Numerical Solution of the Initial Value Problem of the Ordinary Differential Equation with Singular Point by Multi-Step Integration Method
|
|
|
- Bryan Carr
- 5 years ago
- Views:
Transcription
1 Nuerical o te Initial Value Proble o te Ordinar Dierential Equation wit Singular Point b MultiStep Integration Metod KANOKNAPA ERAWUN Kase Bundit Universit Suanluang Bangkok Tailand Abstract. In [] M. Podisuk U. Candang and W. Sanprasert introduced te ultistep integration etod to ind te nuerical solution o te initial value proble o te ordinar dierential equation. W. Sanprasert U. Candang and M. Podisuk in [] used te integration etod wit ortogonal polnoials to ind te nuerical solution o te initial value proble o ordinar dierential equation wit singular point. In tis paper we use te ultistep integration etod in [] to ind te proble in []. Te results b coputer prograing will be illustrated and copare wit te results in []. Kewords: ultistepetod integrationetod Talor series epansion I. Introduction. Te initial value proble o ordinar dierential equation is o te ro () ( ) [a b] () wit te initial condition ( a) () Te above equations are equivalent to te integral equation () (t )dt () a Instead o solving te equations () and () we a solve equation (). We rewrite te equation () as ( )dt () II. Forulation Twopoint orula. We begin b solving or wit te ollowing orula () we use te Talor epansion to ind te value o. Ten we solve or... b te ollowing orula [ ( )]. () Now we solve or te values o... as te sae as inding te values o... Treepoint orula. We begin b solving or and wit te ollowing orulas (7) (8) we use te Talor epansion to ind te value o and. Ten we solve or... b te ollowing orula [ ( ) ( )]. (9) Now we solve or te values o... as te sae as inding te values o.... Fourpoint orula. ISSN: 79 9 ISBN: 97897
2 We begin b solving or and wit te ollowing orula () () () we use te Talor epansion to ind te values o and. Ten we solve or... b te ollowing orula [ ] () Now we solve or te values o... as te sae as inding te values o.... Fivepoint orula. We begin b solving or and wit te ollowing orulas ( ) () () (7) we use te Talor epansion to ind te values o and. Ten we solve or... b te ollowing orula ADVANCES in MATHEMATICAL and COMPUTATIONAL METHODS ISSN: 79 ISBN: 97897
3 [ ( ) ] (8) Now we solve or te values o... as te sae as inding te values o... Sipoint orula. We begin b solving or and wit te ollowing orulas (9 ) () () () () we use te Talor epansion to ind te values o and. Ten we solve or 7... b te ollowing orula [ ] () Now we solve or te values o... as te sae as inding te values o.... ADVANCES in MATHEMATICAL and COMPUTATIONAL METHODS ISSN: 79 ISBN: 97897
4 III. Eaple. Find te nuerical solution o te equation sin ( ) [ ] () wit te initial condition.8888 () Let n and. Te analtical solution o te equations () A() () is () e were 7 A()... (7)!! 7 7! Ten te initial to reer tat equation () (7) we use ind results b tat orulas ()() Te results o te indicated orulas are sown in Tables Table Twopoint orula. H.... Eact Approiated Table Treepoint orula. H.... Eact Approiated ISSN: 79 ISBN: 97897
5 Table Fourpoint orula. Table Sipoint orula. H.... Eact Approiated H.... Eact Approiated Table Fivepoint orula. H.... Eact Approiated Table Global error. Forula..7 Approiated Twopoint Treepoint.7.8 Fourpoint.7.88 Fivepoint.7. Si point.7.8 ISSN: 79 ISBN: 97897
6 IV. Conclusion. Te results are as good as epected. However te twopoint etod is good as te rest. Tere are no signiicant dierent between te. So we recoend te twopoint etod. Note tat our result is not as good as te result in [] since our procedure use te Talor epansion onl at te beginning wic saved uc o coputer eecution tie. V. Reerences. Integration Metod or Solving te Initial Value Proble o Ordinar Dierential Equation (Maitree Podisuk and Wannaporn Sanprasert) KMITL SCIENCE JOURNAL Vol. No. Februar page 9 Integration Metod wit Ortogonal Polnoials (Wannaporn Sanprasert Ungsana Cundang and Maitree Podisuk)Proceeding o te t WSEAS International Conerence on Applied Coputer and Applied Coputational Science Hangzou Cina Ma 9 page ISSN: 79 ISBN: 97897
MA2264 -NUMERICAL METHODS UNIT V : INITIAL VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL. By Dr.T.Kulandaivel Department of Applied Mathematics SVCE
MA64 -NUMERICAL METHODS UNIT V : INITIAL VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS B Dr.T.Kulandaivel Department of Applied Matematics SVCE Numerical ordinar differential equations is te part
5.1 The derivative or the gradient of a curve. Definition and finding the gradient from first principles
Capter 5: Dierentiation In tis capter, we will study: 51 e derivative or te gradient o a curve Deinition and inding te gradient ro irst principles 5 Forulas or derivatives 5 e equation o te tangent line
Derivative at a point
Roberto s Notes on Differential Calculus Capter : Definition of derivative Section Derivative at a point Wat you need to know already: Te concept of liit and basic etods for coputing liits. Wat you can
lecture 35: Linear Multistep Mehods: Truncation Error
88 lecture 5: Linear Multistep Meods: Truncation Error 5.5 Linear ultistep etods One-step etods construct an approxiate solution x k+ x(t k+ ) using only one previous approxiation, x k. Tis approac enoys
(4.2) -Richardson Extrapolation
(.) -Ricardson Extrapolation. Small-O Notation: Recall tat te big-o notation used to define te rate of convergence in Section.: Suppose tat lim G 0 and lim F L. Te function F is said to converge to L as
Solving Second Order Linear Dirichlet and Neumann Boundary Value Problems by Block Method
IAENG International Journal o Applied Matematics 43: IJAM_43 04 Solving Second Order Linear Diriclet and Neumann Boundar Value Problems b Block Metod Zanaria Abdul Majid Mod Mugti Hasni and Norazak Senu
MATH1901 Differential Calculus (Advanced)
MATH1901 Dierential Calculus (Advanced) Capter 3: Functions Deinitions : A B A and B are sets assigns to eac element in A eactl one element in B A is te domain o te unction B is te codomain o te unction
4.2 - Richardson Extrapolation
. - Ricardson Extrapolation. Small-O Notation: Recall tat te big-o notation used to define te rate of convergence in Section.: Definition Let x n n converge to a number x. Suppose tat n n is a sequence
232 Calculus and Structures
3 Calculus and Structures CHAPTER 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS FOR EVALUATING BEAMS Calculus and Structures 33 Copyrigt Capter 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS 17.1 THE
Numerical solution of Boundary Value Problems by Piecewise Analysis Method
ISSN 4-86 (Paper) ISSN 5-9 (Online) Vol., No.4, Nuerical solution of Boundar Value Probles b Piecewise Analsis Method + O. A. TAIWO; A.O ADEWUMI and R. A. RAJI * Departent of Matheatics, Universit of Ilorin,
INTRODUCTION TO CALCULUS LIMITS
Calculus can be divided into two ke areas: INTRODUCTION TO CALCULUS Differential Calculus dealing wit its, rates of cange, tangents and normals to curves, curve sketcing, and applications to maima and
Solving Continuous Linear Least-Squares Problems by Iterated Projection
Solving Continuous Linear Least-Squares Problems by Iterated Projection by Ral Juengling Department o Computer Science, Portland State University PO Box 75 Portland, OR 977 USA Email: juenglin@cs.pdx.edu
Logarithmic functions
Roberto s Notes on Differential Calculus Capter 5: Derivatives of transcendental functions Section Derivatives of Logaritmic functions Wat ou need to know alread: Definition of derivative and all basic
3.1 Extreme Values of a Function
.1 Etreme Values of a Function Section.1 Notes Page 1 One application of te derivative is finding minimum and maimum values off a grap. In precalculus we were only able to do tis wit quadratics by find
2 4 Substituting 2 into gives 2(2) 4. + x + x +...= + + x ! 4! Substituting x= 0 and =1 into (1 + x ) + x = 0, gives 0
Talor Series 6C 1 Differentiating = +, with respect to, gives = 1+ (1) Differentiating (1) gives = () =, = 1 into = +, gives = + (1), so = 1 1 = into (1) gives =1+ = into gives () = () = = Sousing the
Sixth and Fourth Order Compact Finite Difference Schemes for Two and Three Dimension Poisson Equation with Two Methods to derive These Schemes
Basra Journal of Scienec (A) Vol.(),-0, 00 Sit and Fourt Order Compact Finite Difference Scemes for Two and Tree Dimension Poisson Equation wit Two Metods to derive Tese Scemes Akil J. Harfas Huda A. Jalob
LAB #3: ELECTROSTATIC FIELD COMPUTATION
ECE 306 Revised: 1-6-00 LAB #3: ELECTROSTATIC FIELD COMPUTATION Purpose During tis lab you will investigate te ways in wic te electrostatic field can be teoretically predicted. Bot analytic and nuerical
Section 2.1 The Definition of the Derivative. We are interested in finding the slope of the tangent line at a specific point.
Popper 6: Review of skills: Find tis difference quotient. f ( x ) f ( x) if f ( x) x Answer coices given in audio on te video. Section.1 Te Definition of te Derivative We are interested in finding te slope
Integral Calculus, dealing with areas and volumes, and approximate areas under and between curves.
Calculus can be divided into two ke areas: Differential Calculus dealing wit its, rates of cange, tangents and normals to curves, curve sketcing, and applications to maima and minima problems Integral
MAT 1800 FINAL EXAM HOMEWORK
MAT 800 FINAL EXAM HOMEWORK Read te directions to eac problem careully ALL WORK MUST BE SHOWN DO NOT USE A CALCULATOR Problems come rom old inal eams (SS4, W4, F, SS, W) Solving Equations: Let 5 Find all
Analysis of Analytical and Numerical Methods of Epidemic Models
Iteratioal Joural of Egieerig Reearc ad Geeral Sciece Volue, Iue, Noveber-Deceber, 05 ISSN 09-70 Aalyi of Aalytical ad Nuerical Metod of Epideic Model Pooa Kuari Aitat Profeor, Departet of Mateatic Magad
Differentiation. introduction to limits
9 9A Introduction to limits 9B Limits o discontinuous, rational and brid unctions 9C Dierentiation using i rst principles 9D Finding derivatives b rule 9E Antidierentiation 9F Deriving te original unction
Taylor Series and the Mean Value Theorem of Derivatives
1 - Taylor Series and te Mean Value Teorem o Derivatives Te numerical solution o engineering and scientiic problems described by matematical models oten requires solving dierential equations. Dierential
Continuity and Differentiability
Continuity and Dierentiability Tis capter requires a good understanding o its. Te concepts o continuity and dierentiability are more or less obvious etensions o te concept o its. Section - INTRODUCTION
Physics 6C. De Broglie Wavelength Uncertainty Principle. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Pyic 6C De Broglie Wavelengt Uncertainty Principle De Broglie Wavelengt Bot ligt and atter ave bot particle and wavelike propertie. We can calculate te wavelengt of eiter wit te ae forula: p v For large
We name Functions f (x) or g(x) etc.
Section 2 1B: Function Notation Bot of te equations y 2x +1 and y 3x 2 are functions. It is common to ave two or more functions in terms of x in te same problem. If I ask you wat is te value for y if x
روش نصف کردن. Bisection Method Basis of Bisection Method
10/18/01 روش نصف کردن Bisection Method http://www.pedra-payvandy.co 1 Basis o Bisection Method Theore An equation ()=0, where () is a real continuous unction, has at least one root between l and u i (
Nonlinear Integro-differential Equations by Differential Transform Method with Adomian Polynomials
Math Sci Lett No - Mathematical Science Letters An International Journal http://ddoior//msl/ Nonlinear Intero-dierential Equations b Dierential Transorm Method with Adomian Polnomials S H Behir General
1 + t5 dt with respect to x. du = 2. dg du = f(u). du dx. dg dx = dg. du du. dg du. dx = 4x3. - page 1 -
Eercise. Find te derivative of g( 3 + t5 dt wit respect to. Solution: Te integrand is f(t + t 5. By FTC, f( + 5. Eercise. Find te derivative of e t2 dt wit respect to. Solution: Te integrand is f(t e t2.
LIMITS AND DERIVATIVES CONDITIONS FOR THE EXISTENCE OF A LIMIT
LIMITS AND DERIVATIVES Te limit of a function is defined as te value of y tat te curve approaces, as x approaces a particular value. Te limit of f (x) as x approaces a is written as f (x) approaces, as
Eva Stanová 1. mail:
Te International Journal of TRANSPORT &LOGISTICS Medzinárodný časopis DOPRAVA A LOGISTIKA ISSN 45-7X GEOMETRY OF OVAL STRAND CREATED OF n +n +n WIRES Eva Stanová Civil engineering facult, Tecnical Universit
Modern Control Systems (ECEG-4601) Instructor: Andinet Negash. Chapter 1 Lecture 3: State Space, II
Modern Control Systes (ECEG-46) Instructor: Andinet Negash Chapter Lecture 3: State Space, II Eaples Eaple 5: control o liquid levels: in cheical plants, it is oten necessary to aintain the levels o liquids.
pancakes. A typical pancake also appears in the sketch above. The pancake at height x (which is the fraction x of the total height of the cone) has
Volumes One can epress volumes of regions in tree dimensions as integrals using te same strateg as we used to epress areas of regions in two dimensions as integrals approimate te region b a union of small,
Algebraic Multigrid. Multigrid
Algebraic Mltigrid We re going to discss algebraic ltigrid bt irst begin b discssing ordinar ltigrid. Both o these deal with scale space eaining the iage at ltiple scales. This is iportant or segentation
1 Proving the Fundamental Theorem of Statistical Learning
THEORETICAL MACHINE LEARNING COS 5 LECTURE #7 APRIL 5, 6 LECTURER: ELAD HAZAN NAME: FERMI MA ANDDANIEL SUO oving te Fundaental Teore of Statistical Learning In tis section, we prove te following: Teore.
( ) ! = = = = = = = 0. ev nm. h h hc 5-4. (from Equation 5-2) (a) For an electron:! = = 0. % & (b) For a proton: (c) For an alpha particle:
5-3. ( c) 1 ( 140eV nm) (. )(. ) Ek = evo = = V 940 o = = V 5 m mc e 5 11 10 ev 0 04nm 5-4. c = = = mek mc Ek (from Equation 5-) 140eV nm (a) For an electron: = = 0. 0183nm ( )( 0 511 10 )( 4 5 10 3 )
Observability Analysis of Nonlinear Systems Using Pseudo-Linear Transformation
9t IFAC Symposium on Nonlinear Control Systems Toulouse, France, September 4-6, 2013 TC3.4 Observability Analysis o Nonlinear Systems Using Pseudo-Linear Transormation Yu Kawano Tosiyui Otsua Osaa University,
Midterm #1B. x 8 < < x 8 < 11 3 < x < x > x < 5 or 3 2x > 5 2x < 8 2x > 2
Mat 30 College Algebra Februar 2, 2016 Midterm #1B Name: Answer Ke David Arnold Instructions. ( points) For eac o te ollowing questions, select te best answer and darken te corresponding circle on our
Numerical Solution for Non-Stationary Heat Equation in Cooling of Computer Radiator System
(JZS) Journal of Zankoy Sulaiani, 9, 1(1) Part A (97-1) A119 Nuerical Solution for Non-Stationary Heat Equation in Cooling of Coputer Radiator Syste Aree A. Maad*, Faraidun K. Haa Sal**, and Najadin W.
MATH1151 Calculus Test S1 v2a
MATH5 Calculus Test 8 S va January 8, 5 Tese solutions were written and typed up by Brendan Trin Please be etical wit tis resource It is for te use of MatSOC members, so do not repost it on oter forums
Physics Teach Yourself Series Topic 15: Wavelike nature of matter (Unit 4)
Pysics Teac Yourself Series Topic 15: Wavelie nature of atter (Unit 4) A: Level 14, 474 Flinders Street Melbourne VIC 3000 T: 1300 134 518 W: tss.co.au E: info@tss.co.au TSSM 2017 Page 1 of 8 Contents
UNIT #6 EXPONENTS, EXPONENTS, AND MORE EXPONENTS REVIEW QUESTIONS
Answer Key Name: Date: UNIT # EXPONENTS, EXPONENTS, AND MORE EXPONENTS REVIEW QUESTIONS Part I Questions. Te epression 0 can be simpliied to () () 0 0. Wic o te ollowing is equivalent to () () 8 8? 8.
Section 12. Afocal Systems
OPTI-0/0 Geoetrical and Instruental Optics Copyrigt 08 Jon E. Greivenkap - Section Aocal Systes Gaussian Optics Teores In te initial discussion o Gaussian optics, one o te teores deined te two dierent
Finite fields. and we ve used it in various examples and homework problems. In these notes I will introduce more finite fields
Finite fields I talked in class about the field with two eleents F 2 = {, } and we ve used it in various eaples and hoework probles. In these notes I will introduce ore finite fields F p = {,,...,p } for
Numerical Analysis MTH603. dy dt = = (0) , y n+1. We obtain yn. Therefore. and. Copyright Virtual University of Pakistan 1
Numerical Analysis MTH60 PREDICTOR CORRECTOR METHOD Te metods presented so far are called single-step metods, were we ave seen tat te computation of y at t n+ tat is y n+ requires te knowledge of y n only.
Computer Derivations of Numerical Differentiation Formulae. Int. J. of Math. Education in Sci. and Tech., V 34, No 2 (March-April 2003), pp
Computer Derivations o Numerical Dierentiation Formulae By Jon H. Matews Department o Matematics Caliornia State University Fullerton USA Int. J. o Mat. Education in Sci. and Tec. V No (Marc-April ) pp.8-87.
TABLE FOR UPPER PERCENTAGE POINTS OF THE LARGEST ROOT OF A DETERMINANTAL EQUATION WITH FIVE ROOTS. By William W. Chen
TABLE FOR UPPER PERCENTAGE POINTS OF THE LARGEST ROOT OF A DETERMINANTAL EQUATION WITH FIVE ROOTS By Willia W. Chen The distribution of the non-null characteristic roots of a atri derived fro saple observations
11.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR
SECTION 11.6 DIRECTIONAL DERIVATIVES AND THE GRADIENT VECTOR 633 wit speed v o along te same line from te opposite direction toward te source, ten te frequenc of te sound eard b te observer is were c is
Lesson 6: The Derivative
Lesson 6: Te Derivative Def. A difference quotient for a function as te form f(x + ) f(x) (x + ) x f(x + x) f(x) (x + x) x f(a + ) f(a) (a + ) a Notice tat a difference quotient always as te form of cange
Bulletin of the Seismological Society of America, Vol. 92, No. 8, pp , December 2002
Bulletin of te Seisological Society of erica, Vol. 9, No. 8, pp. 304 3066, Deceber 00 3D eterogeneous Staggered-Grid Finite-Difference Modeling of Seisic Motion wit Volue aronic ritetic veraging of Elastic
158 Calculus and Structures
58 Calculus and Structures CHAPTER PROPERTIES OF DERIVATIVES AND DIFFERENTIATION BY THE EASY WAY. Calculus and Structures 59 Copyrigt Capter PROPERTIES OF DERIVATIVES. INTRODUCTION In te last capter you
c hc h c h. Chapter Since E n L 2 in Eq. 39-4, we see that if L is doubled, then E 1 becomes (2.6 ev)(2) 2 = 0.65 ev.
Capter 39 Since n L in q 39-4, we see tat if L is doubled, ten becoes (6 ev)() = 065 ev We first note tat since = 666 0 34 J s and c = 998 0 8 /s, 34 8 c6 66 0 J sc 998 0 / s c 40eV n 9 9 60 0 J / ev 0
On the Convergence of Non-Polynomial Spline Finite Difference Method for Quasi-Linear Elliptic Boundary Value Problems in Two-Space Dimensions
Journal of Advances in Applied Mateatics Vol. No. January 06 ttps://d.doi.org/0.606/jaa.06.006 59 On te Convergence of Non-Polynoial Spline Finite Difference Metod for Quasi-Linear Elliptic Boundary Value
d y f f dy Numerical Solution of Ordinary Differential Equations Consider the 1 st order ordinary differential equation (ODE) . dx
umerical Solutio o Ordiar Dieretial Equatios Cosider te st order ordiar dieretial equatio ODE d. d Te iitial coditio ca be tae as. Te we could use a Talor series about ad obtai te complete solutio or......!!!
Numerical Approach for Solving Stiff Differential Equations Through the Extended Trapezoidal Rule Formulae
Aerican Journal o Matheatical and Coputer Modelling 7; (4): -6 http://wwwciencepublihinggroupco//ac doi: 648/ac74 Nuerical Approach or Solving Sti Dierential Equation Through the Extended Trapezoidal Rule
On the Concept of Returns to Scale: Revisited
3 J. Asian Dev. Stud, Vol. 5, Issue, (Marc 206) ISSN 2304-375X On te Concept of Returns to Scale: Revisited Parvez Azim Abstract Tis paper sows w it is tat in Economics text books and literature we invariabl
The Schrödinger Equation and the Scale Principle
Te Scrödinger Equation and te Scale Princile RODOLFO A. FRINO Jul 014 Electronics Engineer Degree fro te National Universit of Mar del Plata - Argentina rodolfo_frino@aoo.co.ar Earlier tis ear (Ma) I wrote
MATH CALCULUS I 2.1: Derivatives and Rates of Change
MATH 12002 - CALCULUS I 2.1: Derivatives and Rates of Cange Professor Donald L. Wite Department of Matematical Sciences Kent State University D.L. Wite (Kent State University) 1 / 1 Introduction Our main
University Mathematics 2
University Matematics 2 1 Differentiability In tis section, we discuss te differentiability of functions. Definition 1.1 Differentiable function). Let f) be a function. We say tat f is differentiable at
DIFFERENTIAL POLYNOMIALS GENERATED BY SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS
Journal o Applied Analysis Vol. 14, No. 2 2008, pp. 259 271 DIFFERENTIAL POLYNOMIALS GENERATED BY SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS B. BELAÏDI and A. EL FARISSI Received December 5, 2007 and,
1. Questions (a) through (e) refer to the graph of the function f given below. (A) 0 (B) 1 (C) 2 (D) 4 (E) does not exist
Mat 1120 Calculus Test 2. October 18, 2001 Your name Te multiple coice problems count 4 points eac. In te multiple coice section, circle te correct coice (or coices). You must sow your work on te oter
On One Justification on the Use of Hybrids for the Solution of First Order Initial Value Problems of Ordinary Differential Equations
Pure and Applied Matematics Journal 7; 6(5: 74 ttp://wwwsciencepublisinggroupcom/j/pamj doi: 648/jpamj765 ISSN: 6979 (Print; ISSN: 698 (Online On One Justiication on te Use o Hybrids or te Solution o First
Order of Accuracy. ũ h u Ch p, (1)
Order of Accuracy 1 Terminology We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, wic can be for instance te grid size or time step in a numerical
An Accurate Self-Starting Initial Value Solvers for. Second Order Ordinary Differential Equations
International Journal of Contemporar Matematical Sciences Vol. 9, 04, no. 5, 77-76 HIKARI Ltd, www.m-iari.com ttp://dx.doi.org/0.988/icms.04.4554 An Accurate Self-Starting Initial Value Solvers for Second
MTH-112 Quiz 1 Name: # :
MTH- Quiz Name: # : Please write our name in te provided space. Simplif our answers. Sow our work.. Determine weter te given relation is a function. Give te domain and range of te relation.. Does te equation
The Derivative as a Function
Section 2.2 Te Derivative as a Function 200 Kiryl Tsiscanka Te Derivative as a Function DEFINITION: Te derivative of a function f at a number a, denoted by f (a), is if tis limit exists. f (a) f(a + )
Pre-Calculus Review Preemptive Strike
Pre-Calculus Review Preemptive Strike Attaced are some notes and one assignment wit tree parts. Tese are due on te day tat we start te pre-calculus review. I strongly suggest reading troug te notes torougly
Chapter 8. Numerical Solution of Ordinary Differential Equations. Module No. 2. Predictor-Corrector Methods
Numerical Analysis by Dr. Anita Pal Assistant Professor Department of Matematics National Institute of Tecnology Durgapur Durgapur-7109 email: anita.buie@gmail.com 1 . Capter 8 Numerical Solution of Ordinary
Numerical Differentiation
Numerical Differentiation Finite Difference Formulas for te first derivative (Using Taylor Expansion tecnique) (section 8.3.) Suppose tat f() = g() is a function of te variable, and tat as 0 te function
1 2 x Solution. The function f x is only defined when x 0, so we will assume that x 0 for the remainder of the solution. f x. f x h f x.
Problem. Let f x x. Using te definition of te derivative prove tat f x x Solution. Te function f x is only defined wen x 0, so we will assume tat x 0 for te remainder of te solution. By te definition of
The Verlet Algorithm for Molecular Dynamics Simulations
Cemistry 380.37 Fall 2015 Dr. Jean M. Standard November 9, 2015 Te Verlet Algoritm for Molecular Dynamics Simulations Equations of motion For a many-body system consisting of N particles, Newton's classical
1. State whether the function is an exponential growth or exponential decay, and describe its end behaviour using limits.
Questions 1. State weter te function is an exponential growt or exponential decay, and describe its end beaviour using its. (a) f(x) = 3 2x (b) f(x) = 0.5 x (c) f(x) = e (d) f(x) = ( ) x 1 4 2. Matc te
Outline. Review Numerical Approach. Schedule for April and May. Review Simple Methods. Review Notation and Order
Sstes of Ordnar Dfferental Equatons Aprl, Solvng Sstes of Ordnar Dfferental Equatons Larr Caretto Mecancal Engneerng 9 Nuercal Analss of Engneerng Sstes Aprl, Outlne Revew bascs of nuercal solutons of
Homotopy analysis of 1D unsteady, nonlinear groundwater flow through porous media
Hootopy analysis of D unsteady, nonlinear groundwater flow troug porous edia Autor Song, Hao, Tao, Longbin Publised 7 Journal Title Journal of Coastal Researc Copyrigt Stateent 7 CERF. Te attaced file
Nonlinear free flexural vibrations of functionally graded rectangular and skew plates under. thermal environments
Nonlinear free fleural vibrations of functionall graded rectangular and sew plates under teral environents N. Sundararajan. Praas and M. Ganapati 3 * 4 Sout car street V oil irucirapalli ailnadu India.
More on generalized inverses of partitioned matrices with Banachiewicz-Schur forms
More on generalized inverses of partitioned matrices wit anaciewicz-scur forms Yongge Tian a,, Yosio Takane b a Cina Economics and Management cademy, Central University of Finance and Economics, eijing,
Chapter 10 Light- Reflectiion & Refraction
Capter 0 Ligt- Relectiion & Reraction Intext Questions On Page 68 Question : Deine te principal ocus o a concae irror. Principal ocus o te concae irror: A point on principal axis on wic parallel ligt rays
Math Spring 2013 Solutions to Assignment # 3 Completion Date: Wednesday May 15, (1/z) 2 (1/z 1) 2 = lim
Mat 311 - Spring 013 Solutions to Assignment # 3 Completion Date: Wednesday May 15, 013 Question 1. [p 56, #10 (a)] 4z Use te teorem of Sec. 17 to sow tat z (z 1) = 4. We ave z 4z (z 1) = z 0 4 (1/z) (1/z
DELFT UNIVERSITY OF TECHNOLOGY Faculty of Electrical Engineering, Mathematics and Computer Science
DELFT UNIVERSITY OF TECHNOLOGY Faculty of Electrical Engineering, Matematics and Computer Science. ANSWERS OF THE TEST NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS (WI3097 TU) Tuesday January 9 008, 9:00-:00
Chapter 1D - Rational Expressions
- Capter 1D Capter 1D - Rational Expressions Definition of a Rational Expression A rational expression is te quotient of two polynomials. (Recall: A function px is a polynomial in x of degree n, if tere
Conductance from Transmission Probability
Conductance rom Transmission Probability Kelly Ceung Department o Pysics & Astronomy University o Britis Columbia Vancouver, BC. Canada, V6T1Z1 (Dated: November 5, 005). ntroduction For large conductors,
Neural Networks Trained with the EEM Algorithm: Tuning the Smoothing Parameter
eural etworks Trained wit te EEM Algorit: Tuning te Sooting Paraeter JORGE M. SATOS,2, JOAQUIM MARQUES DE SÁ AD LUÍS A. ALEXADRE 3 Intituto de Engenaria Bioédica, Porto, Portugal 2 Instituto Superior de
Physically Based Modeling: Principles and Practice Implicit Methods for Differential Equations
Pysically Based Modeling: Principles and Practice Implicit Metods for Differential Equations David Baraff Robotics Institute Carnegie Mellon University Please note: Tis document is 997 by David Baraff
1 1. Rationalize the denominator and fully simplify the radical expression 3 3. Solution: = 1 = 3 3 = 2
MTH - Spring 04 Exam Review (Solutions) Exam : February 5t 6:00-7:0 Tis exam review contains questions similar to tose you sould expect to see on Exam. Te questions included in tis review, owever, are
µ Differential Equations MAΘ National Convention 2017 For all questions, answer choice E) NOTA means that none of the above answers is correct.
µ Differential Equations MAΘ National Convention 07 For all questions, answer choice E) NOTA means that none of the above answers is correct.. C) Since f (, ) = +, (0) = and h =. Use Euler s method, we
NUMERICAL DIFFERENTIAL 1
NUMERICAL DIFFERENTIAL Ruge-Kutta Metods Ruge-Kutta metods are ver popular ecause o teir good eiciec; ad are used i most computer programs or dieretial equatios. Te are sigle-step metods as te Euler metods.
. Compute the following limits.
Today: Tangent Lines and te Derivative at a Point Warmup:. Let f(x) =x. Compute te following limits. f( + ) f() (a) lim f( +) f( ) (b) lim. Let g(x) = x. Compute te following limits. g(3 + ) g(3) (a) lim
Lecture XVII. Abstract We introduce the concept of directional derivative of a scalar function and discuss its relation with the gradient operator.
Lecture XVII Abstract We introduce te concept of directional derivative of a scalar function and discuss its relation wit te gradient operator. Directional derivative and gradient Te directional derivative
Pre-lab Quiz/PHYS 224 Earth s Magnetic Field. Your name Lab section
Pre-lab Quiz/PHYS 4 Eart s Magnetic Field Your name Lab section 1. Wat do you investigate in tis lab?. For a pair of Helmoltz coils described in tis manual and sown in Figure, r=.15 m, N=13, I =.4 A, wat
Section 15.6 Directional Derivatives and the Gradient Vector
Section 15.6 Directional Derivatives and te Gradient Vector Finding rates of cange in different directions Recall tat wen we first started considering derivatives of functions of more tan one variable,
Problem Score 1 /25 2 /16 3 /20 4 /24 5 /15 Total /100
PHYS 4 Exa 3 Noveber 8, 6 Tie o Discussion Section: Nae: Instructions Clear your desk o everything but pens/pencils and erasers. Take o all caps, hats, visors, etc., and place the on the loor under your
Fall 2014 MAT 375 Numerical Methods. Numerical Differentiation (Chapter 9)
Fall 2014 MAT 375 Numerical Metods (Capter 9) Idea: Definition of te derivative at x Obviuos approximation: f (x) = lim 0 f (x + ) f (x) f (x) f (x + ) f (x) forward-difference formula? ow good is tis
AVL trees. AVL trees
Dnamic set DT dnamic set DT is a structure tat stores a set of elements. Eac element as a (unique) ke and satellite data. Te structure supports te following operations. Searc(S, k) Return te element wose
Math 31A Discussion Notes Week 4 October 20 and October 22, 2015
Mat 3A Discussion Notes Week 4 October 20 and October 22, 205 To prepare for te first midterm, we ll spend tis week working eamples resembling te various problems you ve seen so far tis term. In tese notes
Name: Sept 21, 2017 Page 1 of 1
MATH 111 07 (Kunkle), Eam 1 100 pts, 75 minutes No notes, books, electronic devices, or outside materials of an kind. Read eac problem carefull and simplif our answers. Name: Sept 21, 2017 Page 1 of 1
MA119-A Applied Calculus for Business Fall Homework 4 Solutions Due 9/29/ :30AM
MA9-A Applied Calculus for Business 006 Fall Homework Solutions Due 9/9/006 0:0AM. #0 Find te it 5 0 + +.. #8 Find te it. #6 Find te it 5 0 + + = (0) 5 0 (0) + (0) + =.!! r + +. r s r + + = () + 0 () +
WYSE Academic Challenge 2004 Sectional Mathematics Solution Set
WYSE Academic Callenge 00 Sectional Matematics Solution Set. Answer: B. Since te equation can be written in te form x + y, we ave a major 5 semi-axis of lengt 5 and minor semi-axis of lengt. Tis means
CHAPTER 2 MODELING OF THREE-TANK SYSTEM
7 CHAPTER MODELING OF THREE-TANK SYSTEM. INTRODUCTION Te interacting tree-tank system is a typical example of a nonlinear MIMO system. Heiming and Lunze (999) ave regarded treetank system as a bencmark
Finite Difference Formulae for Unequal Sub- Intervals Using Lagrange s Interpolation Formula
Int. Journal o Mat. Analyi, Vol., 9, no. 7, 85-87 Finite Dierence Formulae or Unequal Sub- Interval Uing Lagrange Interpolation Formula Aok K. Sing a and B. S. Badauria b Department o Matematic, Faculty
Proceedings of the 35th Hawaii International Conference on System Sciences
-7695-45-9/ $7. (c) IEEE roceedings of the 5th Annual Hawaii International Conference on Syste Sciences (HICSS-5 ) -7695-45-9/ $7. IEEE roceedings of the 5th Hawaii International Conference on Syste Sciences