i, j, n topic0_ 8 4
5 The derivatives o the original are approximated using the smbol v i, j, n and the discretization intervals x,, t. The procedure leads to a set o algebraic equations o which are then solved. Fine grids can be used to obtain solutions u i, j, n. Examples o s common in engineering v i, j, n v i, j, n that are close to.unstead heat conduction equation D orm: T T k = c p ρ x x t topic0_ 9 T - temperature k - thermal conductivit ρ - densit c p - speciic heat I k is a constant, the equation becomes Where α k c ρ p T α = x T t is the thermal diusivit topic0_ 0 5
6 In CFD, normalization o variables are oten used to improve the solution (or proper scaling o the variables). Let ξ = x αt, τ = L L or heat conduction in a rod o length L. Figure. The then becomes T = ξ T τ It is also possible to non-dimensionalize the dependent variable T topic0_ Talor s Expansion. d Let ( x, ). = h ( ) ( ) ( x + h) = ( x ) + h x, x + x, x! h x x ( 0 ( 0) ) +, +...! Where d (, ) = (, ) x x x x Talor series is as ollows. () () d x x x x (, ) = (, ) topic0_ 6
7 The higher order derivatives in Eq.() can be determined b Dierentiating Eq.() b chain rule. i. e., d d = +. x Example : (x,) is a unction o x alone. d = x topic0_ ( x ) ( x ) x, = x, =, = n ( x ), = 0 ( x0 0) ( x ) x0, 0 = x0, =, = n ( x ), = topic0_ 4 7
8 The unction at the neighboring point (x = x0+h) becomes x + h = x + hx + h x Example : (x,) is a unction o alone. d ( x, ) = = h topic0_ 5 x d x, = + = 0 + = 4 Similarl x, = 8 x, = ( + ) n x, = n topic0_ 6 8
9 h h x h x h x x x!! ( + ) = ( h) ( h) = ( x0 ) + h !! topic0_ 7 Talor series can also be used or non-linear higher order equations. Example: d d d x + = 0 d = ( x, ) = + x = x = x topic0_ 8 9
10 iv = ( + x + x ) iv = 4 x x Consider the initial conditions At x = = = 0, (0), (0) topic0_ 9 (0) = (0) 0 (0) = (0) = (0) (0) (0) = = iv (0) (0) 4 = + 4 = = topic0_ 0 0
11 Since Talor series gives h h h k k k k k = ! +! +! + For k = 0, becomes 4 h h h = h Letting h=0., we get = 0. + = topic0_ Calculation o Need Calculate w.r.t. h. b dierentiating the expression or h = h h = =.097 iv,, etc. can now be calculated as beore and topic0_ can be obtained.
Boundary-Fitted Coordinates!
Computational Fluid Dnamics http:wwwndedu~gtrggvacfdcourse Computational Fluid Dnamics Computational Methods or Domains with Comple BoundariesI Grétar Trggvason Spring For most engineering problems it
More informationComputational Methods for Domains with! Complex Boundaries-I!
http://www.nd.edu/~gtrggva/cfd-course/ Computational Methods or Domains with Comple Boundaries-I Grétar Trggvason Spring For most engineering problems it is necessar to deal with comple geometries, consisting
More informationPaper-II Chapter- TS-equation, Maxwell s equation. z = z(x, y) dz = dx + dz = Mdx + Ndy. dy Now. = 2 z
aper-ii Chapter- S-equation, Maxwell s equation Let heorem: Condition o exact dierential: Where M Hence, z x dz dx and N Q. Derive Maxwell s equations z x z zx, z dx + dz Mdx + Nd z d Now 2 z x M N x x
More informationFLUID MECHANICS. Lecture 7 Exact solutions
FLID MECHANICS Lecture 7 Eact solutions 1 Scope o Lecture To present solutions or a ew representative laminar boundary layers where the boundary conditions enable eact analytical solutions to be obtained.
More informationQuadratic Functions. The graph of the function shifts right 3. The graph of the function shifts left 3.
Quadratic Functions The translation o a unction is simpl the shiting o a unction. In this section, or the most part, we will be graphing various unctions b means o shiting the parent unction. We will go
More informationAE/ME 339. K. M. Isaac Professor of Aerospace Engineering. December 21, 2001 topic13_grid_generation 1
AE/ME 339 Professor of Aerospace Engineering December 21, 2001 topic13_grid_generation 1 The basic idea behind grid generation is the creation of the transformation laws between the phsical space and the
More informationSaturday X-tra X-Sheet: 8. Inverses and Functions
Saturda X-tra X-Sheet: 8 Inverses and Functions Ke Concepts In this session we will ocus on summarising what ou need to know about: How to ind an inverse. How to sketch the inverse o a graph. How to restrict
More informationPhysics 5153 Classical Mechanics. Solution by Quadrature-1
October 14, 003 11:47:49 1 Introduction Physics 5153 Classical Mechanics Solution by Quadrature In the previous lectures, we have reduced the number o eective degrees o reedom that are needed to solve
More informationNON-SIMILAR SOLUTIONS FOR NATURAL CONVECTION FROM A MOVING VERTICAL PLATE WITH A CONVECTIVE THERMAL BOUNDARY CONDITION
NON-SIMILAR SOLUTIONS FOR NATURAL CONVECTION FROM A MOVING VERTICAL PLATE WITH A CONVECTIVE THERMAL BOUNDARY CONDITION by Asterios Pantokratoras School o Engineering, Democritus University o Thrace, 67100
More informationSolving Partial Differential Equations Numerically. Miklós Bergou with: Gary Miller, David Cardoze, Todd Phillips, Mark Olah
Solving Partial Dierential Equations Numerically Miklós Bergou with: Gary Miller, David Cardoze, Todd Phillips, Mark Olah Overview What are partial dierential equations? How do we solve them? (Example)
More informationSliding Mode Control and Feedback Linearization for Non-regular Systems
Sliding Mode Control and Feedback Linearization or Non-regular Systems Fu Zhang Benito R. Fernández Pieter J. Mosterman Timothy Josserand The MathWorks, Inc. Natick, MA, 01760 University o Texas at Austin,
More information7. Two Random Variables
7. Two Random Variables In man eeriments the observations are eressible not as a single quantit but as a amil o quantities. or eamle to record the height and weight o each erson in a communit or the number
More information1 Relative degree and local normal forms
THE ZERO DYNAMICS OF A NONLINEAR SYSTEM 1 Relative degree and local normal orms The purpose o this Section is to show how single-input single-output nonlinear systems can be locally given, by means o a
More informationFeedback Linearization
Feedback Linearization Peter Al Hokayem and Eduardo Gallestey May 14, 2015 1 Introduction Consider a class o single-input-single-output (SISO) nonlinear systems o the orm ẋ = (x) + g(x)u (1) y = h(x) (2)
More informationReceived: 30 July 2017; Accepted: 29 September 2017; Published: 8 October 2017
mathematics Article Least-Squares Solution o Linear Dierential Equations Daniele Mortari ID Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA; mortari@tamu.edu; Tel.: +1-979-845-734
More informationNatural convection in a vertical strip immersed in a porous medium
European Journal o Mechanics B/Fluids 22 (2003) 545 553 Natural convection in a vertical strip immersed in a porous medium L. Martínez-Suástegui a,c.treviño b,,f.méndez a a Facultad de Ingeniería, UNAM,
More information1/f noise from the nonlinear transformations of the variables
Modern Physics Letters B Vol. 9, No. 34 (015) 15503 (6 pages) c World Scientiic Publishing Company DOI: 10.114/S0179849155031 1/ noise rom the nonlinear transormations o the variables Bronislovas Kaulakys,
More informationMath 754 Chapter III: Fiber bundles. Classifying spaces. Applications
Math 754 Chapter III: Fiber bundles. Classiying spaces. Applications Laurențiu Maxim Department o Mathematics University o Wisconsin maxim@math.wisc.edu April 18, 2018 Contents 1 Fiber bundles 2 2 Principle
More informationEnhancement Using Local Histogram
Enhancement Using Local Histogram Used to enhance details over small portions o the image. Deine a square or rectangular neighborhood hose center moves rom piel to piel. Compute local histogram based on
More informationNumerical Solution of Ordinary Differential Equations in Fluctuationlessness Theorem Perspective
Numerical Solution o Ordinary Dierential Equations in Fluctuationlessness Theorem Perspective NEJLA ALTAY Bahçeşehir University Faculty o Arts and Sciences Beşiktaş, İstanbul TÜRKİYE TURKEY METİN DEMİRALP
More informationLecture : Feedback Linearization
ecture : Feedbac inearization Niola Misovic, dipl ing and Pro Zoran Vuic June 29 Summary: This document ollows the lectures on eedbac linearization tought at the University o Zagreb, Faculty o Electrical
More informationImplicit Second Derivative Hybrid Linear Multistep Method with Nested Predictors for Ordinary Differential Equations
American Scientiic Research Journal or Engineering, Technolog, and Sciences (ASRJETS) ISSN (Print) -44, ISSN (Online) -44 Global Societ o Scientiic Research and Researchers http://asretsournal.org/ Implicit
More informationFluctuationlessness Theorem and its Application to Boundary Value Problems of ODEs
Fluctuationlessness Theorem and its Application to Boundary Value Problems o ODEs NEJLA ALTAY İstanbul Technical University Inormatics Institute Maslak, 34469, İstanbul TÜRKİYE TURKEY) nejla@be.itu.edu.tr
More informationLecture Outline. Basics of Spatial Filtering Smoothing Spatial Filters. Sharpening Spatial Filters
1 Lecture Outline Basics o Spatial Filtering Smoothing Spatial Filters Averaging ilters Order-Statistics ilters Sharpening Spatial Filters Laplacian ilters High-boost ilters Gradient Masks Combining Spatial
More informationComptes rendus de l Academie bulgare des Sciences, Tome 59, 4, 2006, p POSITIVE DEFINITE RANDOM MATRICES. Evelina Veleva
Comtes rendus de l Academie bulgare des ciences Tome 59 4 6 353 36 POITIVE DEFINITE RANDOM MATRICE Evelina Veleva Abstract: The aer begins with necessary and suicient conditions or ositive deiniteness
More informationChapter 6: Functions with severable variables and Partial Derivatives:
Chapter 6: Functions with severable variables and Partial Derivatives: Functions o several variables: A unction involving more than one variable is called unction with severable variables. Eamples: y (,
More information1036: Probability & Statistics
1036: Probabilit & Statistics Lecture 4 Mathematical pectation Prob. & Stat. Lecture04 - mathematical epectation cwliu@twins.ee.nctu.edu.tw 4-1 Mean o a Random Variable Let be a random variable with probabilit
More informationCHAPTER-III CONVECTION IN A POROUS MEDIUM WITH EFFECT OF MAGNETIC FIELD, VARIABLE FLUID PROPERTIES AND VARYING WALL TEMPERATURE
CHAPER-III CONVECION IN A POROUS MEDIUM WIH EFFEC OF MAGNEIC FIELD, VARIABLE FLUID PROPERIES AND VARYING WALL EMPERAURE 3.1. INRODUCION Heat transer studies in porous media ind applications in several
More information. This is the Basic Chain Rule. x dt y dt z dt Chain Rule in this context.
Math 18.0A Gradients, Chain Rule, Implicit Dierentiation, igher Order Derivatives These notes ocus on our things: (a) the application o gradients to ind normal vectors to curves suraces; (b) the generaliation
More information8.4 Inverse Functions
Section 8. Inverse Functions 803 8. Inverse Functions As we saw in the last section, in order to solve application problems involving eponential unctions, we will need to be able to solve eponential equations
More information39.1 Gradually Varied Unsteady Flow
39.1 Gradually Varied Unsteady Flow Gradually varied unsteady low occurs when the low variables such as the low depth and velocity do not change rapidly in time and space. Such lows are very common in
More information9.1 The Square Root Function
Section 9.1 The Square Root Function 869 9.1 The Square Root Function In this section we turn our attention to the square root unction, the unction deined b the equation () =. (1) We begin the section
More informationCurve Sketching. The process of curve sketching can be performed in the following steps:
Curve Sketching So ar you have learned how to ind st and nd derivatives o unctions and use these derivatives to determine where a unction is:. Increasing/decreasing. Relative extrema 3. Concavity 4. Points
More informationMath-3 Lesson 1-4. Review: Cube, Cube Root, and Exponential Functions
Math- Lesson -4 Review: Cube, Cube Root, and Eponential Functions Quiz - Graph (no calculator):. y. y ( ) 4. y What is a power? vocabulary Power: An epression ormed by repeated Multiplication o the same
More informationApplication of Wavelet Transform Modulus Maxima in Raman Distributed Temperature Sensors
PHOTONIC SENSORS / Vol. 4, No. 2, 2014: 142 146 Application o Wavelet Transorm Modulus Maxima in Raman Distributed Temperature Sensors Zongliang WANG, Jun CHANG *, Sasa ZHANG, Sha LUO, Cuanwu JIA, Boning
More informationRoot Arrangements of Hyperbolic Polynomial-like Functions
Root Arrangements o Hyperbolic Polynomial-like Functions Vladimir Petrov KOSTOV Université de Nice Laboratoire de Mathématiques Parc Valrose 06108 Nice Cedex France kostov@mathunicer Received: March, 005
More informationAsymptotics of integrals
December 9, Asymptotics o integrals Paul Garrett garrett@math.umn.e http://www.math.umn.e/ garrett/ [This document is http://www.math.umn.e/ garrett/m/v/basic asymptotics.pd]. Heuristic or the main term
More informationControlling the Heat Flux Distribution by Changing the Thickness of Heated Wall
J. Basic. Appl. Sci. Res., 2(7)7270-7275, 2012 2012, TextRoad Publication ISSN 2090-4304 Journal o Basic and Applied Scientiic Research www.textroad.com Controlling the Heat Flux Distribution by Changing
More informationSUBJECT:ENGINEERING MATHEMATICS-I SUBJECT CODE :SMT1101 UNIT III FUNCTIONS OF SEVERAL VARIABLES. Jacobians
SUBJECT:ENGINEERING MATHEMATICS-I SUBJECT CODE :SMT0 UNIT III FUNCTIONS OF SEVERAL VARIABLES Jacobians Changing ariable is something e come across er oten in Integration There are man reasons or changing
More informationAsymptotics of integrals
October 3, 4 Asymptotics o integrals Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/complex/notes 4-5/4c basic asymptotics.pd].
More informationCHAPTER 1: INTRODUCTION. 1.1 Inverse Theory: What It Is and What It Does
Geosciences 567: CHAPTER (RR/GZ) CHAPTER : INTRODUCTION Inverse Theory: What It Is and What It Does Inverse theory, at least as I choose to deine it, is the ine art o estimating model parameters rom data
More informationModule 1 : The equation of continuity. Lecture 4: Fourier s Law of Heat Conduction
1 Module 1 : The equation of continuit Lecture 4: Fourier s Law of Heat Conduction NPTEL, IIT Kharagpur, Prof. Saikat Chakrabort, Department of Chemical Engineering Fourier s Law of Heat Conduction According
More informationMath Review and Lessons in Calculus
Math Review and Lessons in Calculus Agenda Rules o Eponents Functions Inverses Limits Calculus Rules o Eponents 0 Zero Eponent Rule a * b ab Product Rule * 3 5 a / b a-b Quotient Rule 5 / 3 -a / a Negative
More information( x) f = where P and Q are polynomials.
9.8 Graphing Rational Functions Lets begin with a deinition. Deinition: Rational Function A rational unction is a unction o the orm ( ) ( ) ( ) P where P and Q are polynomials. Q An eample o a simple rational
More informationAnalysis of Non-Thermal Equilibrium in Porous Media
Analysis o Non-Thermal Equilibrium in Porous Media A. Nouri-Borujerdi, M. Nazari 1 School o Mechanical Engineering, Shari University o Technology P.O Box 11365-9567, Tehran, Iran E-mail: anouri@shari.edu
More informationClassification of effective GKM graphs with combinatorial type K 4
Classiication o eective GKM graphs with combinatorial type K 4 Shintarô Kuroki Department o Applied Mathematics, Faculty o Science, Okayama Uniervsity o Science, 1-1 Ridai-cho Kita-ku, Okayama 700-0005,
More informationFlip-Flop Functions KEY
For each rational unction, list the zeros o the polynomials in the numerator and denominator. Then, using a calculator, sketch the graph in a window o [-5.75, 6] by [-5, 5], and provide an end behavior
More information1 Categories, Functors, and Natural Transformations. Discrete categories. A category is discrete when every arrow is an identity.
MacLane: Categories or Working Mathematician 1 Categories, Functors, and Natural Transormations 1.1 Axioms or Categories 1.2 Categories Discrete categories. A category is discrete when every arrow is an
More informationMathematical Notation Math Calculus & Analytic Geometry III
Mathematical Notation Math 221 - alculus & Analytic Geometry III Use Word or WordPerect to recreate the ollowing documents. Each article is worth 10 points and should be emailed to the instructor at james@richland.edu.
More informationChapter 8: MULTIPLE CONTINUOUS RANDOM VARIABLES
Charles Boncelet Probabilit Statistics and Random Signals" Oord Uniersit Press 06. ISBN: 978-0-9-0005-0 Chapter 8: MULTIPLE CONTINUOUS RANDOM VARIABLES Sections 8. Joint Densities and Distribution unctions
More informationMath-3 Lesson 8-5. Unit 4 review: a) Compositions of functions. b) Linear combinations of functions. c) Inverse Functions. d) Quadratic Inequalities
Math- Lesson 8-5 Unit 4 review: a) Compositions o unctions b) Linear combinations o unctions c) Inverse Functions d) Quadratic Inequalities e) Rational Inequalities 1. Is the ollowing relation a unction
More informationIncreasing and Decreasing Functions and the First Derivative Test. Increasing and Decreasing Functions. Video
SECTION and Decreasing Functions and the First Derivative Test 79 Section and Decreasing Functions and the First Derivative Test Determine intervals on which a unction is increasing or decreasing Appl
More informationFeedback Linearization Lectures delivered at IIT-Kanpur, TEQIP program, September 2016.
Feedback Linearization Lectures delivered at IIT-Kanpur, TEQIP program, September 216 Ravi N Banavar banavar@iitbacin September 24, 216 These notes are based on my readings o the two books Nonlinear Control
More informationLocal Features (contd.)
Motivation Local Features (contd.) Readings: Mikolajczyk and Schmid; F&P Ch 10 Feature points are used also or: Image alignment (homography, undamental matrix) 3D reconstruction Motion tracking Object
More informationThe Ascent Trajectory Optimization of Two-Stage-To-Orbit Aerospace Plane Based on Pseudospectral Method
Available online at www.sciencedirect.com ScienceDirect Procedia Engineering 00 (014) 000 000 www.elsevier.com/locate/procedia APISAT014, 014 Asia-Paciic International Symposium on Aerospace Technology,
More informationAsymptote. 2 Problems 2 Methods
Asymptote Problems Methods Problems Assume we have the ollowing transer unction which has a zero at =, a pole at = and a pole at =. We are going to look at two problems: problem is where >> and problem
More informationCategories and Natural Transformations
Categories and Natural Transormations Ethan Jerzak 17 August 2007 1 Introduction The motivation or studying Category Theory is to ormalise the underlying similarities between a broad range o mathematical
More informationIntroduction to Computational Fluid Dynamics
AML2506 Biomechanics and Flow Simulation Day Introduction to Computational Fluid Dynamics Session Speaker Dr. M. D. Deshpande M.S. Ramaiah School of Advanced Studies - Bangalore 1 Session Objectives At
More informationA Function of Two Random Variables
akultät Inormatik Institut ür Sstemarchitektur Proessur Rechnernete A unction o Two Random Variables Waltenegus Dargie Slides are based on the book: A. Papoulis and S.U. Pillai "Probabilit random variables
More informationReview of Prerequisite Skills for Unit # 2 (Derivatives) U2L2: Sec.2.1 The Derivative Function
UL1: Review o Prerequisite Skills or Unit # (Derivatives) Working with the properties o exponents Simpliying radical expressions Finding the slopes o parallel and perpendicular lines Simpliying rational
More informationRoberto s Notes on Differential Calculus Chapter 8: Graphical analysis Section 1. Extreme points
Roberto s Notes on Dierential Calculus Chapter 8: Graphical analysis Section 1 Extreme points What you need to know already: How to solve basic algebraic and trigonometric equations. All basic techniques
More informationEstimation of the Vehicle Sideslip Angle by Means of the State Dependent Riccati Equation Technique
Proceedings o the World Congress on Engineering 7 Vol II WCE 7, Jul 5-7, 7, London, U.K. Estimation o the Vehicle Sideslip Angle b Means o the State Dependent Riccati Equation Technique S. Strano, M. Terzo
More informationModule 1: Introduction to Finite Difference Method and Fundamentals of CFD Lecture 1: Finite Difference Method
file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture1/1_1.htm 1 of 1 6/19/2012 4:29 PM The Lecture deals with: Classification of Partial Differential Equations Boundary and Initial Conditions Finite Differences
More informationA Wavelet Collocation Method for Optimal Control. Problems
A Wavelet ollocation ethod or Optimal ontrol Problems Ran Dai * and John E. ochran Jr. Abstract A Haar wavelet technique is discussed as a method or discretizing the nonlinear sstem equations or optimal
More informationOE4625 Dredge Pumps and Slurry Transport. Vaclav Matousek October 13, 2004
OE465 Vaclav Matousek October 13, 004 1 Dredge Vermelding Pumps onderdeel and Slurry organisatie Transport OE465 Vaclav Matousek October 13, 004 Dredge Vermelding Pumps onderdeel and Slurry organisatie
More informationNon-newtonian Rabinowitsch Fluid Effects on the Lubrication Performances of Sine Film Thrust Bearings
International Journal o Mechanical Engineering and Applications 7; 5(): 6-67 http://www.sciencepublishinggroup.com/j/ijmea doi:.648/j.ijmea.75.4 ISSN: -X (Print); ISSN: -48 (Online) Non-newtonian Rabinowitsch
More informationNEWTONS LAWS OF MOTION AND FRICTIONS STRAIGHT LINES
EWTOS LAWS O OTIO AD RICTIOS STRAIGHT LIES ITRODUCTIO In this chapter, we shall study the motion o bodies along with the causes o their motion assuming that mass is constant. In addition, we are going
More informationMAGNETOHYDRODYNAMIC GO-WATER NANOFLUID FLOW AND HEAT TRANSFER BETWEEN TWO PARALLEL MOVING DISKS
THERMAL SCIENCE: Year 8, Vol., No. B, pp. 383-39 383 MAGNETOHYDRODYNAMIC GO-WATER NANOFLUID FLOW AND HEAT TRANSFER BETWEEN TWO PARALLEL MOVING DISKS Introduction by Mohammadreza AZIMI and Rouzbeh RIAZI
More informationLecture 25: Heat and The 1st Law of Thermodynamics Prof. WAN, Xin
General Physics I Lecture 5: Heat and he 1st Law o hermodynamics Pro. WAN, Xin xinwan@zju.edu.cn http://zimp.zju.edu.cn/~xinwan/ Latent Heat in Phase Changes Latent Heat he latent heat o vaporization or
More informationA NOVEL METHOD OF INTERPOLATION AND EXTRAPOLATION OF FUNCTIONS BY A LINEAR INITIAL VALUE PROBLEM
A OVEL METHOD OF ITERPOLATIO AD EXTRAPOLATIO OF FUCTIOS BY A LIEAR IITIAL VALUE PROBLEM Michael Shatalov Sensor Science and Technolog o CSIR Manuacturing and Materials, P.O.Bo 395, Pretoria, CSIR and Department
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University The Implicit Schemes for the Model Problem The Crank-Nicolson scheme and θ-scheme
More informationUltra Fast Calculation of Temperature Profiles of VLSI ICs in Thermal Packages Considering Parameter Variations
Ultra Fast Calculation o Temperature Proiles o VLSI ICs in Thermal Packages Considering Parameter Variations Je-Hyoung Park, Virginia Martín Hériz, Ali Shakouri, and Sung-Mo Kang Dept. o Electrical Engineering,
More informationChapter 3: Image Enhancement in the. Office room : 841
Chapter 3: Image Enhancement in the Spatial Domain Lecturer: Jianbing Shen Email : shenjianbing@bit.edu.cn Oice room : 841 http://cs.bit.edu.cn/shenjianbing cn/shenjianbing Principle Objective o Enhancement
More informationFunctions of Several Variables
Chapter 1 Functions of Several Variables 1.1 Introduction A real valued function of n variables is a function f : R, where the domain is a subset of R n. So: for each ( 1,,..., n ) in, the value of f is
More informationApplication of Method of Lines on the Bifurcation Analysis of a Horizontal Rijke Tube
th ASPACC July 9, 5 Beijing, China Application o Method o Lines on the Biurcation Analysis o a Horizontal Rijke Tube Xiaochuan Yang, Ali Turan, Adel Nasser, Shenghui Lei School o Mechanical, Aerospace
More informationInternational Journal of Scientific Research and Reviews
Research article Available online www.ijsrr.org ISSN: 2279 543 International Journal of Scientific Research and Reviews Soret effect on Magneto Hdro Dnamic convective immiscible Fluid flow in a Horizontal
More informationCONVECTIVE HEAT TRANSFER CHARACTERISTICS OF NANOFLUIDS. Convective heat transfer analysis of nanofluid flowing inside a
Chapter 4 CONVECTIVE HEAT TRANSFER CHARACTERISTICS OF NANOFLUIDS Convective heat transer analysis o nanoluid lowing inside a straight tube o circular cross-section under laminar and turbulent conditions
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Numerical Methods for Partial Differential Equations Finite Difference Methods
More information3. Several Random Variables
. Several Random Variables. Two Random Variables. Conditional Probabilit--Revisited. Statistical Independence.4 Correlation between Random Variables. Densit unction o the Sum o Two Random Variables. Probabilit
More information0,0 B 5,0 C 0, 4 3,5. y x. Recitation Worksheet 1A. 1. Plot these points in the xy plane: A
Math 13 Recitation Worksheet 1A 1 Plot these points in the y plane: A 0,0 B 5,0 C 0, 4 D 3,5 Without using a calculator, sketch a graph o each o these in the y plane: A y B 3 Consider the unction a Evaluate
More informationy2 = 0. Show that u = e2xsin(2y) satisfies Laplace's equation.
Review 1 1) State the largest possible domain o deinition or the unction (, ) = 3 - ) Determine the largest set o points in the -plane on which (, ) = sin-1( - ) deines a continuous unction 3) Find the
More informationPartial Averaging of Fuzzy Differential Equations with Maxima
Advances in Dynamical Systems and Applications ISSN 973-5321, Volume 6, Number 2, pp. 199 27 211 http://campus.mst.edu/adsa Partial Averaging o Fuzzy Dierential Equations with Maxima Olga Kichmarenko and
More informationDefinition: Let f(x) be a function of one variable with continuous derivatives of all orders at a the point x 0, then the series.
2.4 Local properties o unctions o several variables In this section we will learn how to address three kinds o problems which are o great importance in the ield o applied mathematics: how to obtain the
More informationSWEEP METHOD IN ANALYSIS OPTIMAL CONTROL FOR RENDEZ-VOUS PROBLEMS
J. Appl. Math. & Computing Vol. 23(2007), No. 1-2, pp. 243-256 Website: http://jamc.net SWEEP METHOD IN ANALYSIS OPTIMAL CONTROL FOR RENDEZ-VOUS PROBLEMS MIHAI POPESCU Abstract. This paper deals with determining
More informationModern Ab-initio Calculations Based on Tomas-Fermi-Dirac Theory with High-Pressure Environment
American Journal o Quantum Chemistry and Molecular Spectroscopy 016; 1(1): 7-1 http://www.sciencepublishinggroup.com/j/ajqcms doi: 10.11648/j.ajqcms.0160101.1 Modern Ab-initio Calculations Based on Tomas-Fermi-Dirac
More informationExtreme Values of Functions
Extreme Values o Functions When we are using mathematics to model the physical world in which we live, we oten express observed physical quantities in terms o variables. Then, unctions are used to describe
More informationProbabilistic Engineering Mechanics
Probabilistic Engineering Mechanics 6 (11) 174 181 Contents lists available at ScienceDirect Probabilistic Engineering Mechanics journal homepage: www.elsevier.com/locate/probengmech Variability response
More informationy,z the subscript y, z indicating that the variables y and z are kept constant. The second partial differential with respect to x is written x 2 y,z
8 Partial dierentials I a unction depends on more than one variable, its rate o change with respect to one o the variables can be determined keeping the others ied The dierential is then a partial dierential
More informationGlobal Weak Solution of Planetary Geostrophic Equations with Inviscid Geostrophic Balance
Global Weak Solution o Planetary Geostrophic Equations with Inviscid Geostrophic Balance Jian-Guo Liu 1, Roger Samelson 2, Cheng Wang 3 Communicated by R. Temam) Abstract. A reormulation o the planetary
More informationMathematical Preliminaries. Developed for the Members of Azera Global By: Joseph D. Fournier B.Sc.E.E., M.Sc.E.E.
Mathematical Preliminaries Developed or the Members o Azera Global B: Joseph D. Fournier B.Sc.E.E., M.Sc.E.E. Outline Chapter One, Sets: Slides: 3-27 Chapter Two, Introduction to unctions: Slides: 28-36
More informationEarlier Lecture. Basics of Refrigeration/Liquefaction, coefficient of performance and importance of Carnot COP.
9 1 Earlier Lecture Basics o Rerigeration/Liqueaction, coeicient o perormance and importance o Carnot COP. Throttling, heat exchanger, compression/expansion systems. Deinition o a rerigerator, liqueier
More informationIntroduction to Simulation - Lecture 2. Equation Formulation Methods. Jacob White. Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy
Introduction to Simulation - Lecture Equation Formulation Methods Jacob White Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy Outline Formulating Equations rom Schematics Struts and Joints
More informationComputing proximal points of nonconvex functions
Mathematical Programming manuscript No. (will be inserted by the editor) Warren Hare Claudia Sagastizábal Computing proximal points o nonconvex unctions the date o receipt and acceptance should be inserted
More informationA stochastic representation for the Poisson-Vlasov equation
A stochastic representation for the Poisson-Vlasov equation R. Vilela Mendes and F. Cipriano 2 Centro de Matemática e Aplicações and Centro de Fusão Nuclear, Lisbon 2 Departamento de Matemática, Universidade
More informationFINAL EXAM Spring 2010 TFY4235 Computational Physics
FINAL EXAM Spring 2010 TFY4235 Computational Physics This exam is published on Saturday, May 29, 2010 at 09:00 hours. The solutions should be mailed to me at Alex.Hansen@ntnu.no on Tuesday, June 1 at 23:00
More informationNONLINEAR CONTROL OF POWER NETWORK MODELS USING FEEDBACK LINEARIZATION
NONLINEAR CONTROL OF POWER NETWORK MODELS USING FEEDBACK LINEARIZATION Steven Ball Science Applications International Corporation Columbia, MD email: sball@nmtedu Steve Schaer Department o Mathematics
More informationFeedback linearization control of systems with singularities: a ball-beam revisit
Chapter 1 Feedback linearization control o systems with singularities: a ball-beam revisit Fu Zhang The Mathworks, Inc zhang@mathworks.com Benito Fernndez-Rodriguez Department o Mechanical Engineering,
More informationOn Picard value problem of some difference polynomials
Arab J Math 018 7:7 37 https://doiorg/101007/s40065-017-0189-x Arabian Journal o Mathematics Zinelâabidine Latreuch Benharrat Belaïdi On Picard value problem o some dierence polynomials Received: 4 April
More informationChapter 2. Basic concepts of probability. Summary. 2.1 Axiomatic foundation of probability theory
Chapter Basic concepts o probability Demetris Koutsoyiannis Department o Water Resources and Environmental Engineering aculty o Civil Engineering, National Technical University o Athens, Greece Summary
More information