Lecture 4.2 Finite Difference Approximation

Size: px
Start display at page:

Download "Lecture 4.2 Finite Difference Approximation"

Transcription

1 Lecture 4. Finite Difference Approimation 1

2 Discretization As stated in Lecture 1.0, there are three steps in numerically solving the differential equations. They are: 1. Discretization of the domain by placing a large number of nodes with help of grid generation techniques, as discussed in Module 3.. Approimating (or discretizing) the governing equations at the nodes identified in Step Solving the algebraic equations obtained in Step using direct or iterative methods. In this and in the following few lectures, finite difference method is chosen for the purpose of discretizing the governing equations.

3 Finite Difference Method Finite difference method computes the solution numerically at a predefined set of discrete points in the structured grid of a computational domain. These discrete points along with their inter connections are called nodal points of the grid or mesh. The procedure of identifying the grid points (or grid) for a given domain is called the discretization of the domain, which is the first step in the finite difference method. 3

4 In Module 3, various grid generation techniques have been highlighted. However for the regular domains chosen to eplain the finite difference method, placing nodal points uniformly and connecting them by straight lines gives the required grid therefore, we may not use the procedures given in the Module 3. For eample, the discretization of a one dimensional domain that is, an interval, can be realized as shown in Fig Fig Discretization of an interval 4

5 In this discretization, a set of uniformly distributed points 0, 1,. n are identified such that 0 = a, n = b and i i-1 = for i = 1,, n, as shown in Fig Where, is the step length. A uniform grid means the distance between any two consecutive points i and i-1 is constant. Otherwise, the discretization is called non-uniform. The coordinate of any mesh point in Fig. 4..1, is computed using i a i* i 0,1,... n (4..1) Function value u at any point i is represented by u i ui i 0,1,... n 5

6 If Ω is a rectangular two-dimension domain bounded by [a,b] [c,d], then it can be discretized as shown in Fig Fig. 4.. Discretization of a rectangular domain The coordinates of any point p( i,y j ) are obtained using i 0 i *, i 0,1,,..., n, y j y 0 j * y, j 0,1,,..., m (4..) 6

7 Further, the dependent variable u at any point P is represented using u i,j = u( i, y j ). A similar etension can be carried out for higher dimensional domains. For grid generation in arbitrary domains, refer to Module 3. The second step is the discretization of the governing equations. To realize this, there are several methods. In the present lecture, the Taylor series based method is highlighted. Consider a function u which depends on the independent variable in the interval [a, b]. 7

8 Let the function u be sufficiently smooth (differentiable) and it has values u i and u i + 1 at any two neighboring points, i and i + 1 respectively. By using Taylor series epansion, u i + 1 can be epressed in terms of u i and its higher derivatives as 3 ui 1 u i ui ui ui ui! 3! where the superscript stands for a derivative with respect to.... (4..3) From equation (4..3), u i can be written as u 1 u u i i i ui ui...! 3! ui 1 ui 1 O( ) ui O( ) (4..4) 8

9 where δ + is the forward difference operator defined by δ + u i = u i+1 u i Equation (4..4) is the first approimation to u at the node i. In this approimation ui! 3! ui... is the error. This error is called Truncation error, in which leading term. ui! is the Since the degree of the step size is one in the leading term of the error, (4..4) is a first order approimation for u. 9

10 The order of approimation is an important concept in the process of discretization which gives an immediate insight about what kind of accuracy can be epected from the scheme. Similarly, one can also write 3 u i 1 ui 1 ui ui ui ui! 3!... (4..5) ' 1 1 u u i u i i ui ui... ui O( )! 3! (4..6) where δ - is the backward difference operator defined by δ - u i = u i u i-1. Eq. (4..5) is again first order as the degree of the step size in the leading term of the error is also one. 10

11 From (4..5) and (4..4) (subtracting (4..5) from (4..4)) one can write 3 5 u 1 1 v i ui ui ui u... 3! 5! i (4..7) 1 1 u 0 ( i ui 1 ui 1 ui ui O ) 6 (4..8) Here, δ 0 is called the central difference operator defined by δ 0 u i = u i+1 u i-1. Equation (4..8) is a second order accurate approimation for u as the leading term of the error has in second degree. 11

12 Alternatively, adding (4..4) and (4..5) gives 4 u 1 1 iv i ui ui ui u...! 4! i ui 1 ui ui 1 iv 1 i i i u u u O( ) 1 (4..9) (4..10) where δ is the central difference operator for second derivative which is a second order accurate approimation. 1

13 Numerical Implementation The finite difference approimations (4..4), (4..6), (4..8), and (4..10) can be used to replace the first and second order derivative terms of a differential equation to convert it into a difference (possibly linear algebraic) equation at every nodal point of the interior of the domain. Note that the central approimation (4..8) is one order better accurate than the forward (4..4) and backward (4..6) approimations. To understand the implementation, consider the following boundary value problem (BVP). d u du d d u sin(ln ) 0, 1, u(1) 1and u() (4..11) 13

14 The analytical solution of the BVP Eq. (4..11) is u( ) sin log( ) 5cos log 34 (4..1) The finite difference solution of Eq. (4..11) is obtained by replacing its first and second order derivative terms of with (4..7) and (4..8), respectively to get ui 1 ui ui 1 ui 1 ui 1 O( ) u sin log( ) 0 i i i i for i 1,,..., n 1 (4..13) Note that at node numbers 0 and n, we have boundary conditions and the given differential equation may not be valid at these points, therefore, at these nodes we have (from the boundary conditions) u 0 = 1.0 and u n =.0 (4..14)

15 Equation (4..13) has n-1 equations in n+1 variables, therefore adding Eq. (4..14) closes the system to solve for the unknowns u i, for i = 1,,..., n-1. Rearranging the terms in (4..13) gives u 1 1 sin log( ) i u ( ) i u i i i i i i 1,,... n 1 (4..15) Equation (4..15) is a linear system with tri-diagonal coefficient matri. Solving such linear algebraic systems is the third and final step of the numerical schemes. (Refer Module 5). 15

16 After solving Eq. (4..15) with step lengths ( ) 1/40, 1/80 and 1/160, the absolute errors in the numerical solution are computed and compared in the Fig Fig Comparison absolute errors with three distinct discretizations 16

17 The corresponding infinity error norms, obtained using are u u Eact , , , respectively for the step lengths 1/40, 1/80 and 1/160. It is clear from this demonstration that the accuracy is improved as the number of points is increased. The decimal place accuracy obtained with second order approimations is quite satisfactory, however, if one wishes to further improve the accuracy, higher order approimations to the derivatives can be obtained. Therefore, the topic of discussion in the net lecture is the higher 17 order finite difference approimation.

18 Eercise Problems Find a second order finite difference approimation for the PDE u uv v c c y c y Find the truncation error in the above problem Generate a uniform grid in the circular region r <= 1. Describe the implementation of derivative boundary conditions using finite differences 18

19 Summary of Lecture 4. Finite difference approimations for partial derivatives are introduced in this lecture. END OF LECTURE 4. 19

Lecture 4.5 Schemes for Parabolic Type Equations

Lecture 4.5 Schemes for Parabolic Type Equations Lecture 4.5 Schemes for Parabolic Type Equations 1 Difference Schemes for Parabolic Equations One-dimensional problems: Consider the unsteady diffusion problem (parabolic in nature) in a thin wire governed

More information

Finite Difference Method for PDE. Y V S S Sanyasiraju Professor, Department of Mathematics IIT Madras, Chennai 36

Finite Difference Method for PDE. Y V S S Sanyasiraju Professor, Department of Mathematics IIT Madras, Chennai 36 Finite Difference Method for PDE Y V S S Sanyasiraju Professor, Department of Mathematics IIT Madras, Chennai 36 1 Classification of the Partial Differential Equations Consider a scalar second order partial

More information

Exact and Approximate Numbers:

Exact and Approximate Numbers: Eact and Approimate Numbers: The numbers that arise in technical applications are better described as eact numbers because there is not the sort of uncertainty in their values that was described above.

More information

(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2)

(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2) . f() = 4 cosec 4 +, where is in radians. (a) Show that there is a root α of f () = 0 in the interval [.,.3]. Show that the equation f() = 0 can be written in the form = + sin 4 Use the iterative formula

More information

The Fundamental Theorem of Calculus Part 3

The Fundamental Theorem of Calculus Part 3 The Fundamental Theorem of Calculus Part FTC Part Worksheet 5: Basic Rules, Initial Value Problems, Rewriting Integrands A. It s time to find anti-derivatives algebraically. Instead of saying the anti-derivative

More information

MATH 108 REVIEW TOPIC 6 Radicals

MATH 108 REVIEW TOPIC 6 Radicals Math 08 T6-Radicals Page MATH 08 REVIEW TOPIC 6 Radicals I. Computations with Radicals II. III. IV. Radicals Containing Variables Rationalizing Radicals and Rational Eponents V. Logarithms Answers to Eercises

More information

Notation Nodes are data points at which functional values are available or at which you wish to compute functional values At the nodes fx i

Notation Nodes are data points at which functional values are available or at which you wish to compute functional values At the nodes fx i LECTURE 6 NUMERICAL DIFFERENTIATION To find discrete approximations to differentiation (since computers can only deal with functional values at discrete points) Uses of numerical differentiation To represent

More information

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com . f() = 4 cosec 4 +, where is in radians. (a) Show that there is a root α of f () = 0 in the interval [.,.3]. Show that the equation f() = 0 can be written in the form = + sin 4 (c) Use the iterative formula

More information

Relevant self-assessment exercises: [LIST SELF-ASSESSMENT EXERCISES HERE]

Relevant self-assessment exercises: [LIST SELF-ASSESSMENT EXERCISES HERE] Chapter 5 Fourier Analysis of Finite Difference Methods In this lecture, we determine the stability of PDE discretizations using Fourier analysis. First, we begin with Fourier analysis of PDE s, and then

More information

Numerical Solutions to Partial Differential Equations

Numerical 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 information

Numerical Solution Techniques in Mechanical and Aerospace Engineering

Numerical Solution Techniques in Mechanical and Aerospace Engineering Numerical Solution Techniques in Mechanical and Aerospace Engineering Chunlei Liang LECTURE 3 Solvers of linear algebraic equations 3.1. Outline of Lecture Finite-difference method for a 2D elliptic PDE

More information

TImath.com Calculus. Topic: Techniques of Integration Derive the formula for integration by parts and use it to compute integrals

TImath.com Calculus. Topic: Techniques of Integration Derive the formula for integration by parts and use it to compute integrals Integration by Parts ID: 985 Time required 45 minutes Activity Overview In previous activities, students have eplored the differential calculus through investigations of the methods of first principles,

More information

Exact Differential Equations. The general solution of the equation is f x, y C. If f has continuous second partials, then M y 2 f

Exact Differential Equations. The general solution of the equation is f x, y C. If f has continuous second partials, then M y 2 f APPENDIX C Additional Topics in Differential Equations APPENDIX C. Eact First-Order Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Chapter 6, ou studied applications

More information

Finite Difference Methods for Boundary Value Problems

Finite Difference Methods for Boundary Value Problems Finite Difference Methods for Boundary Value Problems October 2, 2013 () Finite Differences October 2, 2013 1 / 52 Goals Learn steps to approximate BVPs using the Finite Difference Method Start with two-point

More information

Explicit Jump Immersed Interface Method: Documentation for 2D Poisson Code

Explicit Jump Immersed Interface Method: Documentation for 2D Poisson Code Eplicit Jump Immersed Interface Method: Documentation for 2D Poisson Code V. Rutka A. Wiegmann November 25, 2005 Abstract The Eplicit Jump Immersed Interface method is a powerful tool to solve elliptic

More information

2.29 Numerical Fluid Mechanics Fall 2009 Lecture 13

2.29 Numerical Fluid Mechanics Fall 2009 Lecture 13 2.29 Numerical Fluid Mechanics Fall 2009 Lecture 13 REVIEW Lecture 12: Classification of Partial Differential Equations (PDEs) and eamples with finite difference discretizations Parabolic PDEs Elliptic

More information

Numerical Integration (Quadrature) Another application for our interpolation tools!

Numerical Integration (Quadrature) Another application for our interpolation tools! Numerical Integration (Quadrature) Another application for our interpolation tools! Integration: Area under a curve Curve = data or function Integrating data Finite number of data points spacing specified

More information

Taylor Series and Series Convergence (Online)

Taylor Series and Series Convergence (Online) 7in 0in Felder c02_online.te V3 - February 9, 205 9:5 A.M. Page CHAPTER 2 Taylor Series and Series Convergence (Online) 2.8 Asymptotic Epansions In introductory calculus classes the statement this series

More information

Multiple Integration

Multiple Integration Contents 7 Multiple Integration 7. Introduction to Surface Integrals 7. Multiple Integrals over Non-rectangular Regions 7. Volume Integrals 4 7.4 Changing Coordinates 66 Learning outcomes In this Workbook

More information

EDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 4 - CALCULUS TUTORIAL 2 - INTEGRATION

EDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 4 - CALCULUS TUTORIAL 2 - INTEGRATION EDEXCEL NATIONAL CERTIFICATE UNIT 8 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME - CALCULUS TUTORIAL - INTEGRATION CONTENTS Be able to apply calculus Differentiation: review of standard derivatives, differentiation

More information

Further factorising, simplifying, completing the square and algebraic proof

Further factorising, simplifying, completing the square and algebraic proof Further factorising, simplifying, completing the square and algebraic proof 8 CHAPTER 8. Further factorising Quadratic epressions of the form b c were factorised in Section 8. by finding two numbers whose

More information

Numerical Solutions to Partial Differential Equations

Numerical Solutions to Partial Differential Equations Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Discretization of Boundary Conditions Discretization of Boundary Conditions On

More information

Numerical Analysis & Computer Programming

Numerical Analysis & Computer Programming ++++++++++ Numerical Analysis & Computer Programming Previous year Questions from 07 to 99 Ramanasri Institute W E B S I T E : M A T H E M A T I C S O P T I O N A L. C O M C O N T A C T : 8 7 5 0 7 0 6

More information

The embedded finite difference method for the Poisson equation in a domain with an irregular boundary and Dirichlet boundary conditions

The embedded finite difference method for the Poisson equation in a domain with an irregular boundary and Dirichlet boundary conditions The embedded finite difference method for the Poisson equation in a domain with an irregular boundary and Dirichlet boundary conditions Z. Jomaa and C. Macaskill School of Mathematics & Statistics University

More information

Finite difference method for elliptic problems: I

Finite difference method for elliptic problems: I Finite difference method for elliptic problems: I Praveen. C praveen@math.tifrbng.res.in Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore 560065 http://math.tifrbng.res.in/~praveen

More information

APPLICATIONS OF FD APPROXIMATIONS FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS

APPLICATIONS OF FD APPROXIMATIONS FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS LECTURE 10 APPLICATIONS OF FD APPROXIMATIONS FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS Ordinary Differential Equations Initial Value Problems For Initial Value problems (IVP s), conditions are specified

More information

lecture 7: Trigonometric Interpolation

lecture 7: Trigonometric Interpolation lecture : Trigonometric Interpolation 9 Trigonometric interpolation for periodic functions Thus far all our interpolation schemes have been based on polynomials However, if the function f is periodic,

More information

A Propagating Wave Packet The Group Velocity

A Propagating Wave Packet The Group Velocity Lecture 7 A Propagating Wave Pacet The Group Velocity Phys 375 Overview and Motivation: Last time we looed at a solution to the Schrödinger equation (SE) with an initial condition (,) that corresponds

More information

Finite Difference Methods (FDMs) 1

Finite Difference Methods (FDMs) 1 Finite Difference Methods (FDMs) 1 1 st - order Approxima9on Recall Taylor series expansion: Forward difference: Backward difference: Central difference: 2 nd - order Approxima9on Forward difference: Backward

More information

Euler-Maclaurin summation formula

Euler-Maclaurin summation formula Physics 4 Spring 6 Euler-Maclaurin summation formula Lecture notes by M. G. Rozman Last modified: March 9, 6 Euler-Maclaurin summation formula gives an estimation of the sum N in f i) in terms of the integral

More information

Lesson 5.1 Exercises, pages

Lesson 5.1 Exercises, pages Lesson 5.1 Eercises, pages 346 352 A 4. Use the given graphs to write the solutions of the corresponding quadratic inequalities. a) 2 2-8 - 10 < 0 The solution is the values of for which y

More information

A Propagating Wave Packet The Group Velocity

A Propagating Wave Packet The Group Velocity Lecture 7 A Propagating Wave Packet The Group Velocity Phys 375 Overview and Motivation: Last time we looked at a solution to the Schrödinger equation (SE) with an initial condition (,) that corresponds

More information

Trigonometric. equations. Topic: Periodic functions and applications. Simple trigonometric. equations. Equations using radians Further trigonometric

Trigonometric. equations. Topic: Periodic functions and applications. Simple trigonometric. equations. Equations using radians Further trigonometric Trigonometric equations 6 sllabusref eferenceence Topic: Periodic functions and applications In this cha 6A 6B 6C 6D 6E chapter Simple trigonometric equations Equations using radians Further trigonometric

More information

Lesson 10.1 Polynomials

Lesson 10.1 Polynomials Lesson 10.1 Polynomials Objectives Classify polynomials. Use algebra tiles to add polynomials. Add and subtract polynomials. A contractor is buying paint to cover the interior of two cubical storage tanks.

More information

Basic Aspects of Discretization

Basic Aspects of Discretization Basic Aspects of Discretization Solution Methods Singularity Methods Panel method and VLM Simple, very powerful, can be used on PC Nonlinear flow effects were excluded Direct numerical Methods (Field Methods)

More information

MA3232 Numerical Analysis Week 9. James Cooley (1926-)

MA3232 Numerical Analysis Week 9. James Cooley (1926-) MA umerical Analysis Week 9 James Cooley (96-) James Cooley is an American mathematician. His most significant contribution to the world of mathematics and digital signal processing is the Fast Fourier

More information

Interpolation Models CHAPTER 3

Interpolation Models CHAPTER 3 CHAPTER 3 CHAPTER OUTLINE 3. Introduction 75 3.2 Polynomial Form of Interpolation Functions 77 3.3 Simple, Comple, and Multiple Elements 78 3.4 Interpolation Polynomial in Terms of Nodal Degrees of Freedom

More information

2.29 Numerical Fluid Mechanics Spring 2015 Lecture 4

2.29 Numerical Fluid Mechanics Spring 2015 Lecture 4 2.29 Spring 2015 Lecture 4 Review Lecture 3 Truncation Errors, Taylor Series and Error Analysis Taylor series: 2 3 n n i1 i i i i i n f( ) f( ) f '( ) f ''( ) f '''( )... f ( ) R 2! 3! n! n1 ( n1) Rn f

More information

ME Computational Fluid Mechanics Lecture 5

ME Computational Fluid Mechanics Lecture 5 ME - 733 Computational Fluid Mechanics Lecture 5 Dr./ Ahmed Nagib Elmekawy Dec. 20, 2018 Elliptic PDEs: Finite Difference Formulation Using central difference formulation, the so called five-point formula

More information

3.1: 1, 3, 5, 9, 10, 12, 14, 18

3.1: 1, 3, 5, 9, 10, 12, 14, 18 3.:, 3, 5, 9,,, 4, 8 ) We want to solve d d c() d = f() with c() = c = constant and f() = for different boundary conditions to get w() and u(). dw d = dw d d = ( )d w() w() = w() = w() ( ) c d d = u()

More information

MATHEMATICS FOR ENGINEERING

MATHEMATICS FOR ENGINEERING MATHEMATICS FOR ENGINEERING INTEGRATION TUTORIAL FURTHER INTEGRATION This tutorial is essential pre-requisite material for anyone studying mechanical engineering. This tutorial uses the principle of learning

More information

SOLVING QUADRATICS. Copyright - Kramzil Pty Ltd trading as Academic Teacher Resources

SOLVING QUADRATICS. Copyright - Kramzil Pty Ltd trading as Academic Teacher Resources SOLVING QUADRATICS Copyright - Kramzil Pty Ltd trading as Academic Teacher Resources SOLVING QUADRATICS General Form: y a b c Where a, b and c are constants To solve a quadratic equation, the equation

More information

Graphing Exponential Functions

Graphing Exponential Functions MHF UI Unit Da Graphing Eponential Functions. Using a table of values (no decimals), graph the function.. For the function, state: a) domain b) range c) equation of the asmptote d) -intercept e) -intercept

More information

SYLLABUS FOR ENTRANCE EXAMINATION NANYANG TECHNOLOGICAL UNIVERSITY FOR INTERNATIONAL STUDENTS A-LEVEL MATHEMATICS

SYLLABUS FOR ENTRANCE EXAMINATION NANYANG TECHNOLOGICAL UNIVERSITY FOR INTERNATIONAL STUDENTS A-LEVEL MATHEMATICS SYLLABUS FOR ENTRANCE EXAMINATION NANYANG TECHNOLOGICAL UNIVERSITY FOR INTERNATIONAL STUDENTS A-LEVEL MATHEMATICS STRUCTURE OF EXAMINATION PAPER. There will be one -hour paper consisting of 4 questions..

More information

Regent College Maths Department. Core Mathematics 4 Trapezium Rule. C4 Integration Page 1

Regent College Maths Department. Core Mathematics 4 Trapezium Rule. C4 Integration Page 1 Regent College Maths Department Core Mathematics Trapezium Rule C Integration Page Integration It might appear to be a bit obvious but you must remember all of your C work on differentiation if you are

More information

Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 584 Mark Sparks 2012

Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 584 Mark Sparks 2012 The Second Fundamental Theorem of Calculus Functions Defined by Integrals Given the functions, f(t), below, use F( ) f ( t) dt to find F() and F () in terms of.. f(t) = 4t t. f(t) = cos t Given the functions,

More information

Monte Carlo Method for Finding the Solution of Dirichlet Partial Differential Equations

Monte Carlo Method for Finding the Solution of Dirichlet Partial Differential Equations Applied Mathematical Sciences, Vol. 1, 2007, no. 10, 453-462 Monte Carlo Method for Finding the Solution of Dirichlet Partial Differential Equations Behrouz Fathi Vajargah Department of Mathematics Guilan

More information

MARIS STELLA HIGH SCHOOL PRELIMINARY EXAMINATION 2

MARIS STELLA HIGH SCHOOL PRELIMINARY EXAMINATION 2 Class Inde Number Name MRIS STELL HIGH SCHOOL PRELIMINRY EXMINTION DDITIONL MTHEMTICS 406/0 8 September 008 Paper hours 0minutes dditional Materials: nswer Paper (6 Sheets RED THESE INSTRUCTIONS FIRST

More information

Chapter XX: 1: Functions. XXXXXXXXXXXXXXX <CT>Chapter 1: Data representation</ct> 1.1 Mappings

Chapter XX: 1: Functions. XXXXXXXXXXXXXXX <CT>Chapter 1: Data representation</ct> 1.1 Mappings 978--08-8-8 Cambridge IGCSE and O Level Additional Mathematics Practice Book Ecerpt Chapter XX: : Functions XXXXXXXXXXXXXXX Chapter : Data representation This section will show you how to: understand

More information

GG303 Lecture 15 10/6/09 1 FINITE STRAIN AND INFINITESIMAL STRAIN

GG303 Lecture 15 10/6/09 1 FINITE STRAIN AND INFINITESIMAL STRAIN GG303 Lecture 5 0609 FINITE STRAIN AND INFINITESIMAL STRAIN I Main Topics on infinitesimal strain A The finite strain tensor [E] B Deformation paths for finite strain C Infinitesimal strain and the infinitesimal

More information

CE 601: Numerical Methods Lecture 7. Course Coordinator: Dr. Suresh A. Kartha, Associate Professor, Department of Civil Engineering, IIT Guwahati.

CE 601: Numerical Methods Lecture 7. Course Coordinator: Dr. Suresh A. Kartha, Associate Professor, Department of Civil Engineering, IIT Guwahati. CE 60: Numerical Methods Lecture 7 Course Coordinator: Dr. Suresh A. Kartha, Associate Professor, Department of Civil Engineering, IIT Guwahati. Drawback in Elimination Methods There are various drawbacks

More information

Application of modified Leja sequences to polynomial interpolation

Application of modified Leja sequences to polynomial interpolation Special Issue for the years of the Padua points, Volume 8 5 Pages 66 74 Application of modified Leja sequences to polynomial interpolation L. P. Bos a M. Caliari a Abstract We discuss several applications

More information

Markscheme November 2016 Mathematics Standard level Paper 1

Markscheme November 2016 Mathematics Standard level Paper 1 N6/5/MATME/SP/ENG/TZ0/XX/M Markscheme November 06 Mathematics Standard level Paper 6 pages N6/5/MATME/SP/ENG/TZ0/XX/M This markscheme is the property of the International Baccalaureate and must not be

More information

3.6 Determinants. 3.6 Determinants 1

3.6 Determinants. 3.6 Determinants 1 3.6 Determinants 1 3.6 Determinants We said in Section 3.3 that a 2 2 matri a b c d is invertible if and only if its erminant, ad - bc, is nonzero, and we saw the erminant used in the formula for the inverse

More information

Lecture 18. Double Integrals (cont d) Electrostatic field near an infinite flat charged plate

Lecture 18. Double Integrals (cont d) Electrostatic field near an infinite flat charged plate Lecture 18 ouble Integrals (cont d) Electrostatic field near an infinite flat charged plate Consider a thin, flat plate of infinite size that is charged, with constant charge density ρ (in appropriate

More information

Fixed Point Theorem and Sequences in One or Two Dimensions

Fixed Point Theorem and Sequences in One or Two Dimensions Fied Point Theorem and Sequences in One or Two Dimensions Dr. Wei-Chi Yang Let us consider a recursive sequence of n+ = n + sin n and the initial value can be an real number. Then we would like to ask

More information

Study Guide and Intervention. The Quadratic Formula and the Discriminant. Quadratic Formula. Replace a with 1, b with -5, and c with -14.

Study Guide and Intervention. The Quadratic Formula and the Discriminant. Quadratic Formula. Replace a with 1, b with -5, and c with -14. Study Guide and Intervention Quadratic Formula The Quadratic Formula can be used to solve any quadratic equation once it is written in the form a 2 + b + c = 0. Quadratic Formula The solutions of a 2 +

More information

Module 3: BASICS OF CFD. Part A: Finite Difference Methods

Module 3: BASICS OF CFD. Part A: Finite Difference Methods Module 3: BASICS OF CFD Part A: Finite Difference Methods THE CFD APPROACH Assembling the governing equations Identifying flow domain and boundary conditions Geometrical discretization of flow domain Discretization

More information

CHAPTER 2 DIFFERENTIATION 2.1 FIRST ORDER DIFFERENTIATION. What is Differentiation?

CHAPTER 2 DIFFERENTIATION 2.1 FIRST ORDER DIFFERENTIATION. What is Differentiation? BA01 ENGINEERING MATHEMATICS 01 CHAPTER DIFFERENTIATION.1 FIRST ORDER DIFFERENTIATION What is Differentiation? Differentiation is all about finding rates of change of one quantity compared to another.

More information

High-order solution of elliptic partial differential equations in domains containing conical singularities

High-order solution of elliptic partial differential equations in domains containing conical singularities High-order solution of elliptic partial differential equations in domains containing conical singularities Thesis by Zhiyi Li In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

More information

The Control-Volume Finite-Difference Approximation to the Diffusion Equation

The Control-Volume Finite-Difference Approximation to the Diffusion Equation The Control-Volume Finite-Difference Approimation to the Diffusion Equation ME 448/548 Notes Gerald Recktenwald Portland State Universit Department of Mechanical Engineering gerr@mepdedu ME 448/548: D

More information

Errors Intensive Computation

Errors Intensive Computation Errors Intensive Computation Annalisa Massini - 2015/2016 OVERVIEW Sources of Approimation Before computation modeling empirical measurements previous computations During computation truncation or discretization

More information

High-order ADI schemes for convection-diffusion equations with mixed derivative terms

High-order ADI schemes for convection-diffusion equations with mixed derivative terms High-order ADI schemes for convection-diffusion equations with mixed derivative terms B. Düring, M. Fournié and A. Rigal Abstract We consider new high-order Alternating Direction Implicit ADI) schemes

More information

Kasetsart University Workshop. Multigrid methods: An introduction

Kasetsart University Workshop. Multigrid methods: An introduction Kasetsart University Workshop Multigrid methods: An introduction Dr. Anand Pardhanani Mathematics Department Earlham College Richmond, Indiana USA pardhan@earlham.edu A copy of these slides is available

More information

Linear Equations and Vectors

Linear Equations and Vectors Chapter Linear Equations and Vectors Linear Algebra, Fall 6 Matrices and Systems of Linear Equations Figure. Linear Algebra, Fall 6 Figure. Linear Algebra, Fall 6 Figure. Linear Algebra, Fall 6 Unique

More information

arxiv:math-ph/ v1 10 Jan 2005

arxiv:math-ph/ v1 10 Jan 2005 Asymptotic and eact series representations for the incomplete Gamma function arxiv:math-ph/5119v1 1 Jan 5 Paolo Amore Facultad de Ciencias, Universidad de Colima, Bernal Díaz del Castillo 34, Colima, Colima,

More information

α x x 0 α x x f(x) α x x α x ( 1) f(x) x f(x) x f(x) α x = α x x 2

α x x 0 α x x f(x) α x x α x ( 1) f(x) x f(x) x f(x) α x = α x x 2 Quadratic speedup for unstructured search - Grover s Al- CS 94- gorithm /8/07 Spring 007 Lecture 11 01 Unstructured Search Here s the problem: You are given an efficient boolean function f : {1,,} {0,1},

More information

Sparse Linear Systems. Iterative Methods for Sparse Linear Systems. Motivation for Studying Sparse Linear Systems. Partial Differential Equations

Sparse Linear Systems. Iterative Methods for Sparse Linear Systems. Motivation for Studying Sparse Linear Systems. Partial Differential Equations Sparse Linear Systems Iterative Methods for Sparse Linear Systems Matrix Computations and Applications, Lecture C11 Fredrik Bengzon, Robert Söderlund We consider the problem of solving the linear system

More information

University of Alberta ENGM 541: Modeling and Simulation of Engineering Systems Laboratory #5

University of Alberta ENGM 541: Modeling and Simulation of Engineering Systems Laboratory #5 University of Alberta ENGM 54: Modeling and Simulation of Engineering Systems Laboratory #5 M.G. Lipsett, Updated 00 Integration Methods with Higher-Order Truncation Errors with MATLAB MATLAB is capable

More information

Course. Print and use this sheet in conjunction with MathinSite s Maclaurin Series applet and worksheet.

Course. Print and use this sheet in conjunction with MathinSite s Maclaurin Series applet and worksheet. Maclaurin Series Learning Outcomes After reading this theory sheet, you should recognise the difference between a function and its polynomial epansion (if it eists!) understand what is meant by a series

More information

Maths A Level Summer Assignment & Transition Work

Maths A Level Summer Assignment & Transition Work Maths A Level Summer Assignment & Transition Work The summer assignment element should take no longer than hours to complete. Your summer assignment for each course must be submitted in the relevant first

More information

Computer Problems for Taylor Series and Series Convergence

Computer Problems for Taylor Series and Series Convergence Computer Problems for Taylor Series and Series Convergence The two problems below are a set; the first should be done without a computer and the second is a computer-based follow up. 1. The drawing below

More information

AE/ME 339. K. M. Isaac Professor of Aerospace Engineering. December 21, 2001 topic13_grid_generation 1

AE/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 information

Geometric Modeling Summer Semester 2010 Mathematical Tools (1)

Geometric Modeling Summer Semester 2010 Mathematical Tools (1) Geometric Modeling Summer Semester 2010 Mathematical Tools (1) Recap: Linear Algebra Today... Topics: Mathematical Background Linear algebra Analysis & differential geometry Numerical techniques Geometric

More information

MTH 3311 Test #1. order 3, linear. The highest order of derivative of y is 2. Furthermore, y and its derivatives are all raised to the

MTH 3311 Test #1. order 3, linear. The highest order of derivative of y is 2. Furthermore, y and its derivatives are all raised to the MTH 3311 Test #1 F 018 Pat Rossi Name Show CLEARLY how you arrive at your answers. 1. Classify the following according to order and linearity. If an equation is not linear, eplain why. (a) y + y y = 4

More information

Page No.1. MTH603-Numerical Analysis_ Muhammad Ishfaq

Page No.1. MTH603-Numerical Analysis_ Muhammad Ishfaq Page No.1 File Version v1.5.3 Update: (Dated: 3-May-011) This version of file contains: Content of the Course (Done) FAQ updated version.(these must be read once because some very basic definition and

More information

Chapter 5 HIGH ACCURACY CUBIC SPLINE APPROXIMATION FOR TWO DIMENSIONAL QUASI-LINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS

Chapter 5 HIGH ACCURACY CUBIC SPLINE APPROXIMATION FOR TWO DIMENSIONAL QUASI-LINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS Chapter 5 HIGH ACCURACY CUBIC SPLINE APPROXIMATION FOR TWO DIMENSIONAL QUASI-LINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS 5.1 Introduction When a physical system depends on more than one variable a general

More information

Numerical Method for Solving Second-Order. Fuzzy Boundary Value Problems. by Using the RPSM

Numerical Method for Solving Second-Order. Fuzzy Boundary Value Problems. by Using the RPSM International Mathematical Forum, Vol., 26, no. 4, 643-658 HIKARI Ltd, www.m-hikari.com http://d.doi.org/.2988/imf.26.6338 Numerical Method for Solving Second-Order Fuzzy Boundary Value Problems by Using

More information

Fourier Analysis Fourier Series C H A P T E R 1 1

Fourier Analysis Fourier Series C H A P T E R 1 1 C H A P T E R Fourier Analysis 474 This chapter on Fourier analysis covers three broad areas: Fourier series in Secs...4, more general orthonormal series called Sturm iouville epansions in Secs..5 and.6

More information

OCR 06 Algebra (Higher)

OCR 06 Algebra (Higher) OCR 06 Algebra (Higher) 1. Simplify. Simplify 5 5 4. 1 3 8y y. 3. A function is given by y 5 3. Write an epression for the inverse of this function. 4. A value,, is input into this function. y The output,

More information

Suggested Problems for Math 122

Suggested Problems for Math 122 Suggested Problems for Math 22 Note: This file will grow as the semester evolves and more sections are added. CCA = Contemporary College Algebra, SIA = Shaum s Intermediate Algebra SIA(.) Rational Epressions

More information

Chapter 5 FINITE DIFFERENCE METHOD (FDM)

Chapter 5 FINITE DIFFERENCE METHOD (FDM) MEE7 Computer Modeling Tecniques in Engineering Capter 5 FINITE DIFFERENCE METHOD (FDM) 5. Introduction to FDM Te finite difference tecniques are based upon approximations wic permit replacing differential

More information

APPLICATIONS OF DIFFERENTIATION

APPLICATIONS OF DIFFERENTIATION 4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION 4.8 Newton s Method In this section, we will learn: How to solve high degree equations using Newton s method. INTRODUCTION Suppose that

More information

Time: 1 hour 30 minutes

Time: 1 hour 30 minutes Paper Reference(s) 6665/01 Edecel GCE Core Mathematics C Silver Level S Time: 1 hour 0 minutes Materials required for eamination papers Mathematical Formulae (Green) Items included with question Nil Candidates

More information

18.303: Introduction to Green s functions and operator inverses

18.303: Introduction to Green s functions and operator inverses 8.33: Introduction to Green s functions and operator inverses S. G. Johnson October 9, 2 Abstract In analogy with the inverse A of a matri A, we try to construct an analogous inverse  of differential

More information

Lab on Taylor Polynomials. This Lab is accompanied by an Answer Sheet that you are to complete and turn in to your instructor.

Lab on Taylor Polynomials. This Lab is accompanied by an Answer Sheet that you are to complete and turn in to your instructor. Lab on Taylor Polynomials This Lab is accompanied by an Answer Sheet that you are to complete and turn in to your instructor. In this Lab we will approimate complicated unctions by simple unctions. The

More information

CHAPTER 3a. INTRODUCTION TO NUMERICAL METHODS

CHAPTER 3a. INTRODUCTION TO NUMERICAL METHODS CHAPTER 3a. INTRODUCTION TO NUMERICAL METHODS A. J. Clark School of Engineering Department of Civil and Environmental Engineering by Dr. Ibrahim A. Assakkaf Spring 1 ENCE 3 - Computation in Civil Engineering

More information

Numerical Analysis of Differential Equations Numerical Solution of Elliptic Boundary Value

Numerical Analysis of Differential Equations Numerical Solution of Elliptic Boundary Value Numerical Analysis of Differential Equations 188 5 Numerical Solution of Elliptic Boundary Value Problems 5 Numerical Solution of Elliptic Boundary Value Problems TU Bergakademie Freiberg, SS 2012 Numerical

More information

1A (13) 1. Find an equation for the tangent line to the graph of y = 3 3y +3at the point ( ; 1). The first thing to do is to check that the values =, y =1satisfy the given equation. They do. Differentiating

More information

Composition of and the Transformation of Functions

Composition of and the Transformation of Functions 1 3 Specific Outcome Demonstrate an understanding of operations on, and compositions of, functions. Demonstrate an understanding of the effects of horizontal and vertical translations on the graphs of

More information

Mathematical Methods for Numerical Analysis and Optimization

Mathematical Methods for Numerical Analysis and Optimization Biyani's Think Tank Concept based notes Mathematical Methods for Numerical Analysis and Optimization (MCA) Varsha Gupta Poonam Fatehpuria M.Sc. (Maths) Lecturer Deptt. of Information Technology Biyani

More information

Chapter 4 Page 1 of 16. Lecture Guide. Math College Algebra Chapter 4. to accompany. College Algebra by Julie Miller

Chapter 4 Page 1 of 16. Lecture Guide. Math College Algebra Chapter 4. to accompany. College Algebra by Julie Miller Chapter 4 Page 1 of 16 Lecture Guide Math 105 - College Algebra Chapter 4 to accompan College Algebra b Julie Miller Corresponding Lecture Videos can be found at Prepared b Stephen Toner & Nichole DuBal

More information

CLASS IX MATHS CHAPTER REAL NUMBERS

CLASS IX MATHS CHAPTER REAL NUMBERS Previous knowledge question Ques. Define natural numbers? CLASS IX MATHS CHAPTER REAL NUMBERS counting numbers are known as natural numbers. Thus,,3,4,. etc. are natural numbers. Ques. Define whole numbers?

More information

= V I = Bus Admittance Matrix. Chapter 6: Power Flow. Constructing Ybus. Example. Network Solution. Triangular factorization. Let

= V I = Bus Admittance Matrix. Chapter 6: Power Flow. Constructing Ybus. Example. Network Solution. Triangular factorization. Let Chapter 6: Power Flow Network Matrices Network Solutions Newton-Raphson Method Fast Decoupled Method Bus Admittance Matri Let I = vector of currents injected into nodes V = vector of node voltages Y bus

More information

1. Introduction. Consider the Navier Stokes equations ηu t ν 2 u + (u grad) u + grad p = f div u = 0 (1.1)

1. Introduction. Consider the Navier Stokes equations ηu t ν 2 u + (u grad) u + grad p = f div u = 0 (1.1) University of Maryland Department of Computer Science TR-C4930 University of Maryland Institute for Advanced Computer Studies TR-009-0 BOUNDARY CONDITIONS IN APPROXIMATE COMMUTATOR PRECONDITIONERS FOR

More information

Name of the Student: Unit I (Solution of Equations and Eigenvalue Problems)

Name of the Student: Unit I (Solution of Equations and Eigenvalue Problems) Engineering Mathematics 8 SUBJECT NAME : Numerical Methods SUBJECT CODE : MA6459 MATERIAL NAME : University Questions REGULATION : R3 UPDATED ON : November 7 (Upto N/D 7 Q.P) (Scan the above Q.R code for

More information

Core Mathematics 2 Unit C2 AS

Core Mathematics 2 Unit C2 AS Core Mathematics 2 Unit C2 AS compulsory unit for GCE AS and GCE Mathematics, GCE AS and GCE Pure Mathematics C2.1 Unit description Algebra and functions; coordinate geometry in the (, y) plane; sequences

More information

The Simple Harmonic Oscillator

The Simple Harmonic Oscillator The Simple Harmonic Oscillator Michael Fowler, University of Virginia Einstein s Solution of the Specific Heat Puzzle The simple harmonic oscillator, a nonrelativistic particle in a potential ½C, is a

More information

Lecture 9 Approximations of Laplace s Equation, Finite Element Method. Mathématiques appliquées (MATH0504-1) B. Dewals, C.

Lecture 9 Approximations of Laplace s Equation, Finite Element Method. Mathématiques appliquées (MATH0504-1) B. Dewals, C. Lecture 9 Approximations of Laplace s Equation, Finite Element Method Mathématiques appliquées (MATH54-1) B. Dewals, C. Geuzaine V1.2 23/11/218 1 Learning objectives of this lecture Apply the finite difference

More information

AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals

AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals 8. Basic Integration Rules In this section we will review various integration strategies. Strategies: I. Separate

More information