FINITE DIFFERENCE METHODS

 To view this video please enable JavaScript, and consider upgrading to a web browser that supports HTML5 video
Save this PDF as:

Size: px
Start display at page:

Transcription

1 FINITE DIFFERENCE METHODS LONG CHEN Te best known metods, finite difference, consists of replacing eac derivative by a difference quotient in te classic formulation. It is simple to code and economic to compute. In a sense, a finite difference formulation offers a more direct approac to te numerical solution of partial differential equations tan does a metod based on oter formulations. Te drawback of te finite difference metods is accuracy and flexibility. Standard finite difference metods requires more regularity of te solution (e.g. u C 2 (Ω)) and te triangulation (e.g. uniform grids). Difficulties also arises in imposing boundary conditions. 1. FINITE DIFFERENCE FORMULA In tis section, for simplicity, we discuss Poisson equation posed on te unit square Ω = (0, 1) (0, 1). Variable coefficients and more complex domains will be discussed in finite element metods. Furtermore we assume u is smoot enoug to enable us use Taylor expansion freely. Given two integer m, n 2, we construct a rectangular grids T by te tensor product of two grids of (0, 1): {x i = (i 1) x, i = 1, m, x = 1/(m 1)} and {y j = (j 1) y, j = 1, n, y = 1/(n 1)}. Let = max{ x, y } denote te size of T. We denote Ω = {(x i, y j ) Ω} and boundary Γ = {(x i, y j ) Ω}. We consider te discrete function space given by V = {u (x i, y j ), 1 i m, 1 j n} wic is isomorpism to R N wit N = m n. It is more convenient to use sub-index (i, j) for te discrete function: u i,j := u (x i, y j ). For a continuous function u C(Ω), te interpolation operator I : C(Ω) V maps u to a discrete function and will be denoted by u I. By te definition (u I ) i,j = u(x i, y j ). Note tat te value of a discrete function is only defined at grid points. Values inside eac cell can be obtained by interpolation of values at grid points. Similar definitions can be applied to one dimensional case. Coose a mes size and u V (0, 1). Popular difference formulas at an interior node x j for a discrete function u V include: Te backward difference: (D u) j = u j u j 1 ; Te forward difference: (D + u) j = u j+1 u j ; Te centered difference: (D ± u) j = u j+1 u j 1 ; 2 Te centered second difference: (D 2 u) j = u j+1 2u j + u j 1 2. It is easy to prove by Talyor expansion tat (D u) j u (x j ) = O(), (D + u) j u (x j ) = O(), (D ± u) j u (x j ) = O( 2 ), (D 2 u) j u (x j ) = O( 2 ). Date: Updated October 12,

2 2 LONG CHEN We sall use tese difference formulation, especially te centered second difference to approximate te Laplace operator at an interior node (x i, y j ): ( u) i,j = (D 2 xxu) i,j + (D 2 yyu) i,j = u i+1,j 2u i,j + u i 1,i 2 x + u i,j+1 2u i,j + u i,j 1 2. y It is called five point stencil since only five points are involved. Wen x = y, it is simplified to (1) ( u) i,j = 4u i,j u i+1,j u i 1,i u i,j+1 u i,j 1 2 and can be denoted by te following stencil symbol For te rigt and side, we simply take node values i.e. f i,j = (f I ) i,j = f(x i, y j ). Te finite difference metods for solving Poisson equation is simply (2) ( u) i,j = f i,j, 1 i m, 1 j n, wit appropriate processing of boundary conditions. Here in (2), we also use (1) for boundary points but drop terms involving grid points outside of te domain. Let us give an ordering of N = m n grids and use a single index k = 1 to N for u k = u i(k),j(k). For example, te index map k (i(k), j(k)) can be easily written out for te lexicograpical ordering. Wit any coosing ordering, (2) can be written as a linear algebraic equation: (3) Au = f, were A R N N, u R N and f R N. Remark 1.1. Tere exist different orderings for te grid points. Altoug tey give equivalent matrixes up to permutations, different ordering does matter wen solving linear algebraic equations. 2. BOUNDARY CONDITIONS We sall discuss ow to deal wit boundary conditions in finite difference metods. Te Diriclet boundary condition is relatively easy and te Neumann boundary condition requires te gost points. Diriclet boundary condition. For te Poisson equation wit Diriclet boundary condition (4) u = f in Ω, u = g on Γ = Ω, te value on te boundary is given by te boundary conditions. Namely u i,j = g(x i, y j ) for (x i, y j ) Ω and tus not unknowns in te equation. Tere are several ways to impose te Diriclet boundary condition. One approac is to let a ii = 1, a ij = 0, j i and f i = g(x i ) for nodes x i Γ. Note tat tis will destroy te symmetry of te corresponding matrix. Anoter approac is to modify te rigt and side at interior nodes and solve only equations at interior nodes. Let

3 FINITE DIFFERENCE METHODS 3 us consider a simple example wit 9 nodes. Te only unknowns is u 5 wit te lexicograpical ordering. By te formula of discrete Laplace operator at tat node, we obtain te adjusted equation 4 2 u 5 = f (u 2 + u 4 + u 6 + u 8 ). We use te following Matlab code to illustrate te implementation of Diriclet boundary condition. Let bdnode be a logic array representing boundary nodes: bdnode(k)=1 if (x k, y k ) Ω and bdnode(k)=0 oterwise. 1 freenode = bdnode; 2 u = zeros(n,1); 3 u(bdnode) = g(node(bdnode,:)); 4 f = f-a*u; 5 u(freenode) = A(freeNode,freeNode)\f(freeNode); Te matrix A(freeNode,freeNode) is symmetric and positive definite (SPD) (see Exercise 1) and tus ensure te existence of te solution. Neumann boundary condition. For te Poisson equation wit Neumann boundary condition u = f in Ω, = g on Γ, n tere is a compatible condition for f and g: (5) f dx = u dx = n ds = g ds. Ω Ω Ω A natural approximation to te normal derivative is a one sided difference, for example: Ω n (x 1, y j ) = u 1,j u 2,j + O(). But tis is only a first order approximation. To treat Neumann boundary condition more accurately, we introduce te gost points outside of te domain and next to te boundary. We extend te lattice by allowing te index 0 i, j n + 1. Ten we can use center difference sceme: n (x 1, y j ) = u 0,j u 2,j + O( 2 ). 2 Te value u 0,j is not well defined. We need to eliminate it from te equation. Tis is possible since on te boundary point (x 1, y j ), we ave two equations: (6) (7) 4u 1,j u 2,j u 0,j u 1,j+1 u 1,j 1 = 2 f 1,j u 0,j u 2,j = 2 g 1,j. From (7), we get u 0,j = 2 g 1,j + u 2,j. Substituting it into (6) and scaling by a factor 1/2, we get an equation at point (x 1, y j ): 2u 1,j u 2,j 0.5 u 1,j u j 1 = f 1,j + g 1,j. Te scaling is to preserve te symmetry of te matrix. We can deal wit oter boundary points by te same tecnique except te four corner points. At corner points, even te

4 4 LONG CHEN norm vector is not well defined. We will use average of two directional derivatives to get an approximation. Taking (0, 0) as an example, we ave (8) (9) (10) 4u 1,1 u 2,1 u 0,1 u 1,1 u 1,0 = 2 f 1,1, u 0,1 u 2,1 = 2 g 1,1, u 1,0 u 1,2 = 2 g 1,1. So we can solve u 0,1 and u 1,0 from (9) and (10), and substitute tem into (8). Again to maintain te symmetric of te matrix, we multiply (8) by 1/4. Tis gives an equation for te corner point (x 1, y 1 ) u 1,1 0.5 u 2,1 0.5 u 1,1 = f 1,1 + g 1,1. Similar tecnique will be used to deal wit oter corner points. We ten end wit a linear algebraic equation Au = f. It can be sown tat te corresponding matrix A is still symmetric but only semi-definite (see Exercise 2). Te kernel of A consists of constant vectors i.e. Au = 0 if and only if u = c. Tis requires a discrete version of te compatible condition (5): N (11) f i = 0 i=1 and can be satisfied by te modification f = f - mean(f). 3. ERROR ESTIMATE In order to analyze te error, we need to put te problem into a norm space. A natural norm for te finite linear space V is te maximum norm: for v V, v,ω = max 1 i n+1, 1 j m+1 { v i,j }. Te subscript indicates tis norm depends on te triangulation since for different, we ave different numbers of v i,j. Note tat tis is te l norm for R N. We sall prove 1 : (V,,Ω ) (V,,Ω ) is stable uniform to. Te proof will use te discrete maximal principal and barrier functions. Teorem 3.1 (Discrete Maximum Principle). Let v V satisfy v 0. Ten max v max v, Ω Γ and te equality olds if and only if v is constant. Proof. Suppose max Ω v > max Γ v. Ten we can take an interior node x 0 were te maximum is acieved. Let x 1, x 2, x 3, and x 4 be te four neigbors used in te stencil. Ten 4 4 4v(x 0 ) = v(x i ) 2 v(x 0 ) v(x i ) 4v(x 0 ). i=1 Tus equality olds trougout and v acieves its maximum at all te nearest neigbors of x 0 as well. Applying te same argument to te neigbors in te interior, and ten to teir neigbors, etc, we conclude tat v is constant wic contradicts to te assumption i=1

5 FINITE DIFFERENCE METHODS 5 max Ω v > max Γ v. Te second statement can be proved easily by a similar argument. Teorem 3.2. Let u be te equation of (12) u = f I at Ω \Γ, u = g I at Γ. Ten (13) u,ω 1 8 f I,Ω \Γ + g I Γ,. Proof. We introduce te comparison function φ = 1 [ (x )2 + (y 1 2 )2], wic satisfies φ I = 1 at Ω \Γ and 0 φ 1/8. Set M = f I,Ω \Γ. Ten so max Ω u max Ω (u + Mφ I ) = u + M 0, (u + Mφ I ) max(u + Mφ I ) max g I + 1 Γ 8 M. Γ Tus u is bounded above by te rigt-and side of (13). A similar argument applies to u giving te teorem. Corollary 3.3. Let u be te solution of te Diriclet problem (4) and u te solution of te discrete problem (12). Ten u I u,ω 1 8 u I ( u) I,Ω \Γ. Te next step is to study te consistence error u I ( u) I,. Te following Lemma can be easily proved by Taylor expansion. Lemma 3.4. If u C 4 (Ω), ten u I ( u) I,Ω \Γ 2 6 max{ 4 u x 4,Ω \Γ, 4 u y 4,Ω \Γ }. We summarize te convergence results on te finite difference metods in te following teorem. Teorem 3.5. Let u be te solution of te Diriclet problem (4) and u te solution of te discrete problem (12). If u C 4 (Ω), ten wit constant u I u,ω C 2, C = 1 48 max{ 4 u x 4, 4 u y 4 }. In practice, te second order of convergence can be observed even te solution u is less smoot tan C 4 (Ω), i.e. te requirement u C 4 (Ω). Tis restriction comes from te point-wise estimate. In finite element metod, we sall use integral norms to find te rigt setting of function spaces.

6 ~u =0 were ~u =(u, v) t, and ~ f =(f 1,f 2 ) t. 2. MAC DISCRETIZATION 6 LONG CHEN j j j i i i (A) index for p FIGURE 1. A cell centered uniform grid 4. CELL CENTERED FINITE DIFFERENCE METHODS (B) index for u FIGURE 1. Index for p, u, v. (C) index for v 2.1. MAC Sceme. Suppose we ave a rectangular decomposition, for eac cell, gree of freedoms for u and v are located on te vertical edge centers and orizont centers, respectively, and te degree of freedoms for pressure p are located at cell c Te MAC sceme is written as (µ =1) In some applications, notable te computational fluid dynamics (CFD), te Poisson equation is solved on sligtly different grids. In tis section, we consider FDM for te Poisson equation at cell centers; see Fig 4. At interior nodes, te standard (4, 1, 1, 1, 1) but boundary conditions will be treated differently. Te distance in axis direction between interior nodes is still but te near boundary nodes (centers of te cells toucing boundary) is /2 away from te boundary. One can ten easily verify tat for Neumann boundary condition, te stencil for near boundary nodes is (3, 1, 1, 1) and for corner cells (2, 1, 1). Of course te boundary condition g sould be evaluated and moved to te rigt and side. Te Diriclet boundary condition is more subtle for cell centered difference. We can still introduce te gost grid points and use standard (4, 1, 1, 1) stencil for near boundary nodes. But no grid points are on te boundary. Te gost value can be eliminated by linear extrapolation, i.e, requiring (u 0,j + u 1,j )/2 = g(0, y j ) := g 1/2,j. (14) (2.1) 4u i,j u i 1,j u i+1,j u i,j 1 u i,j pi,j p i,j 1 (2.2) 4v i,j v i 1,j v i+1,j v i,j 1 v i,j pi 1,j p i,j (2.3) u i,j+1 u i,j + vi,j v i+1,j =0 It s easy to see tat te above sceme as second order truncation error. 5u 1,j u 2,j u 1,j 1 u 1,j+1 2 = f 1,j + 2g 1/2,j 2. Te stencil will be (5, 1, 1, 1, 2) for near boundary nodes and (6, 1, 1, 2, 2) for corner nodes. Te symmetry of te corresponding matrix is still preserved. However, tis treatment is of low order (see Exercise 3). To obtain a better truncation error, we can use te quadratic extrapolation, tat is, use u 1/2,j, u 1,j, u 2,j to fit a quadratic function and evaluate at u 0,j, we get u 0,j = 2u 1,j u 2,j u 1/2,j, and obtain te modified boundary sceme sould be: (15) 6u 1,j 4 3 u 2,j u 1,j 1 u 1,j+1 2 = f 1,j g 1/2,j 2. We denote te near boundary stencil by (6, 4 3, 1, 1, 8 3 ). Te quadratic extrapolation will lead to a better rate of convergence since te truncation error is improved. Te disadvantage of tis treatment is tat te symmetry of te matrix is destroyed. For te Poisson equation, tere is anoter way to keep te second order truncation error and symmetry. For simplicity, let us consider te omogenous Diriclet boundary condition, i.e., u Ω = 0. Ten te tangential derivatives along te boundary is vanised, in particular, 2 t u = 0. Assume te equation u = f olds also on te boundary condition. Note tat on te boundary, te operator can be written as 2 t + 2 n. We ten get 2 nu = ±f on Ω. Te sign is determined by if te norm direction is te same as te axis direction. Ten we can use u 1, u 1/2 = 0 and 2 nu = f to fit a quadratic function and 1 = f i,j 1 = f i,j 2

7 FINITE DIFFERENCE METHODS 7 extrapolate to get an equation for te gost point u 1,j + u 0,j = 2 4 f 1/2,j and modify te boundary stencil as 5u 1,j u 2,j u 1,j 1 u 1,j+1 (16) 2 = f 1,j f 1/2,j. 5. EXERCISES (1) Prove te following properties of te matrix A formed in te finite difference metods for Poisson equation wit Diriclet boundary condition: (a) it is symmetric: a ij = a ji ; (b) it is diagonally dominant: a ii N j=1,j i a ij; (c) it is positive definite: u t Au 0 for any u R N and u t Au = 0 if and only if u = 0. (2) Let us consider te finite difference discritization of Poisson equation wit Neumann boundary condition. (a) Write out te 9 9 matrix A for = 1/2. (b) Prove tat in general te matrix corresponding to Neumann boundary condition is only semi-positive definite. (c) Sow tat te kernel of A consists of constant vectors: Au = 0 if and only if u = c. (3) Ceck te truncation error of scemes (14), (15) and (16) for different treatments of Diriclet boundary condition in te cell centered finite difference metods.

Verifying Numerical Convergence Rates

1 Order of accuracy Verifying Numerical Convergence Rates We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, suc as te grid size or time step, and

Finite Difference Approximations

Capter Finite Difference Approximations Our goal is to approximate solutions to differential equations, i.e., to find a function (or some discrete approximation to tis function) tat satisfies a given relationsip

Finite Volume Discretization of the Heat Equation

Lecture Notes 3 Finite Volume Discretization of te Heat Equation We consider finite volume discretizations of te one-dimensional variable coefficient eat equation, wit Neumann boundary conditions u t x

In other words the graph of the polynomial should pass through the points

Capter 3 Interpolation Interpolation is te problem of fitting a smoot curve troug a given set of points, generally as te grap of a function. It is useful at least in data analysis (interpolation is a form

Trapezoid Rule. y 2. y L

Trapezoid Rule and Simpson s Rule c 2002, 2008, 200 Donald Kreider and Dwigt Lar Trapezoid Rule Many applications of calculus involve definite integrals. If we can find an antiderivative for te integrand,

Derivatives Math 120 Calculus I D Joyce, Fall 2013

Derivatives Mat 20 Calculus I D Joyce, Fall 203 Since we ave a good understanding of its, we can develop derivatives very quickly. Recall tat we defined te derivative f x of a function f at x to be te

This supplement is meant to be read after Venema s Section 9.2. Throughout this section, we assume all nine axioms of Euclidean geometry.

Mat 444/445 Geometry for Teacers Summer 2008 Supplement : Similar Triangles Tis supplement is meant to be read after Venema s Section 9.2. Trougout tis section, we assume all nine axioms of uclidean geometry.

Instantaneous Rate of Change:

Instantaneous Rate of Cange: Last section we discovered tat te average rate of cange in F(x) can also be interpreted as te slope of a scant line. Te average rate of cange involves te cange in F(x) over

7.6 Complex Fractions

Section 7.6 Comple Fractions 695 7.6 Comple Fractions In tis section we learn ow to simplify wat are called comple fractions, an eample of wic follows. 2 + 3 Note tat bot te numerator and denominator are

Understanding the Derivative Backward and Forward by Dave Slomer

Understanding te Derivative Backward and Forward by Dave Slomer Slopes of lines are important, giving average rates of cange. Slopes of curves are even more important, giving instantaneous rates of cange.

OPTIMAL DISCONTINUOUS GALERKIN METHODS FOR THE ACOUSTIC WAVE EQUATION IN HIGHER DIMENSIONS

OPTIMAL DISCONTINUOUS GALERKIN METHODS FOR THE ACOUSTIC WAVE EQUATION IN HIGHER DIMENSIONS ERIC T. CHUNG AND BJÖRN ENGQUIST Abstract. In tis paper, we developed and analyzed a new class of discontinuous

SAT Subject Math Level 1 Facts & Formulas

Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses

Part II: Finite Difference/Volume Discretisation for CFD

Part II: Finite Difference/Volume Discretisation for CFD Finite Volume Metod of te Advection-Diffusion Equation A Finite Difference/Volume Metod for te Incompressible Navier-Stokes Equations Marker-and-Cell

f(a + h) f(a) f (a) = lim

Lecture 7 : Derivative AS a Function In te previous section we defined te derivative of a function f at a number a (wen te function f is defined in an open interval containing a) to be f (a) 0 f(a + )

Differentiable Functions

Capter 8 Differentiable Functions A differentiable function is a function tat can be approximated locally by a linear function. 8.. Te derivative Definition 8.. Suppose tat f : (a, b) R and a < c < b.

Math 113 HW #5 Solutions

Mat 3 HW #5 Solutions. Exercise.5.6. Suppose f is continuous on [, 5] and te only solutions of te equation f(x) = 6 are x = and x =. If f() = 8, explain wy f(3) > 6. Answer: Suppose we ad tat f(3) 6. Ten

Module 1: Introduction to Finite Element Analysis Lecture 1: Introduction

Module : Introduction to Finite Element Analysis Lecture : Introduction.. Introduction Te Finite Element Metod (FEM) is a numerical tecnique to find approximate solutions of partial differential equations.

Lecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function

Lecture 10: Wat is a Function, definition, piecewise defined functions, difference quotient, domain of a function A function arises wen one quantity depends on anoter. Many everyday relationsips between

A Multigrid Tutorial part two

A Multigrid Tutorial part two William L. Briggs Department of Matematics University of Colorado at Denver Van Emden Henson Center for Applied Scientific Computing Lawrence Livermore National Laboratory

Høgskolen i Narvik Sivilingeniørutdanningen STE6237 ELEMENTMETODER. Oppgaver

Høgskolen i Narvik Sivilingeniørutdanningen STE637 ELEMENTMETODER Oppgaver Klasse: 4.ID, 4.IT Ekstern Professor: Gregory A. Chechkin e-mail: chechkin@mech.math.msu.su Narvik 6 PART I Task. Consider two-point

1 Computational Fluid Dynamics Reading Group: Finite Element Methods for Stokes & The Infamous Inf-Sup Condition

1 Computational Fluid Dynamics Reading Group: Finite Element Metods for Stokes & Te Infamous Inf-Sup Condition Fall 009. 09/11/09 Darsi Devendran, Sandra May & Eduardo Corona 1.1 Introduction Last week

Surface Areas of Prisms and Cylinders

12.2 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G.10.B G.11.C Surface Areas of Prisms and Cylinders Essential Question How can you find te surface area of a prism or a cylinder? Recall tat te surface area of

Tangent Lines and Rates of Change

Tangent Lines and Rates of Cange 9-2-2005 Given a function y = f(x), ow do you find te slope of te tangent line to te grap at te point P(a, f(a))? (I m tinking of te tangent line as a line tat just skims

Solutions by: KARATUĞ OZAN BiRCAN. PROBLEM 1 (20 points): Let D be a region, i.e., an open connected set in

KOÇ UNIVERSITY, SPRING 2014 MATH 401, MIDTERM-1, MARCH 3 Instructor: BURAK OZBAGCI TIME: 75 Minutes Solutions by: KARATUĞ OZAN BiRCAN PROBLEM 1 (20 points): Let D be a region, i.e., an open connected set

Projective Geometry. Projective Geometry

Euclidean versus Euclidean geometry describes sapes as tey are Properties of objects tat are uncanged by rigid motions» Lengts» Angles» Parallelism Projective geometry describes objects as tey appear Lengts,

Optimal Pricing Strategy for Second Degree Price Discrimination

Optimal Pricing Strategy for Second Degree Price Discrimination Alex O Brien May 5, 2005 Abstract Second Degree price discrimination is a coupon strategy tat allows all consumers access to te coupon. Purcases

Geometric Stratification of Accounting Data

Stratification of Accounting Data Patricia Gunning * Jane Mary Horgan ** William Yancey *** Abstract: We suggest a new procedure for defining te boundaries of te strata in igly skewed populations, usual

1.6. Analyse Optimum Volume and Surface Area. Maximum Volume for a Given Surface Area. Example 1. Solution

1.6 Analyse Optimum Volume and Surface Area Estimation and oter informal metods of optimizing measures suc as surface area and volume often lead to reasonable solutions suc as te design of te tent in tis

Area of a Parallelogram

Area of a Parallelogram Focus on After tis lesson, you will be able to... φ develop te φ formula for te area of a parallelogram calculate te area of a parallelogram One of te sapes a marcing band can make

ME422 Mechanical Control Systems Modeling Fluid Systems

Cal Poly San Luis Obispo Mecanical Engineering ME422 Mecanical Control Systems Modeling Fluid Systems Owen/Ridgely, last update Mar 2003 Te dynamic euations for fluid flow are very similar to te dynamic

Chapter 7 Numerical Differentiation and Integration

45 We ave a abit in writing articles publised in scientiþc journals to make te work as Þnised as possible, to cover up all te tracks, to not worry about te blind alleys or describe ow you ad te wrong idea

Introduction to the Finite Element Method

Introduction to the Finite Element Method 09.06.2009 Outline Motivation Partial Differential Equations (PDEs) Finite Difference Method (FDM) Finite Element Method (FEM) References Motivation Figure: cross

CHAPTER 7. Di erentiation

CHAPTER 7 Di erentiation 1. Te Derivative at a Point Definition 7.1. Let f be a function defined on a neigborood of x 0. f is di erentiable at x 0, if te following it exists: f 0 fx 0 + ) fx 0 ) x 0 )=.

The EOQ Inventory Formula

Te EOQ Inventory Formula James M. Cargal Matematics Department Troy University Montgomery Campus A basic problem for businesses and manufacturers is, wen ordering supplies, to determine wat quantity of

Research on the Anti-perspective Correction Algorithm of QR Barcode

Researc on te Anti-perspective Correction Algoritm of QR Barcode Jianua Li, Yi-Wen Wang, YiJun Wang,Yi Cen, Guoceng Wang Key Laboratory of Electronic Tin Films and Integrated Devices University of Electronic

Proof of the Power Rule for Positive Integer Powers

Te Power Rule A function of te form f (x) = x r, were r is any real number, is a power function. From our previous work we know tat x x 2 x x x x 3 3 x x In te first two cases, te power r is a positive

Inner product. Definition of inner product

Math 20F Linear Algebra Lecture 25 1 Inner product Review: Definition of inner product. Slide 1 Norm and distance. Orthogonal vectors. Orthogonal complement. Orthogonal basis. Definition of inner product

ACT Math Facts & Formulas

Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Rationals: fractions, tat is, anyting expressable as a ratio of integers Reals: integers plus rationals plus special numbers suc as

ON LOCAL LIKELIHOOD DENSITY ESTIMATION WHEN THE BANDWIDTH IS LARGE

ON LOCAL LIKELIHOOD DENSITY ESTIMATION WHEN THE BANDWIDTH IS LARGE Byeong U. Park 1 and Young Kyung Lee 2 Department of Statistics, Seoul National University, Seoul, Korea Tae Yoon Kim 3 and Ceolyong Park

Section 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations

Difference Equations to Differential Equations Section 3.3 Differentiation of Polynomials an Rational Functions In tis section we begin te task of iscovering rules for ifferentiating various classes of

New Vocabulary volume

-. Plan Objectives To find te volume of a prism To find te volume of a cylinder Examples Finding Volume of a Rectangular Prism Finding Volume of a Triangular Prism 3 Finding Volume of a Cylinder Finding

Determine the perimeter of a triangle using algebra Find the area of a triangle using the formula

Student Name: Date: Contact Person Name: Pone Number: Lesson 0 Perimeter, Area, and Similarity of Triangles Objectives Determine te perimeter of a triangle using algebra Find te area of a triangle using

6. Differentiating the exponential and logarithm functions

1 6. Differentiating te exponential and logaritm functions We wis to find and use derivatives for functions of te form f(x) = a x, were a is a constant. By far te most convenient suc function for tis purpose

Differential Calculus: Differentiation (First Principles, Rules) and Sketching Graphs (Grade 12)

OpenStax-CNX moule: m39313 1 Differential Calculus: Differentiation (First Principles, Rules) an Sketcing Graps (Grae 12) Free Hig Scool Science Texts Project Tis work is prouce by OpenStax-CNX an license

Solution Derivations for Capa #7

Solution Derivations for Capa #7 1) Consider te beavior of te circuit, wen various values increase or decrease. (Select I-increases, D-decreases, If te first is I and te rest D, enter IDDDD). A) If R1

LECTURE NOTES: FINITE ELEMENT METHOD

LECTURE NOTES: FINITE ELEMENT METHOD AXEL MÅLQVIST. Motivation The finite element method has two main strengths... Geometry. Very complex geometries can be used. This is probably the main reason why finite

Optimized Data Indexing Algorithms for OLAP Systems

Database Systems Journal vol. I, no. 2/200 7 Optimized Data Indexing Algoritms for OLAP Systems Lucian BORNAZ Faculty of Cybernetics, Statistics and Economic Informatics Academy of Economic Studies, Bucarest

POISSON AND LAPLACE EQUATIONS. Charles R. O Neill. School of Mechanical and Aerospace Engineering. Oklahoma State University. Stillwater, OK 74078

21 ELLIPTICAL PARTIAL DIFFERENTIAL EQUATIONS: POISSON AND LAPLACE EQUATIONS Charles R. O Neill School of Mechanical and Aerospace Engineering Oklahoma State University Stillwater, OK 74078 2nd Computer

Can a Lump-Sum Transfer Make Everyone Enjoy the Gains. from Free Trade?

Can a Lump-Sum Transfer Make Everyone Enjoy te Gains from Free Trade? Yasukazu Icino Department of Economics, Konan University June 30, 2010 Abstract I examine lump-sum transfer rules to redistribute te

FOURIER ANALYSIS OF MULTIGRID METHODS ON HEXAGONAL GRIDS

FOURIER ANALYSIS OF MULTIGRID METHODS ON HEXAGONAL GRIDS GUOHUA ZHOU AND SCOTT R. FULTON Abstract. Tis paper applies local Fourier analysis to multigrid metods on exagonal grids. Using oblique coordinates

CHAPTER 8: DIFFERENTIAL CALCULUS

CHAPTER 8: DIFFERENTIAL CALCULUS 1. Rules of Differentiation As we ave seen, calculating erivatives from first principles can be laborious an ifficult even for some relatively simple functions. It is clearly

13 PERIMETER AND AREA OF 2D SHAPES

13 PERIMETER AND AREA OF D SHAPES 13.1 You can find te perimeter of sapes Key Points Te perimeter of a two-dimensional (D) sape is te total distance around te edge of te sape. l To work out te perimeter

2 Limits and Derivatives

2 Limits and Derivatives 2.7 Tangent Lines, Velocity, and Derivatives A tangent line to a circle is a line tat intersects te circle at exactly one point. We would like to take tis idea of tangent line

Sections 3.1/3.2: Introducing the Derivative/Rules of Differentiation

Sections 3.1/3.2: Introucing te Derivative/Rules of Differentiation 1 Tangent Line Before looking at te erivative, refer back to Section 2.1, looking at average velocity an instantaneous velocity. Here

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

The finite element immersed boundary method: model, stability, and numerical results

Te finite element immersed boundary metod: model, stability, and numerical results Lucia Gastaldi Università di Brescia ttp://dm.ing.unibs.it/gastaldi/ INdAM Worksop, Cortona, September 18, 2006 Joint

Iterative Techniques in Matrix Algebra. Jacobi & Gauss-Seidel Iterative Techniques II

Iterative Techniques in Matrix Algebra Jacobi & Gauss-Seidel Iterative Techniques II Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin

Areas and Centroids. Nothing. Straight Horizontal line. Straight Sloping Line. Parabola. Cubic

Constructing Sear and Moment Diagrams Areas and Centroids Curve Equation Sape Centroid (From Fat End of Figure) Area Noting Noting a x 0 Straigt Horizontal line /2 Straigt Sloping Line /3 /2 Paraola /4

Pipe Flow Analysis. Pipes in Series. Pipes in Parallel

Pipe Flow Analysis Pipeline system used in water distribution, industrial application and in many engineering systems may range from simple arrangement to extremely complex one. Problems regarding pipelines

Note nine: Linear programming CSE 101. 1 Linear constraints and objective functions. 1.1 Introductory example. Copyright c Sanjoy Dasgupta 1

Copyrigt c Sanjoy Dasgupta Figure. (a) Te feasible region for a linear program wit two variables (see tet for details). (b) Contour lines of te objective function: for different values of (profit). Te

CITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION

No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August

Lecture 10. Limits (cont d) One-sided limits. (Relevant section from Stewart, Seventh Edition: Section 2.4, pp. 113.)

Lecture 10 Limits (cont d) One-sided its (Relevant section from Stewart, Sevent Edition: Section 2.4, pp. 113.) As you may recall from your earlier course in Calculus, we may define one-sided its, were

Exercises for numerical integration. Øyvind Ryan

Exercises for numerical integration Øyvind Ryan February 25, 21 1. Vi ar r(t) = (t cos t, t sin t, t) Solution: Koden blir % Oppgave.1.11 b) quad(@(x)sqrt(2+t.^2),,2*pi) a. Finn astigeten, farten og akselerasjonen.

Module 2. The Science of Surface and Ground Water. Version 2 CE IIT, Kharagpur

Module Te Science of Surface and Ground Water Version CE IIT, Karagpur Lesson 6 Principles of Ground Water Flow Version CE IIT, Karagpur Instructional Objectives On completion of te lesson, te student

Sum of the generalized harmonic series with even natural exponent

Rendiconti di Matematica, Serie VII Volume 33, Roma (23), 9 26 Sum of te generalized armonic erie wit even natural exponent STEFANO PATRÌ Abtract: In ti paper we deal wit real armonic erie, witout conidering

ACTIVITY: Deriving the Area Formula of a Trapezoid

4.3 Areas of Trapezoids a trapezoid? How can you derive a formula for te area of ACTIVITY: Deriving te Area Formula of a Trapezoid Work wit a partner. Use a piece of centimeter grid paper. a. Draw any

Writing Mathematics Papers

Writing Matematics Papers Tis essay is intended to elp your senior conference paper. It is a somewat astily produced amalgam of advice I ave given to students in my PDCs (Mat 4 and Mat 9), so it s not

5 Numerical Differentiation

D. Levy 5 Numerical Differentiation 5. Basic Concepts This chapter deals with numerical approximations of derivatives. The first questions that comes up to mind is: why do we need to approximate derivatives

Moving Least Squares Approximation

Chapter 7 Moving Least Squares Approimation An alternative to radial basis function interpolation and approimation is the so-called moving least squares method. As we will see below, in this method the

MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION

MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION Tis tutorial is essential pre-requisite material for anyone stuing mecanical engineering. Tis tutorial uses te principle of

TRADING AWAY WIDE BRANDS FOR CHEAP BRANDS. Swati Dhingra London School of Economics and CEP. Online Appendix

TRADING AWAY WIDE BRANDS FOR CHEAP BRANDS Swati Dingra London Scool of Economics and CEP Online Appendix APPENDIX A. THEORETICAL & EMPIRICAL RESULTS A.1. CES and Logit Preferences: Invariance of Innovation

Numerical methods for American options

Lecture 9 Numerical methods for American options Lecture Notes by Andrzej Palczewski Computational Finance p. 1 American options The holder of an American option has the right to exercise it at any moment

Scientific Computing: An Introductory Survey

Scientific Computing: An Introductory Survey Chapter 10 Boundary Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign

A New Cement to Glue Nonconforming Grids with Robin Interface Conditions: The Finite Element Case

A New Cement to Glue Nonconforming Grids wit Robin Interface Conditions: Te Finite Element Case Martin J. Gander, Caroline Japet 2, Yvon Maday 3, and Frédéric Nataf 4 McGill University, Dept. of Matematics

Chapter 10: Refrigeration Cycles

Capter 10: efrigeration Cycles Te vapor compression refrigeration cycle is a common metod for transferring eat from a low temperature to a ig temperature. Te above figure sows te objectives of refrigerators

Fourth-Order Compact Schemes of a Heat Conduction Problem with Neumann Boundary Conditions

Fourth-Order Compact Schemes of a Heat Conduction Problem with Neumann Boundary Conditions Jennifer Zhao, 1 Weizhong Dai, Tianchan Niu 1 Department of Mathematics and Statistics, University of Michigan-Dearborn,

- 1 - Handout #22 May 23, 2012 Huffman Encoding and Data Compression. CS106B Spring 2012. Handout by Julie Zelenski with minor edits by Keith Schwarz

CS106B Spring 01 Handout # May 3, 01 Huffman Encoding and Data Compression Handout by Julie Zelenski wit minor edits by Keit Scwarz In te early 1980s, personal computers ad ard disks tat were no larger

Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

SYSTEMS OF EQUATIONS

SYSTEMS OF EQUATIONS 1. Examples of systems of equations Here are some examples of systems of equations. Each system has a number of equations and a number (not necessarily the same) of variables for which

since by using a computer we are limited to the use of elementary arithmetic operations

> 4. Interpolation and Approximation Most functions cannot be evaluated exactly: x, e x, ln x, trigonometric functions since by using a computer we are limited to the use of elementary arithmetic operations

Average and Instantaneous Rates of Change: The Derivative

9.3 verage and Instantaneous Rates of Cange: Te Derivative 609 OBJECTIVES 9.3 To define and find average rates of cange To define te derivative as a rate of cange To use te definition of derivative to

Math Warm-Up for Exam 1 Name: Solutions

Disclaimer: Tese review problems do not represent te exact questions tat will appear te exam. Tis is just a warm-up to elp you begin studying. It is your responsibility to review te omework problems, webwork

Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 12.

Capter 6. Fluid Mecanics Notes: Most of te material in tis capter is taken from Young and Freedman, Cap. 12. 6.1 Fluid Statics Fluids, i.e., substances tat can flow, are te subjects of tis capter. But

Theoretical calculation of the heat capacity

eoretical calculation of te eat capacity Principle of equipartition of energy Heat capacity of ideal and real gases Heat capacity of solids: Dulong-Petit, Einstein, Debye models Heat capacity of metals

1 The Collocation Method

CS410 Assignment 7 Due: 1/5/14 (Fri) at 6pm You must wor eiter on your own or wit one partner. You may discuss bacground issues and general solution strategies wit oters, but te solutions you submit must

Exam 2 Review. . You need to be able to interpret what you get to answer various questions.

Exam Review Exam covers 1.6,.1-.3, 1.5, 4.1-4., and 5.1-5.3. You sould know ow to do all te omework problems from tese sections and you sould practice your understanding on several old exams in te exam

Pyramidization of Polygonal Prisms and Frustums

European International Journal of cience and ecnology IN: 04-99 www.eijst.org.uk Pyramidization of Polygonal Prisms and Frustums Javad Hamadani Zade Department of Matematics Dalton tate College Dalton,

Math 5311 Gateaux differentials and Frechet derivatives

Math 5311 Gateaux differentials and Frechet derivatives Kevin Long January 26, 2009 1 Differentiation in vector spaces Thus far, we ve developed the theory of minimization without reference to derivatives.

Inner Product Spaces

Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

An inquiry into the multiplier process in IS-LM model

An inquiry into te multiplier process in IS-LM model Autor: Li ziran Address: Li ziran, Room 409, Building 38#, Peing University, Beijing 00.87,PRC. Pone: (86) 00-62763074 Internet Address: jefferson@water.pu.edu.cn

Elasticity Theory Basics

G22.3033-002: Topics in Computer Graphics: Lecture #7 Geometric Modeling New York University Elasticity Theory Basics Lecture #7: 20 October 2003 Lecturer: Denis Zorin Scribe: Adrian Secord, Yotam Gingold

The finite-element method

The finite-element method Version April 19, 21 [1, 2] 1 One-dimensional problems Let us apply the finite element method to the one-dimensional fin equation d 2 T dx 2 T = (1) with the boundary conditions

Fourier Series. Chapter Some Properties of Functions Goal Preliminary Remarks

Chapter 3 Fourier Series 3.1 Some Properties of Functions 3.1.1 Goal We review some results about functions which play an important role in the development of the theory of Fourier series. These results

EECS 556 Image Processing W 09. Interpolation. Interpolation techniques B splines

EECS 556 Image Processing W 09 Interpolation Interpolation techniques B splines What is image processing? Image processing is the application of 2D signal processing methods to images Image representation

f(x + h) f(x) h as representing the slope of a secant line. As h goes to 0, the slope of the secant line approaches the slope of the tangent line.

Derivative of f(z) Dr. E. Jacobs Te erivative of a function is efine as a limit: f (x) 0 f(x + ) f(x) We can visualize te expression f(x+) f(x) as representing te slope of a secant line. As goes to 0,

1 Derivatives of Piecewise Defined Functions

MATH 1010E University Matematics Lecture Notes (week 4) Martin Li 1 Derivatives of Piecewise Define Functions For piecewise efine functions, we often ave to be very careful in computing te erivatives.

Training Robust Support Vector Regression via D. C. Program

Journal of Information & Computational Science 7: 12 (2010) 2385 2394 Available at ttp://www.joics.com Training Robust Support Vector Regression via D. C. Program Kuaini Wang, Ping Zong, Yaoong Zao College