Fourth Order RK-Method
|
|
- Kristin Dalton
- 6 years ago
- Views:
Transcription
1 Fourth Order RK-Method The most commonly used method is Runge-Kutta fourth order method. The fourth order RK-method is y i+1 = y i (k 1 + 2k 2 + 2k 3 + k 4 ), Ordinary Differential Equations (ODE) p.44/89
2 where k 1 = hf(x i, y i ), ( k 2 = hf x i + h 2, y i + k 1 2 ( k 3 = hf x i + h 2, y i + k 2 2 k 4 = hf(x i + h, y i + k 3 ). ) ),, Ordinary Differential Equations (ODE) p.45/89
3 Example 1 Find the approximate solution of the initial value problem dx dt = 1 + x t, 1 t 3 with the initial condition x(1) = 1, using the Runge-Kutta second order and fourth order with step size of h = 1. Ordinary Differential Equations (ODE) p.46/89
4 Solution RK 2nd order method. The formula is y i+1 = y i + h 2 (k 1 + k 2 ), where k 1 = f(x i, t i ), k 2 = f(x i + h, t i + hk 1 ). Here, h = 1 and t 0 = x 0 = 1. Therefore, k 1 = f(t 0, x 0 ) = 1 + x 0 t 0 = 2, k 2 = f(t 0 + h, x 0 + hk 1 ) = f(2, 3) = = 2.5. Ordinary Differential Equations (ODE) p.47/89
5 Thus, y 1 = ( ) = Similary, we calculate y 2. Ordinary Differential Equations (ODE) p.48/89
6 RK 4th order method. The formula is where y i+1 = y i (k 1 + 2k 2 + 2k 3 + k 4 ), k 1 = hf(t i, x i ), ( k 2 = hf t i + h 2, x i + k 1 2 ( k 3 = hf t i + h 2, x i + k 2 2 ) ),, k 4 = hf(t i + h, x i + k 3 ). Ordinary Differential Equations (ODE) p.49/89
7 k 1 = hf(t 0, x 0 ) = f(1, 1) = 2, ( k 2 = hf t 0 + h 2, x 0 + k ) 1 = f(1.5, 2) = , 2 ( k 3 = hf t 0 + h 2, x 0 + k ) 2 = , 2 k 4 = hf(t 0 + h, x 0 + k 3 ) = Ordinary Differential Equations (ODE) p.50/89
8 We find y 1 = ( ) = Similarly, we calculate y 2. Ordinary Differential Equations (ODE) p.51/89
9 System of First Order ODE s We consider the n system of first order equations dy 1 dx dy 2 dx dy n dx = f 1 (x, y 1, y 2,, y n ), y 1 (x 0 ) = y 10, = f 2 (x, y 1, y 2,, y n ), y 2 (x 0 ) = y 20, = f n (x, y 1, y 2,, y n ), y n (x 0 ) = y n0. With the help of single step methods, we find the approximate solution for the system of n equations. Ordinary Differential Equations (ODE) p.52/89
10 We now consider the two first order equations of the form dy 1 dx = f 1(x, y 1, y 2 ), y 1 (x 0 ) = y 10, dy 2 dx = f 2(x, y 1, y 2 ), y 2 (x 0 ) = y 20. We use RK second order and fourth order method to solve the system of equations. Ordinary Differential Equations (ODE) p.53/89
11 2nd Order RK Method The formula for second order RK method with step size h is where y 1,n+1 = = y 1,n (s 1 + k 1 ), y 2,n+1 = y 2,n (s 2 + k 2 ), k 1 = hf 1 (x n, y 1n, y 2n ), k 2 = hf 2 (x n, y 1n, y 2n ), Ordinary Differential Equations (ODE) p.54/89
12 s 1 = hf 1 (x n + h, y 1n + k 1, y 2n + k 2 ), s 2 = hf 2 (x n + h, y 1n + k 1, y 2n + k 2 ). Similarly, the formula for fourth order RK-method is y 1,n+1 = y 1,n (k 1 + 2s 1 + 2l 1 + p 1 ), y 2,n+1 = y 2,n (k 2 + 2s 2 + 2l 2 + p 2 ), Ordinary Differential Equations (ODE) p.55/89
13 where k 1 = hf 1 (x n, y 1n, y 2n ), k 2 = hf 2 (x n, y 1n, y 2n ), s 1 = hf 1 (x n + h 2, y 1n + k 1 2, y 2n + k 2 2 ), s 2 = hf 2 (x n + h 2, y 1n + k 1 2, y 2n + k 2 2 ) Ordinary Differential Equations (ODE) p.56/89
14 l 1 = hf 1 (x n + h 2, y 1n + s 1 2, y 2n + s 2 2 ), l 2 = hf 2 (x n + h 2, y 1n + s 1 2, y 2n + s 2 2 ), p 1 = hf 1 (x n + h, y 1n + l 1, y 2n + l 2 ), p 2 = hf 2 (x n + h, y 1n + l 1, y 2n + l 2 ). Ordinary Differential Equations (ODE) p.57/89
15 Example 2 Use RK-method 2nd order and 4th order to find the approximate solution of y(0.1) and z(0.1) as a solution of pair of equations with the initial conditions dy dx = x + z, dz dx = y x y(0) = 1, z(0) = 1. Take step size h = 0.1. Ordinary Differential Equations (ODE) p.58/89
16 Multistep Methods The general form of a linear m-step multistep method is y n+1 a 1 y n a 2 y n 1 a m y n+1 m h i = b 0 f(x n+1, y n+1 ) + b 1 f(x n, y n ) + + b m f(x n+1 m, y n+1 m ). When b 0 = 0, the method is called explicit, otherwise, it is said to be implicit. In multistep method, we require multiple starting values. Ordinary Differential Equations (ODE) p.64/89
17 Adams-Bashforth Methods We assume a uniform discretization in the x-domain, i.e., we define where h = b a N x i = a + ih,, for some positive integer N. Let the given differential equation is of the form y (x) = f(x, y). We now integrate the given differential equation from x = x i to x = x i+1. Ordinary Differential Equations (ODE) p.65/89
18 This yields the equation y(x i+1 ) y(x i ) = Next, we write xi+1 x i f(x, y(x)) dx. f(x, y(x)) = P m 1 (x) + R m 1 (x), Ordinary Differential Equations (ODE) p.66/89
19 where P m 1 (x) = m j=1 L m 1,j (x)f(x i+1 j, y(x i+1 j )), is the Lagrange form of the polynomial of degree at most m 1 that interpolates f at the m points x i, x i 1, x i 2,, x i+1 m. Ordinary Differential Equations (ODE) p.67/89
20 And R m 1 (x) = f(m) (ξ, y(ξ)) m! Π m j=1 (x x i+1 j) is the corresponding remainder term. Ordinary Differential Equations (ODE) p.68/89
21 Two-Step Adams Bashforth Method Since m = 2, we write f(x, y(x)) = x x i 1 x i x i 1 f(x i, y i ) + x x i x i 1 x i f(x i 1, y i 1 ) + f (ξ, y(ξ)) 2 (x x i )(x x i 1 ). Integrating the Lagrange polynomials yields Ordinary Differential Equations (ODE) p.69/89
22 b 1 = b 2 = xi+1 x i xi+1 x i x x i 1 dx = 3h x i x i 1 2, x x i dx = h x i 1 x i 2. Ordinary Differential Equations (ODE) p.70/89
23 Therefore, the two-step Adams-Bashforth method is y n+1 = y n + h 2 [3f(x n, y n ) f(x n 1, y n 1 )]. Proceeding in a similar manner, the three step Adams-bashforth method is y n+1 = y n + h 12 [23f(x n, y n ) 16f(x n 1, y n 1 ) + 5f(x n 2, y n 2 )]. Ordinary Differential Equations (ODE) p.71/89
24 The four-step Adams-Bashforth method is y n+1 = y n + h 24 [55f(x n, y n ) 59f(x n 1, y n 1 ) + 37f(x n 2, y n 2 ) 9f(x n 3, y n 3 ]. Ordinary Differential Equations (ODE) p.72/89
25 Example 5 Use two-step Adams-Bashforth method to find the approximate solution of dx = 1 + x dt t, x(1) = 1, near x(1.5). Take step size h = 0.5. Ordinary Differential Equations (ODE) p.73/89
26 Solution The two-step Adams-Bashforth formula is x n+1 = x n + h 2 [3f(t n, x n ) f(t n 1, x n 1 )]. We note that for finding x 2, we require x 1 and x 0. We calculate x 1 with the help of second order Taylor s method x 1 = Ordinary Differential Equations (ODE) p.74/89
27 Now, x 2 = x 1 + h 2 [3f(t 1, x 1 ) f(t 0, x 0 )], = Ordinary Differential Equations (ODE) p.75/89
28 Adams-Moulton Methods The derivation of Adams-Moulton methods follows exactly the same procedure as the derivation of the Adams-Bashforth method with one exception. In addition to interpolating f at x i, x i 1, x i 2,, x i+1 m, we also interpolate at x i+1. Ordinary Differential Equations (ODE) p.76/89
29 Hence, we write where f(x, y(x)) = P m (x) + R m (x), P m (x) = m j=0 L m,j (x)f(x i+1 j, y(x i+1 j )), and R m (x) = f(m+1) (ξ, y(ξ)) (m + 1)! Π m j=0(x x i+1 j ). Ordinary Differential Equations (ODE) p.77/89
30 Since P m (x) contains a term involving f(x i+1, y(x i+1 )), the resulting method will be implicit. Furthermore, by using an additional point in the interpolating polynomial the degree of the remainder term is increased by one over the Adams-Bashforth case. Ordinary Differential Equations (ODE) p.78/89
31 2-Step Adams-Moulton Method Since m = 2, we write f(x, y(x)) = (x x i )(x x i 1 ) (x i+1 x i )(x i+1 x i 1 ) f(x i+1, y i+1 ) + (x x i+1)(x x i 1 ) (x i x i+1 )(x i x i 1 ) f(x i, y i ) + (x x i+1)(x x i ) (x i 1 x i+1 )(x i 1 x i ) f(x i 1, y i 1 ) + f (ξ, y(ξ)) 6 (x x i+1 )(x x i )(x x i 1 ). Ordinary Differential Equations (ODE) p.79/89
32 Integrating the Lagrange polynomials yields b 0 = b 1 = b 2 = xi+1 x i (x x i )(x x i 1 ) (x i+1 x i )(x i+1 x i 1 ) xi+1 dt = 5h 12, x i (x x i+1 )(x x i 1 ) (x i x i+1 )(x i x i 1 ) dt = 2h 3, xi+1 x i (x x i+1 )(x x i ) (x i 1 x i+1 )(x i 1 x i ) dt = h 12. Ordinary Differential Equations (ODE) p.80/89
33 Therefore, the two-step Adams-Moulton method is y i+1 y i h = 5 12 f(x i+1, y i+1 ) f(x i, y i ) 1 12 f(x i 1, y i 1 ). To find the approximate solution, we required y 0 and y 1. The initial condition give y 0 and y 1 can be obtained using any third order one-step method. Ordinary Differential Equations (ODE) p.81/89
34 Similarly, we derive the three-step Adams-Moulton method is given by y i+1 = y i + h 24 [9f(x i+1, y i+1 ) +19f(x i, y i ) 5f(x i 1, y i 1 ) +f(x i 2, y i 2 )]. The required starting values for the three step method should be obtained using a fourth-order one-step method. Ordinary Differential Equations (ODE) p.82/89
35 Predictor-Corrector Method The most popular of the predictor-corrector scheme is the Adams fourth-order predictor-corrector method. This uses the four-step, fourth-order Adams-Bashforth method y (0) n+1 = y n + h 24 [55f(x n, y n ) 59f(x n 1, y n 1 ) as a predictor + 37f(x n 2, y n 2 ) 9f(x n 3, y n 3 )], Ordinary Differential Equations (ODE) p.83/89
36 followed by the three-step, fourth-order Adams-Moulton method y (1) n+1 = y n + h 24 [9f(x n+1, y (0) n+1) + 19f(x n, y n ) as a corrector. 5f(x n 1, y n 1 ) + f(x n 2, y n 2 )], Ordinary Differential Equations (ODE) p.84/89
37 Note that the computation of y (0) 4, we require the values (x 0, y 0 ), (x 1, y 1 ), (x 2, y 2 ), (x 3, y 3 ). These values should be given with the given equation or should be computed using single step methods like, Euler s method, Taylor s method or Runge-Kutta methods. To compute the value y (1) 4, we require the values (x 1, y 1 ), (x 2, y 2 ), (x 3, y 3 ) and (x 4, y (0) 4 ). Ordinary Differential Equations (ODE) p.85/89
38 Modifier Let y(x n+1 ) denotes the true solution of y at x n+1. Then y(x n+1 ) = y (0) n h5 f (iv) (ξ 1 ), x n 3 < ξ 1 < x n+1, y(x n+1 ) = y (1) n h5 f (iv) (ξ 2 ), x n 2 < ξ 2 < x n+1. On subtracting, we get 0 = y (0) n+1 y (1) n h5 f (iv) (ξ). Ordinary Differential Equations (ODE) p.86/89
39 This implies that We rewrite as y (1) n+1 y (0) n+1 = h5 f (iv) (ξ). h f(iv) (ξ) = (y(1) n+1 y n+1). (0) Ordinary Differential Equations (ODE) p.87/89
40 Finally, we find the modifier as y(x n+1 ) y (1) n+1 = (y(1) n+1 y (0) n+1) = D n+1. Here, D n+1 is called as modifier. Ordinary Differential Equations (ODE) p.88/89
41 Example Find y(1.4) by Adams Moulton 4th order predictor-corrector pair with modifier as a solution of dy dx = x3 + xy, y(1) = 2, y(1.1) = 1.6, y(1.2) = 0.34 and y(1.3) = with spacing h = 0.1. Ordinary Differential Equations (ODE) p.89/89
Review Higher Order methods Multistep methods Summary HIGHER ORDER METHODS. P.V. Johnson. School of Mathematics. Semester
HIGHER ORDER METHODS School of Mathematics Semester 1 2008 OUTLINE 1 REVIEW 2 HIGHER ORDER METHODS 3 MULTISTEP METHODS 4 SUMMARY OUTLINE 1 REVIEW 2 HIGHER ORDER METHODS 3 MULTISTEP METHODS 4 SUMMARY OUTLINE
More informationNumerical Methods - Initial Value Problems for ODEs
Numerical Methods - Initial Value Problems for ODEs Y. K. Goh Universiti Tunku Abdul Rahman 2013 Y. K. Goh (UTAR) Numerical Methods - Initial Value Problems for ODEs 2013 1 / 43 Outline 1 Initial Value
More informationJim Lambers MAT 772 Fall Semester Lecture 21 Notes
Jim Lambers MAT 772 Fall Semester 21-11 Lecture 21 Notes These notes correspond to Sections 12.6, 12.7 and 12.8 in the text. Multistep Methods All of the numerical methods that we have developed for solving
More informationmultistep methods Last modified: November 28, 2017 Recall that we are interested in the numerical solution of the initial value problem (IVP):
MATH 351 Fall 217 multistep methods http://www.phys.uconn.edu/ rozman/courses/m351_17f/ Last modified: November 28, 217 Recall that we are interested in the numerical solution of the initial value problem
More informationECE257 Numerical Methods and Scientific Computing. Ordinary Differential Equations
ECE257 Numerical Methods and Scientific Computing Ordinary Differential Equations Today s s class: Stiffness Multistep Methods Stiff Equations Stiffness occurs in a problem where two or more independent
More informationChapter 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
More informationInitial-Value Problems for ODEs. Introduction to Linear Multistep Methods
Initial-Value Problems for ODEs Introduction to Linear Multistep Methods Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University
More informationCOSC 3361 Numerical Analysis I Ordinary Differential Equations (II) - Multistep methods
COSC 336 Numerical Analysis I Ordinary Differential Equations (II) - Multistep methods Fall 2005 Repetition from the last lecture (I) Initial value problems: dy = f ( t, y) dt y ( a) = y 0 a t b Goal:
More informationNumerical Differential Equations: IVP
Chapter 11 Numerical Differential Equations: IVP **** 4/16/13 EC (Incomplete) 11.1 Initial Value Problem for Ordinary Differential Equations We consider the problem of numerically solving a differential
More informationApplied Math for Engineers
Applied Math for Engineers Ming Zhong Lecture 15 March 28, 2018 Ming Zhong (JHU) AMS Spring 2018 1 / 28 Recap Table of Contents 1 Recap 2 Numerical ODEs: Single Step Methods 3 Multistep Methods 4 Method
More information9.6 Predictor-Corrector Methods
SEC. 9.6 PREDICTOR-CORRECTOR METHODS 505 Adams-Bashforth-Moulton Method 9.6 Predictor-Corrector Methods The methods of Euler, Heun, Taylor, and Runge-Kutta are called single-step methods because they use
More informationPowerPoints organized by Dr. Michael R. Gustafson II, Duke University
Part 6 Chapter 20 Initial-Value Problems PowerPoints organized by Dr. Michael R. Gustafson II, Duke University All images copyright The McGraw-Hill Companies, Inc. Permission required for reproduction
More informationMultistep Methods for IVPs. t 0 < t < T
Multistep Methods for IVPs We are still considering the IVP dy dt = f(t,y) t 0 < t < T y(t 0 ) = y 0 So far we have looked at Euler s method, which was a first order method and Runge Kutta (RK) methods
More informationAN OVERVIEW. Numerical Methods for ODE Initial Value Problems. 1. One-step methods (Taylor series, Runge-Kutta)
AN OVERVIEW Numerical Methods for ODE Initial Value Problems 1. One-step methods (Taylor series, Runge-Kutta) 2. Multistep methods (Predictor-Corrector, Adams methods) Both of these types of methods are
More informationInitial value problems for ordinary differential equations
Initial value problems for ordinary differential equations Xiaojing Ye, Math & Stat, Georgia State University Spring 2019 Numerical Analysis II Xiaojing Ye, Math & Stat, Georgia State University 1 IVP
More informationHIGHER ORDER METHODS. There are two principal means to derive higher order methods. b j f(x n j,y n j )
HIGHER ORDER METHODS There are two principal means to derive higher order methods y n+1 = p j=0 a j y n j + h p j= 1 b j f(x n j,y n j ) (a) Method of Undetermined Coefficients (b) Numerical Integration
More informationPart IB Numerical Analysis
Part IB Numerical Analysis Definitions Based on lectures by G. Moore Notes taken by Dexter Chua Lent 206 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after
More informationIntegration of Ordinary Differential Equations
Integration of Ordinary Differential Equations Com S 477/577 Nov 7, 00 1 Introduction The solution of differential equations is an important problem that arises in a host of areas. Many differential equations
More informationDifferential Equations
Differential Equations Definitions Finite Differences Taylor Series based Methods: Euler Method Runge-Kutta Methods Improved Euler, Midpoint methods Runge Kutta (2nd, 4th order) methods Predictor-Corrector
More informationPowerPoints organized by Dr. Michael R. Gustafson II, Duke University
Part 6 Chapter 20 Initial-Value Problems PowerPoints organized by Dr. Michael R. Gustafson II, Duke University All images copyright The McGraw-Hill Companies, Inc. Permission required for reproduction
More informationMath 128A Spring 2003 Week 12 Solutions
Math 128A Spring 2003 Week 12 Solutions Burden & Faires 5.9: 1b, 2b, 3, 5, 6, 7 Burden & Faires 5.10: 4, 5, 8 Burden & Faires 5.11: 1c, 2, 5, 6, 8 Burden & Faires 5.9. Higher-Order Equations and Systems
More informationChapter 11 ORDINARY DIFFERENTIAL EQUATIONS
Chapter 11 ORDINARY DIFFERENTIAL EQUATIONS The general form of a first order differential equations is = f(x, y) with initial condition y(a) = y a We seek the solution y = y(x) for x > a This is shown
More informationIntroduction to the Numerical Solution of IVP for ODE
Introduction to the Numerical Solution of IVP for ODE 45 Introduction to the Numerical Solution of IVP for ODE Consider the IVP: DE x = f(t, x), IC x(a) = x a. For simplicity, we will assume here that
More informationMath 128A Spring 2003 Week 11 Solutions Burden & Faires 5.6: 1b, 3b, 7, 9, 12 Burden & Faires 5.7: 1b, 3b, 5 Burden & Faires 5.
Math 128A Spring 2003 Week 11 Solutions Burden & Faires 5.6: 1b, 3b, 7, 9, 12 Burden & Faires 5.7: 1b, 3b, 5 Burden & Faires 5.8: 1b, 3b, 4 Burden & Faires 5.6. Multistep Methods 1. Use all the Adams-Bashforth
More informationConsistency and Convergence
Jim Lambers MAT 77 Fall Semester 010-11 Lecture 0 Notes These notes correspond to Sections 1.3, 1.4 and 1.5 in the text. Consistency and Convergence We have learned that the numerical solution obtained
More informationDo not turn over until you are told to do so by the Invigilator.
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 216 17 INTRODUCTION TO NUMERICAL ANALYSIS MTHE612B Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other questions.
More informationLecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 9
Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T. Heath Chapter 9 Initial Value Problems for Ordinary Differential Equations Copyright c 2001. Reproduction
More informationOrdinary differential equations - Initial value problems
Education has produced a vast population able to read but unable to distinguish what is worth reading. G.M. TREVELYAN Chapter 6 Ordinary differential equations - Initial value problems In this chapter
More informationChap. 20: Initial-Value Problems
Chap. 20: Initial-Value Problems Ordinary Differential Equations Goal: to solve differential equations of the form: dy dt f t, y The methods in this chapter are all one-step methods and have the general
More informationMTH 452/552 Homework 3
MTH 452/552 Homework 3 Do either 1 or 2. 1. (40 points) [Use of ode113 and ode45] This problem can be solved by a modifying the m-files odesample.m and odesampletest.m available from the author s webpage.
More informationOrdinary Differential Equations
CHAPTER 8 Ordinary Differential Equations 8.1. Introduction My section 8.1 will cover the material in sections 8.1 and 8.2 in the book. Read the book sections on your own. I don t like the order of things
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 9 Initial Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign
More information5.6 Multistep Methods
5.6 Multistep Methods 1 Motivation: Consider IVP: yy = ff(tt, yy), aa tt bb, yy(aa) = αα. To compute solution at tt ii+1, approximate solutions at mesh points tt 0, tt 1, tt 2, tt ii are already obtained.
More informationChapter 8. Numerical Solution of Ordinary Differential Equations. Module No. 1. Runge-Kutta Methods
Numerical Analysis by Dr. Anita Pal Assistant Professor Department of Mathematics National Institute of Technology Durgapur Durgapur-71309 email: anita.buie@gmail.com 1 . Chapter 8 Numerical Solution of
More informationSolving PDEs with PGI CUDA Fortran Part 4: Initial value problems for ordinary differential equations
Solving PDEs with PGI CUDA Fortran Part 4: Initial value problems for ordinary differential equations Outline ODEs and initial conditions. Explicit and implicit Euler methods. Runge-Kutta methods. Multistep
More informationODE Runge-Kutta methods
ODE Runge-Kutta methods The theory (very short excerpts from lectures) First-order initial value problem We want to approximate the solution Y(x) of a system of first-order ordinary differential equations
More information8.1 Introduction. Consider the initial value problem (IVP):
8.1 Introduction Consider the initial value problem (IVP): y dy dt = f(t, y), y(t 0)=y 0, t 0 t T. Geometrically: solutions are a one parameter family of curves y = y(t) in(t, y)-plane. Assume solution
More informationKINGS COLLEGE OF ENGINEERING DEPARTMENT OF MATHEMATICS ACADEMIC YEAR / EVEN SEMESTER QUESTION BANK
KINGS COLLEGE OF ENGINEERING MA5-NUMERICAL METHODS DEPARTMENT OF MATHEMATICS ACADEMIC YEAR 00-0 / EVEN SEMESTER QUESTION BANK SUBJECT NAME: NUMERICAL METHODS YEAR/SEM: II / IV UNIT - I SOLUTION OF EQUATIONS
More informationNumerical Methods for the Solution of Differential Equations
Numerical Methods for the Solution of Differential Equations Markus Grasmair Vienna, winter term 2011 2012 Analytical Solutions of Ordinary Differential Equations 1. Find the general solution of the differential
More informationBindel, Fall 2011 Intro to Scientific Computing (CS 3220) Week 12: Monday, Apr 18. HW 7 is posted, and will be due in class on 4/25.
Logistics Week 12: Monday, Apr 18 HW 6 is due at 11:59 tonight. HW 7 is posted, and will be due in class on 4/25. The prelim is graded. An analysis and rubric are on CMS. Problem du jour For implicit methods
More informationPowerPoints organized by Dr. Michael R. Gustafson II, Duke University
Part 5 Chapter 21 Numerical Differentiation PowerPoints organized by Dr. Michael R. Gustafson II, Duke University 1 All images copyright The McGraw-Hill Companies, Inc. Permission required for reproduction
More informationModule 4: Numerical Methods for ODE. Michael Bader. Winter 2007/2008
Outlines Module 4: for ODE Part I: Basic Part II: Advanced Lehrstuhl Informatik V Winter 2007/2008 Part I: Basic 1 Direction Fields 2 Euler s Method Outlines Part I: Basic Part II: Advanced 3 Discretized
More information2.1 NUMERICAL SOLUTION OF SIMULTANEOUS FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS. differential equations with the initial values y(x 0. ; l.
Numerical Methods II UNIT.1 NUMERICAL SOLUTION OF SIMULTANEOUS FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS.1.1 Runge-Kutta Method of Fourth Order 1. Let = f x,y,z, = gx,y,z be the simultaneous first order
More informationLogistic Map, Euler & Runge-Kutta Method and Lotka-Volterra Equations
Logistic Map, Euler & Runge-Kutta Method and Lotka-Volterra Equations S. Y. Ha and J. Park Department of Mathematical Sciences Seoul National University Sep 23, 2013 Contents 1 Logistic Map 2 Euler and
More informationMathematics for chemical engineers. Numerical solution of ordinary differential equations
Mathematics for chemical engineers Drahoslava Janovská Numerical solution of ordinary differential equations Initial value problem Winter Semester 2015-2016 Outline 1 Introduction 2 One step methods Euler
More informationLinear Multistep Methods I: Adams and BDF Methods
Linear Multistep Methods I: Adams and BDF Methods Varun Shankar January 1, 016 1 Introduction In our review of 5610 material, we have discussed polynomial interpolation and its application to generating
More informationApplied Numerical Analysis
Applied Numerical Analysis Using MATLAB Second Edition Laurene V. Fausett Texas A&M University-Commerce PEARSON Prentice Hall Upper Saddle River, NJ 07458 Contents Preface xi 1 Foundations 1 1.1 Introductory
More informationNotes for Numerical Analysis Math 5466 by S. Adjerid Virginia Polytechnic Institute and State University (A Rough Draft) Contents Numerical Methods for ODEs 5. Introduction............................
More informationUnit I (Testing of Hypothesis)
SUBJECT NAME : Statistics and Numerical Methods SUBJECT CODE : MA645 MATERIAL NAME : Part A questions REGULATION : R03 UPDATED ON : November 07 (Upto N/D 07 Q.P) Unit I (Testing of Hypothesis). State level
More informationLecture 10: Linear Multistep Methods (LMMs)
Lecture 10: Linear Multistep Methods (LMMs) 2nd-order Adams-Bashforth Method The approximation for the 2nd-order Adams-Bashforth method is given by equation (10.10) in the lecture note for week 10, as
More informationName 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 informationNumerical Methods for Partial Differential Equations
Numerical Methods for Partial Differential Equations Seongjai Kim Department of Mathematics and Statistics Mississippi State University Mississippi State, MS 39762 USA Email: skim@math.msstate.edu September
More informationYou may not use your books, notes; calculators are highly recommended.
Math 301 Winter 2013-14 Midterm 1 02/06/2014 Time Limit: 60 Minutes Name (Print): Instructor This exam contains 8 pages (including this cover page) and 6 problems. Check to see if any pages are missing.
More informationModeling & Simulation 2018 Lecture 12. Simulations
Modeling & Simulation 2018 Lecture 12. Simulations Claudio Altafini Automatic Control, ISY Linköping University, Sweden Summary of lecture 7-11 1 / 32 Models of complex systems physical interconnections,
More informationSolving Ordinary Differential equations
Solving Ordinary Differential equations Taylor methods can be used to build explicit methods with higher order of convergence than Euler s method. The main difficult of these methods is the computation
More informationNumerical Methods for Differential Equations
Numerical Methods for Differential Equations Chapter 2: Runge Kutta and Linear Multistep methods Gustaf Söderlind and Carmen Arévalo Numerical Analysis, Lund University Textbooks: A First Course in the
More information2.29 Numerical Fluid Mechanics Fall 2011 Lecture 20
2.29 Numerical Fluid Mechanics Fall 2011 Lecture 20 REVIEW Lecture 19: Finite Volume Methods Review: Basic elements of a FV scheme and steps to step-up a FV scheme One Dimensional examples d x j x j 1/2
More informationCS 450 Numerical Analysis. Chapter 9: Initial Value Problems for Ordinary Differential Equations
Lecture slides based on the textbook Scientific Computing: An Introductory Survey by Michael T. Heath, copyright c 2018 by the Society for Industrial and Applied Mathematics. http://www.siam.org/books/cl80
More informationComputational Methods CMSC/AMSC/MAPL 460. Ordinary differential equations
Computational Methods CMSC/AMSC/MAPL 460 Ordinar differential equations Ramani Duraiswami, Dept. of Computer Science Several slides adapted from Prof. ERIC SANDT, TAMU ODE: Previous class Applications
More informationInitial value problems for ordinary differential equations
AMSC/CMSC 660 Scientific Computing I Fall 2008 UNIT 5: Numerical Solution of Ordinary Differential Equations Part 1 Dianne P. O Leary c 2008 The Plan Initial value problems (ivps) for ordinary differential
More informationOn interval predictor-corrector methods
DOI 10.1007/s11075-016-0220-x ORIGINAL PAPER On interval predictor-corrector methods Andrzej Marcinia 1,2 Malgorzata A. Janowsa 3 Tomasz Hoffmann 4 Received: 26 March 2016 / Accepted: 3 October 2016 The
More informationComputational Methods CMSC/AMSC/MAPL 460. Ordinary differential equations
Computational Methods CMSC/AMSC/MAPL 460 Ordinar differential equations Ramani Duraiswami, Dept. of Computer Science Several slides adapted from Prof. ERIC SANDT, TAMU ODE: Previous class Standard form
More informationNumerical Methods for Differential Equations
Numerical Methods for Differential Equations Chapter 2: Runge Kutta and Multistep Methods Gustaf Söderlind Numerical Analysis, Lund University Contents V4.16 1. Runge Kutta methods 2. Embedded RK methods
More informationSolution of First Order Initial Value Problem by Sixth Order Predictor Corrector Method
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 2277 2290 Research India Publications http://www.ripublication.com/gjpam.htm Solution of First Order Initial
More information16.7 Multistep, Multivalue, and Predictor-Corrector Methods
740 Chapter 16. Integration of Ordinary Differential Equations 16.7 Multistep, Multivalue, and Predictor-Corrector Methods The terms multistepand multivaluedescribe two different ways of implementing essentially
More informationEuler s Method, cont d
Jim Lambers MAT 461/561 Spring Semester 009-10 Lecture 3 Notes These notes correspond to Sections 5. and 5.4 in the text. Euler s Method, cont d We conclude our discussion of Euler s method with an example
More informationScientific Computing with Case Studies SIAM Press, Lecture Notes for Unit V Solution of
Scientific Computing with Case Studies SIAM Press, 2009 http://www.cs.umd.edu/users/oleary/sccswebpage Lecture Notes for Unit V Solution of Differential Equations Part 1 Dianne P. O Leary c 2008 1 The
More informationNumerical Analysis. A Comprehensive Introduction. H. R. Schwarz University of Zürich Switzerland. with a contribution by
Numerical Analysis A Comprehensive Introduction H. R. Schwarz University of Zürich Switzerland with a contribution by J. Waldvogel Swiss Federal Institute of Technology, Zürich JOHN WILEY & SONS Chichester
More informationLecture 4: Numerical solution of ordinary differential equations
Lecture 4: Numerical solution of ordinary differential equations Department of Mathematics, ETH Zürich General explicit one-step method: Consistency; Stability; Convergence. High-order methods: Taylor
More informationChapter 5 Exercises. (a) Determine the best possible Lipschitz constant for this function over 2 u <. u (t) = log(u(t)), u(0) = 2.
Chapter 5 Exercises From: Finite Difference Methods for Ordinary and Partial Differential Equations by R. J. LeVeque, SIAM, 2007. http://www.amath.washington.edu/ rjl/fdmbook Exercise 5. (Uniqueness for
More information8 Numerical Integration of Ordinary Differential
8 Numerical Integration of Ordinary Differential Equations 8.1 Introduction Most ordinary differential equations of mathematical physics are secondorder equations. Examples include the equation of motion
More informationNumerical solution of ODEs
Numerical solution of ODEs Arne Morten Kvarving Department of Mathematical Sciences Norwegian University of Science and Technology November 5 2007 Problem and solution strategy We want to find an approximation
More informationNumerical Methods for Partial Differential Equations
Numerical Methods for Partial Differential Equations Seongjai Kim Department of Mathematics and Statistics Mississippi State University Mississippi State, MS 39762 USA Email: skim@math.msstate.edu September
More informationResearch Computing with Python, Lecture 7, Numerical Integration and Solving Ordinary Differential Equations
Research Computing with Python, Lecture 7, Numerical Integration and Solving Ordinary Differential Equations Ramses van Zon SciNet HPC Consortium November 25, 2014 Ramses van Zon (SciNet HPC Consortium)Research
More informationSYSTEMS OF ODES. mg sin ( (x)) dx 2 =
SYSTEMS OF ODES Consider the pendulum shown below. Assume the rod is of neglible mass, that the pendulum is of mass m, and that the rod is of length `. Assume the pendulum moves in the plane shown, and
More informationLecture Notes on Numerical Differential Equations: IVP
Lecture Notes on Numerical Differential Equations: IVP Professor Biswa Nath Datta Department of Mathematical Sciences Northern Illinois University DeKalb, IL. 60115 USA E mail: dattab@math.niu.edu URL:
More informationOne-Step Hybrid Block Method with One Generalized Off-Step Points for Direct Solution of Second Order Ordinary Differential Equations
Applied Mathematical Sciences, Vol. 10, 2016, no. 29, 142-142 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.6127 One-Step Hybrid Block Method with One Generalized Off-Step Points for
More informationSolving Ordinary Differential Equations
Solving Ordinary Differential Equations Sanzheng Qiao Department of Computing and Software McMaster University March, 2014 Outline 1 Initial Value Problem Euler s Method Runge-Kutta Methods Multistep Methods
More informationLecture IV: Time Discretization
Lecture IV: Time Discretization Motivation Kinematics: continuous motion in continuous time Computer simulation: Discrete time steps t Discrete Space (mesh particles) Updating Position Force induces acceleration.
More informationMath Numerical Analysis Homework #4 Due End of term. y = 2y t 3y2 t 3, 1 t 2, y(1) = 1. n=(b-a)/h+1; % the number of steps we need to take
Math 32 - Numerical Analysis Homework #4 Due End of term Note: In the following y i is approximation of y(t i ) and f i is f(t i,y i ).. Consider the initial value problem, y = 2y t 3y2 t 3, t 2, y() =.
More informationNUMERICAL SOLUTION OF ODE IVPs. Overview
NUMERICAL SOLUTION OF ODE IVPs 1 Quick review of direction fields Overview 2 A reminder about and 3 Important test: Is the ODE initial value problem? 4 Fundamental concepts: Euler s Method 5 Fundamental
More informationMini project ODE, TANA22
Mini project ODE, TANA22 Filip Berglund (filbe882) Linh Nguyen (linng299) Amanda Åkesson (amaak531) October 2018 1 1 Introduction Carl David Tohmé Runge (1856 1927) was a German mathematician and a prominent
More information1 Ordinary Differential Equations
Ordinary Differential Equations.0 Mathematical Background.0. Smoothness Definition. A function f defined on [a, b] is continuous at ξ [a, b] if lim x ξ f(x) = f(ξ). Remark Note that this implies existence
More informationMilne s Spreadsheet Calculator Using VBA with Excel Programming for solving Ordinary Differential Equations
Milne s Spreadsheet Calculator Using VBA with Excel Programming for solving Ordinary Differential Equations Miss. S. P. Hingmire 1, Dr. M.R. Gosavi 2, Dr. N. A.Patil 3 1 Research Scholar, Asst.Professor,
More informationMath 216 Final Exam 14 December, 2012
Math 216 Final Exam 14 December, 2012 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material that
More informationSolving scalar IVP s : Runge-Kutta Methods
Solving scalar IVP s : Runge-Kutta Methods Josh Engwer Texas Tech University March 7, NOTATION: h step size x n xt) t n+ t + h x n+ xt n+ ) xt + h) dx = ft, x) SCALAR IVP ASSUMED THROUGHOUT: dt xt ) =
More informationNumerical Methods for Differential Equations
CHAPTER 5 Numerical Methods for Differential Equations In this chapter we will discuss a few of the many numerical methods which can be used to solve initial value problems and one-dimensional boundary
More informationVirtual University of Pakistan
Virtual University of Pakistan File Version v.0.0 Prepared For: Final Term Note: Use Table Of Content to view the Topics, In PDF(Portable Document Format) format, you can check Bookmarks menu Disclaimer:
More informationPhysics 584 Computational Methods
Physics 584 Computational Methods Introduction to Matlab and Numerical Solutions to Ordinary Differential Equations Ryan Ogliore April 18 th, 2016 Lecture Outline Introduction to Matlab Numerical Solutions
More informationCS520: numerical ODEs (Ch.2)
.. CS520: numerical ODEs (Ch.2) Uri Ascher Department of Computer Science University of British Columbia ascher@cs.ubc.ca people.cs.ubc.ca/ ascher/520.html Uri Ascher (UBC) CPSC 520: ODEs (Ch. 2) Fall
More informationSOLUTION OF EQUATION AND EIGENVALUE PROBLEMS PART A( 2 MARKS)
CHENDU COLLEGE OF ENGINEERING AND TECHNOLOGY (Approved by AICTE New Delhi, Affiliated to Anna University Chennai. Zamin Endathur Village, Madurntakam Taluk, Kancheepuram Dist.-603311.) MA6459 - NUMERICAL
More informationThe Definition and Numerical Method of Final Value Problem and Arbitrary Value Problem Shixiong Wang 1*, Jianhua He 1, Chen Wang 2, Xitong Li 1
The Definition and Numerical Method of Final Value Problem and Arbitrary Value Problem Shixiong Wang 1*, Jianhua He 1, Chen Wang 2, Xitong Li 1 1 School of Electronics and Information, Northwestern Polytechnical
More informationFinite Differences for Differential Equations 28 PART II. Finite Difference Methods for Differential Equations
Finite Differences for Differential Equations 28 PART II Finite Difference Methods for Differential Equations Finite Differences for Differential Equations 29 BOUNDARY VALUE PROBLEMS (I) Solving a TWO
More informationInitial Value Problems
Numerical Analysis, lecture 13: Initial Value Problems (textbook sections 10.1-4, 10.7) differential equations standard form existence & uniqueness y 0 y 2 solution methods x 0 x 1 h h x 2 y1 Euler, Heun,
More informationNumerical Methods for Engineers. and Scientists. Applications using MATLAB. An Introduction with. Vish- Subramaniam. Third Edition. Amos Gilat.
Numerical Methods for Engineers An Introduction with and Scientists Applications using MATLAB Third Edition Amos Gilat Vish- Subramaniam Department of Mechanical Engineering The Ohio State University Wiley
More informationOrdinary Differential Equations (ODEs)
Ordinary Differential Equations (ODEs) 1 Computer Simulations Why is computation becoming so important in physics? One reason is that most of our analytical tools such as differential calculus are best
More informationResearch Article Diagonally Implicit Block Backward Differentiation Formulas for Solving Ordinary Differential Equations
International Mathematics and Mathematical Sciences Volume 212, Article ID 767328, 8 pages doi:1.1155/212/767328 Research Article Diagonally Implicit Block Backward Differentiation Formulas for Solving
More informationLecture V: The game-engine loop & Time Integration
Lecture V: The game-engine loop & Time Integration The Basic Game-Engine Loop Previous state: " #, %(#) ( #, )(#) Forces -(#) Integrate velocities and positions Resolve Interpenetrations Per-body change
More informationNumerical Solution of Differential Equations
1 Numerical Solution of Differential Equations A differential equation (or "DE") contains derivatives or differentials. In a differential equation the unknown is a function, and the differential equation
More informationNUMERICAL ANALYSIS SYLLABUS MATHEMATICS PAPER IV (A)
NUMERICAL ANALYSIS SYLLABUS MATHEMATICS PAPER IV (A) Unit - 1 Errors & Their Accuracy Solutions of Algebraic and Transcendental Equations Bisection Method The method of false position The iteration method
More information