Numerical Methods for PDEs
|
|
|
- Lee Flynn
- 6 years ago
- Views:
Transcription
1 Numerical Methods for PDEs Hyperbolic PDEs: Coupled system/noliear coservatio laws/a oliear Lax-Wedroff scheme (Lecture 18, Week 6 Markus Schmuck Departmet of Mathematics ad Maxwell Istitute for Mathematical Scieces Heriot-Watt Uiversity, Ediburgh Ediburgh, February 19, 015 1/ 11
2 Outlie 1 Coupled system of equatios Noliear coservatio laws 3 Noliear coservatio laws: The Lax-Wedroff scheme / 11
3 Wave Equatio as Coupled Advectio Equatios We ca decouple the wave equatio u tt = a u xx as a pair of coupled first order equatios { u t + av x = 0 (CAE v t + au x = 0, which is supplemeted with the physical iitial coditios u(x, 0 ad v(x, 0. Check this: Differetiate the first equatio with respect to t, the secod with respect to x, the elimiate v xt Vector form of (CAE: u t + Au x = 0, where u = ( u v ( 0 a, A = a 0 3/ 11
4 Lax-Wedroff scheme (LW for (CAE Basic idea: Iclude dissipatio by a Taylor expasio i time up to d order ad use the relatio ( m ( t u = a m x u, such that u(x, t + k = u + ku t + 1 k u tt + O(k 3, (x,t = u Aku x + 1 A k u xx + O(k 3, (x,t u Ak D x h u + 1 A k δ x h u. This suggests the followig explicit scheme w +1 j = w j 1 P(w j+1 w j P (w j+1 w j + w j 1 ( ( w u where w =, P = k z v h A. For I the uit matrix (LW w +1 j = (I P w j 1 P(I Pw j P(I + Pw j 1. 4/ 11
5 Noliear coservatio laws So far: Liear hyperbolic problems May real problems: Require oliear equatios, e.g. fluid ad gas flow Noliear (AE: u t + a(uu x = 0 which icludes the Burgers equatio (BE u t + u u x = 0. Properties of (BE: Top moves faster tha the tail o either side ad evetually overtakes the tail formatio of a shock 5/ 11
6 u fastest faster steepes shock slow x flow directio This figure shows a flow i positive x-directio, i.e., a(u > 0. Coservatio law (CL: The umerical approximatio of u t + a(uu x = 0 leads ofte to ustable results. Rewritig it as follows (CL u t + a(uu x = 0 rearrage u t + (F (u = 0 x improves properties of associated umerical shemes where d F F(u = a(u, sice du x = df u du x. Example: Burgers equatio: a(u = u F (u = 1/u 6/ 11
7 Noliear (CL: A Lax-Wedroff scheme Goal: Exted the Lax-Wedroff method to oliear coservatio laws (NLCL u t + [F (u] x = 0. Idea: Apply the same strategy as for the advectio equatio: Step 1: Use a trucated Taylor series for u(x, t + k, i.e., u(x, t + k [u + k u t + 1 ] k u tt. (x,t Step : Replace the time derivatives by space derivatives, that is, u t = [F (u] x u tt = t ( [F (u] x = x ([F (u] t = ( F (u u t x = ( ( F (u[ F (u] x = F (u x x x F (u. 7/ 11
8 Step 3: Defiig leads to [ ] Q(u = F (u x F (u, so that u tt = x Q(u. u(x j, t +1 [ u k x F (u + 1 k ] x Q(u. (x j,t Step 4: Approximate x-derivatives by cetral differeces: x F (u F (w j+1 F (w j 1 = F j+1 F (xj,t h h x Q(u Q Q j+ 1 j 1, (xj,t h where we applied a spacig of h/ w.r.t. Q. j 1, 8/ 11
9 We have also used the otatio Q = F (w j+ 1 j+ 1 x F (u so x Q(u (xj,t (xj+ 1,t ( F F (w j+1 Fj j+ 1 h 1 { ( ( } h F (w F j+ 1 j+1 F j F (w F j 1 j Fj 1. Sice we do ot have values for u at half-iteger poits x j+ 1, we further approximate ad similarly for F (w. j 1 F (w 1 ( F (w j+ 1 j + F (wj+1 9/ 11
10 Step 5: Isertig these fidigs ito [ ] gives ( = wj Fj+1 F j 1 (NLW w +1 j a j 1 k h [ Fj Fj 1 ] } [ ] + {a k F h j+ 1 j+1 F j [ ] where a = 1 j+ 1 F (wj + F (wj+1 F (w. j+ 1 Remark: The (NLW scheme reduces to the liear (LW for F (u = au, a = cost, i.e. whe F = a. Check: w +1 j = wj k h [ ] } a awj awj 1 ( awj+1 aw j 1 + k h { [ ] a awj+1 aw j = wj p ( wj+1 w j 1 + p ( wj+1 w j + wj 1, p = ak h which is the formula derived earlier. 10/ 11
11 Stability of (NLW I the liear scheme we have stability oly for p 1. I the oliear scheme, F (u plays the role of a, so we should require that F (w j k h 1 for all m. For safety we replace 1 by = 0.9 ad require k = 0.9h max j F (w j Note that k will vary from step to step. 11/ 11
Partial Differential Equations
Partial Differetial Equatios Part 2 Massimo Ricotti ricotti@astro.umd.edu Uiversity of Marylad Partial Differetial Equatios p.1/15 Upwid differecig I additio to amplitude errors (istability or dampig),
Math 257: Finite difference methods
Math 257: Fiite differece methods 1 Fiite Differeces Remember the defiitio of a derivative f f(x + ) f(x) (x) = lim 0 Also recall Taylor s formula: (1) f(x + ) = f(x) + f (x) + 2 f (x) + 3 f (3) (x) +...
Finite Dierence Schemes
MATH-459 Numerical Methods for Coservatio Laws by Prof. Ja S. Hesthave Solutio set 2: Fiite Dierece Schemes Exercise 2. Cosistecy A method is cosistet if its local trucatio error T k satises T k (x, t)
Math 312 Lecture Notes One Dimensional Maps
Math 312 Lecture Notes Oe Dimesioal Maps Warre Weckesser Departmet of Mathematics Colgate Uiversity 21-23 February 25 A Example We begi with the simplest model of populatio growth. Suppose, for example,
Computational Fluid Dynamics. Lecture 3
Computatioal Fluid Dyamics Lecture 3 Discretizatio Cotiued. A fourth order approximatio to f x ca be foud usig Taylor Series. ( + ) + ( + ) + + ( ) + ( ) = a f x x b f x x c f x d f x x e f x x f x 0 0
The Advection-Diffusion equation!
ttp://www.d.edu/~gtryggva/cf-course/! Te Advectio-iffusio equatio! Grétar Tryggvaso! Sprig 3! Navier-Stokes equatios! Summary! u t + u u x + v u y = P ρ x + µ u + u ρ y Hyperbolic part! u x + v y = Elliptic
Modified Decomposition Method by Adomian and. Rach for Solving Nonlinear Volterra Integro- Differential Equations
Noliear Aalysis ad Differetial Equatios, Vol. 5, 27, o. 4, 57-7 HIKARI Ltd, www.m-hikari.com https://doi.org/.2988/ade.27.62 Modified Decompositio Method by Adomia ad Rach for Solvig Noliear Volterra Itegro-
Streamfunction-Vorticity Formulation
Streamfuctio-Vorticity Formulatio A. Salih Departmet of Aerospace Egieerig Idia Istitute of Space Sciece ad Techology, Thiruvaathapuram March 2013 The streamfuctio-vorticity formulatio was amog the first
1. Linearization of a nonlinear system given in the form of a system of ordinary differential equations
. Liearizatio of a oliear system give i the form of a system of ordiary differetial equatios We ow show how to determie a liear model which approximates the behavior of a time-ivariat oliear system i a
1 6 = 1 6 = + Factorials and Euler s Gamma function
Royal Holloway Uiversity of Lodo Departmet of Physics Factorials ad Euler s Gamma fuctio Itroductio The is a self-cotaied part of the course dealig, essetially, with the factorial fuctio ad its geeralizatio
Numerical Methods for Partial Differential Equations
Numerical Methods for Partial Differetial Equatios Eric de Sturler Uiversity of Illiois at Urbaa-Champaig Cosider the liear first order hyperbolic equatio Øu Øu Øt + a(x, t) Øx = 0 I spite of its simple
CONTENTS. Course Goals. Course Materials Lecture Notes:
INTRODUCTION Ho Chi Mih City OF Uiversity ENVIRONMENTAL of Techology DESIGN Faculty Chapter of Civil 1: Orietatio. Egieerig Evaluatio Departmet of mathematical of Water Resources skill Egieerig & Maagemet
Numerical Astrophysics: hydrodynamics
Numerical Astrophysics: hydrodyamics Part 1: Numerical solutios to the Euler Equatios Outlie Numerical eperimets are a valuable tool to study astrophysical objects (where we ca rarely do direct eperimets).
Poincaré Problem for Nonlinear Elliptic Equations of Second Order in Unbounded Domains
Advaces i Pure Mathematics 23 3 72-77 http://dxdoiorg/4236/apm233a24 Published Olie Jauary 23 (http://wwwscirporg/oural/apm) Poicaré Problem for Noliear Elliptic Equatios of Secod Order i Ubouded Domais
Numerical Method for Blasius Equation on an infinite Interval
Numerical Method for Blasius Equatio o a ifiite Iterval Alexader I. Zadori Omsk departmet of Sobolev Mathematics Istitute of Siberia Brach of Russia Academy of Scieces, Russia zadori@iitam.omsk.et.ru 1
Lecture 8: Solving the Heat, Laplace and Wave equations using finite difference methods
Itroductory lecture otes o Partial Differetial Equatios - c Athoy Peirce. Not to be copied, used, or revised without explicit writte permissio from the copyright ower. 1 Lecture 8: Solvig the Heat, Laplace
Løsningsførslag i 4M
Norges tekisk aturviteskapelige uiversitet Istitutt for matematiske fag Side 1 av 6 Løsigsførslag i 4M Oppgave 1 a) A sketch of the graph of the give f o the iterval [ 3, 3) is as follows: The Fourier
Finite Difference Derivations for Spreadsheet Modeling John C. Walton Modified: November 15, 2007 jcw
Fiite Differece Derivatios for Spreadsheet Modelig Joh C. Walto Modified: November 15, 2007 jcw Figure 1. Suset with 11 swas o Little Platte Lake, Michiga. Page 1 Modificatio Date: November 15, 2007 Review
Numerical Schemes to Solve Advective Contaminant Transport Problems with Linear Sorption and First Order Decay
Numerical Schemes to Solve Advective Cotamiat Trasport Problems with Liear Sorptio ad First Order Decay Adré Luís Brasil Cavalcate PhD, Associate Professor, Uiversity of Brasilia, Brazil, Departmet of
3 Gauss map and continued fractions
ICTP, Trieste, July 08 Gauss map ad cotiued fractios I this lecture we will itroduce the Gauss map, which is very importat for its coectio with cotiued fractios i umber theory. The Gauss map G : [0, ]
An efficient time integration method for extra-large eddy simulations
A efficiet time itegratio method for extra-large eddy simulatios M.A. Scheibeler Departmet of Mathematics Master s Thesis A efficiet time itegratio method for extra-large eddy simulatios M.A. Scheibeler
Taylor polynomial solution of difference equation with constant coefficients via time scales calculus
TMSCI 3, o 3, 129-135 (2015) 129 ew Treds i Mathematical Scieces http://wwwtmscicom Taylor polyomial solutio of differece equatio with costat coefficiets via time scales calculus Veysel Fuat Hatipoglu
PHY4905: Nearly-Free Electron Model (NFE)
PHY4905: Nearly-Free Electro Model (NFE) D. L. Maslov Departmet of Physics, Uiversity of Florida (Dated: Jauary 12, 2011) 1 I. REMINDER: QUANTUM MECHANICAL PERTURBATION THEORY A. No-degeerate eigestates
Higher-order iterative methods by using Householder's method for solving certain nonlinear equations
Math Sci Lett, No, 7- ( 7 Mathematical Sciece Letters A Iteratioal Joural http://dxdoiorg/785/msl/5 Higher-order iterative methods by usig Householder's method for solvig certai oliear equatios Waseem
( ) (( ) ) ANSWERS TO EXERCISES IN APPENDIX B. Section B.1 VECTORS AND SETS. Exercise B.1-1: Convex sets. are convex, , hence. and. (a) Let.
Joh Riley 8 Jue 03 ANSWERS TO EXERCISES IN APPENDIX B Sectio B VECTORS AND SETS Exercise B-: Covex sets (a) Let 0 x, x X, X, hece 0 x, x X ad 0 x, x X Sice X ad X are covex, x X ad x X The x X X, which
TMA4205 Numerical Linear Algebra. The Poisson problem in R 2 : diagonalization methods
TMA4205 Numerical Liear Algebra The Poisso problem i R 2 : diagoalizatio methods September 3, 2007 c Eiar M Røquist Departmet of Mathematical Scieces NTNU, N-749 Trodheim, Norway All rights reserved A
Some properties of Boubaker polynomials and applications
Some properties of Boubaker polyomials ad applicatios Gradimir V. Milovaović ad Duša Joksimović Citatio: AIP Cof. Proc. 179, 1050 (2012); doi: 10.1063/1.756326 View olie: http://dx.doi.org/10.1063/1.756326
Exact Solutions for a Class of Nonlinear Singular Two-Point Boundary Value Problems: The Decomposition Method
Exact Solutios for a Class of Noliear Sigular Two-Poit Boudary Value Problems: The Decompositio Method Abd Elhalim Ebaid Departmet of Mathematics, Faculty of Sciece, Tabuk Uiversity, P O Box 741, Tabuki
Chapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
Z - Transform. It offers the techniques for digital filter design and frequency analysis of digital signals.
Z - Trasform The -trasform is a very importat tool i describig ad aalyig digital systems. It offers the techiques for digital filter desig ad frequecy aalysis of digital sigals. Defiitio of -trasform:
REGRESSION (Physics 1210 Notes, Partial Modified Appendix A)
REGRESSION (Physics 0 Notes, Partial Modified Appedix A) HOW TO PERFORM A LINEAR REGRESSION Cosider the followig data poits ad their graph (Table I ad Figure ): X Y 0 3 5 3 7 4 9 5 Table : Example Data
k-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c 1. Introduction
Acta Math. Uiv. Comeiaae Vol. LXXXVI, 2 (2017), pp. 279 286 279 k-generalized FIBONACCI NUMBERS CLOSE TO THE FORM 2 a + 3 b + 5 c N. IRMAK ad M. ALP Abstract. The k-geeralized Fiboacci sequece { F (k)
Solution of Linear Constant-Coefficient Difference Equations
ECE 38-9 Solutio of Liear Costat-Coefficiet Differece Equatios Z. Aliyazicioglu Electrical ad Computer Egieerig Departmet Cal Poly Pomoa Solutio of Liear Costat-Coefficiet Differece Equatios Example: Determie
The picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled
1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how
6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
CHAPTER 1 SEQUENCES AND INFINITE SERIES
CHAPTER SEQUENCES AND INFINITE SERIES SEQUENCES AND INFINITE SERIES (0 meetigs) Sequeces ad limit of a sequece Mootoic ad bouded sequece Ifiite series of costat terms Ifiite series of positive terms Alteratig
Numerical Methods for Partial Differential Equations
Numerical Methods for Partial Differetial Equatios CAAM 45 Sprig 005 Lecture 4 -step time-steppig methods: stability, accuracy Ruge-Kutta Methods, Istructor: Tim Warburto Today Recall AB stability regios
On Strongly Consistent Finite Dierence Approximations
D.Michels et al. (KAUST,JINR,SSU) Strogly Cosistet Approximatios 17 April 2018 1 / 25 O Strogly Cosistet Fiite Dierece Approximatios Domiik Michels 1, Vladimir Gerdt 2, Dmitry Lyakhov 1, ad Yuri Blikov
A large class of initial value (time-evolution) PDEs in one space dimension can be cast into the form of a flux-conservative equation, u (19.1.
834 Chapter 19. Partial Differetial Equatios egieerig; these methods allow cosiderable freedom i puttig computatioal elemets where you wat them, importat whe dealig with highly irregular geometries. Spectral
NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE
UPB Sci Bull, Series A, Vol 79, Iss, 207 ISSN 22-7027 NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE Gabriel Bercu We itroduce two ew sequeces of Euler-Mascheroi type which have fast covergece
AE/ME 339 Computational Fluid Dynamics (CFD)
AE/ME 339 Computatioal Fluid Dyamics (CFD 0//004 Topic0_PresCorr_ Computatioal Fluid Dyamics (AE/ME 339 Pressure Correctio Method The pressure correctio formula (6.8.4 Calculatio of p. Coservatio form
The axial dispersion model for tubular reactors at steady state can be described by the following equations: dc dz R n cn = 0 (1) (2) 1 d 2 c.
5.4 Applicatio of Perturbatio Methods to the Dispersio Model for Tubular Reactors The axial dispersio model for tubular reactors at steady state ca be described by the followig equatios: d c Pe dz z =
A NEW CLASS OF 2-STEP RATIONAL MULTISTEP METHODS
Jural Karya Asli Loreka Ahli Matematik Vol. No. (010) page 6-9. Jural Karya Asli Loreka Ahli Matematik A NEW CLASS OF -STEP RATIONAL MULTISTEP METHODS 1 Nazeeruddi Yaacob Teh Yua Yig Norma Alias 1 Departmet
Taylor expansion: Show that the TE of f(x)= sin(x) around. sin(x) = x - + 3! 5! L 7 & 8: MHD/ZAH
Taylor epasio: Let ƒ() be a ifiitely differetiable real fuctio. A ay poit i the eighbourhood of 0, the fuctio ƒ() ca be represeted by a power series of the followig form: X 0 f(a) f() f() ( ) f( ) ( )
Pell and Lucas primes
Notes o Number Theory ad Discrete Mathematics ISSN 30 532 Vol. 2, 205, No. 3, 64 69 Pell ad Lucas primes J. V. Leyedekkers ad A. G. Shao 2 Faculty of Sciece, The Uiversity of Sydey NSW 2006, Australia
Time-Domain Representations of LTI Systems
2.1 Itroductio Objectives: 1. Impulse resposes of LTI systems 2. Liear costat-coefficiets differetial or differece equatios of LTI systems 3. Bloc diagram represetatios of LTI systems 4. State-variable
Chapter 10: Power Series
Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because
FIR Filters. Lecture #7 Chapter 5. BME 310 Biomedical Computing - J.Schesser
FIR Filters Lecture #7 Chapter 5 8 What Is this Course All About? To Gai a Appreciatio of the Various Types of Sigals ad Systems To Aalyze The Various Types of Systems To Lear the Skills ad Tools eeded
CS321. Numerical Analysis and Computing
CS Numerical Aalysis ad Computig Lecture Locatig Roots o Equatios Proessor Ju Zhag Departmet o Computer Sciece Uiversity o Ketucky Leigto KY 456-6 September 8 5 What is the Root May physical system ca
Introduction to Signals and Systems, Part V: Lecture Summary
EEL33: Discrete-Time Sigals ad Systems Itroductio to Sigals ad Systems, Part V: Lecture Summary Itroductio to Sigals ad Systems, Part V: Lecture Summary So far we have oly looked at examples of o-recursive
Fourier Series and the Wave Equation
Fourier Series ad the Wave Equatio We start with the oe-dimesioal wave equatio u u =, x u(, t) = u(, t) =, ux (,) = f( x), u ( x,) = This represets a vibratig strig, where u is the displacemet of the strig
NUMERICAL METHOD FOR SINGULARLY PERTURBED DELAY PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS
THERMAL SCIENCE, Year 07, Vol., No. 4, pp. 595-599 595 NUMERICAL METHOD FOR SINGULARLY PERTURBED DELAY PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS by Yula WANG *, Da TIAN, ad Zhiyua LI Departmet of Mathematics,
Differentiable Convex Functions
Differetiable Covex Fuctios The followig picture motivates Theorem 11. f ( x) f ( x) f '( x)( x x) ˆx x 1 Theorem 11 : Let f : R R be differetiable. The, f is covex o the covex set C R if, ad oly if for
A Lattice Green Function Introduction. Abstract
August 5, 25 A Lattice Gree Fuctio Itroductio Stefa Hollos Exstrom Laboratories LLC, 662 Nelso Park Dr, Logmot, Colorado 853, USA Abstract We preset a itroductio to lattice Gree fuctios. Electroic address:
Sequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018
CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical
Chapter 4 : Laplace Transform
4. Itroductio Laplace trasform is a alterative to solve the differetial equatio by the complex frequecy domai ( s = σ + jω), istead of the usual time domai. The DE ca be easily trasformed ito a algebraic
MATH 10550, EXAM 3 SOLUTIONS
MATH 155, EXAM 3 SOLUTIONS 1. I fidig a approximate solutio to the equatio x 3 +x 4 = usig Newto s method with iitial approximatio x 1 = 1, what is x? Solutio. Recall that x +1 = x f(x ) f (x ). Hece,
SNAP Centre Workshop. Basic Algebraic Manipulation
SNAP Cetre Workshop Basic Algebraic Maipulatio 8 Simplifyig Algebraic Expressios Whe a expressio is writte i the most compact maer possible, it is cosidered to be simplified. Not Simplified: x(x + 4x)
NANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS AO-LEVEL MATHEMATICS
NANYANG TECHNOLOGICAL UNIVERSITY SYLLABUS FOR ENTRANCE EXAMINATION FOR INTERNATIONAL STUDENTS AO-LEVEL MATHEMATICS STRUCTURE OF EXAMINATION PAPER. There will be oe 2-hour paper cosistig of 4 questios.
Solution of Differential Equation from the Transform Technique
Iteratioal Joural of Computatioal Sciece ad Mathematics ISSN 0974-3189 Volume 3, Number 1 (2011), pp 121-125 Iteratioal Research Publicatio House http://wwwirphousecom Solutio of Differetial Equatio from
Kinetics of Complex Reactions
Kietics of Complex Reactios by Flick Colema Departmet of Chemistry Wellesley College Wellesley MA 28 wcolema@wellesley.edu Copyright Flick Colema 996. All rights reserved. You are welcome to use this documet
DYNAMIC ANALYSIS OF BEAM-LIKE STRUCTURES SUBJECT TO MOVING LOADS
DYNAMIC ANALYSIS OF BEAM-LIKE STRUCTURES SUBJECT TO MOVING LOADS Ivaa Štimac 1, Ivica Kožar 1 M.Sc,Assistat, Ph.D. Professor 1, Faculty of Civil Egieerig, Uiverity of Rieka, Croatia INTRODUCTION The vehicle-iduced
INTEGRATION BY PARTS (TABLE METHOD)
INTEGRATION BY PARTS (TABLE METHOD) Suppose you wat to evaluate cos d usig itegratio by parts. Usig the u dv otatio, we get So, u dv d cos du d v si cos d si si d or si si d We see that it is ecessary
Chapter 14: Chemical Equilibrium
hapter 14: hemical Equilibrium 46 hapter 14: hemical Equilibrium Sectio 14.1: Itroductio to hemical Equilibrium hemical equilibrium is the state where the cocetratios of all reactats ad products remai
Assignment Number 3 Solutions
Math 4354, Assigmet Number 3 Solutios 1. u t (x, t) = u xx (x, t), < x (1) u(, t) =, u(, t) = u(x, ) = x ( 1) +1 u(x, t) = e t si(x). () =1 Solutio: Look for simple solutios i the form u(x, t) =
NICK DUFRESNE. 1 1 p(x). To determine some formulas for the generating function of the Schröder numbers, r(x) = a(x) =
AN INTRODUCTION TO SCHRÖDER AND UNKNOWN NUMBERS NICK DUFRESNE Abstract. I this article we will itroduce two types of lattice paths, Schröder paths ad Ukow paths. We will examie differet properties of each,
~j=zhax~ 6=0, t,2... ), k=~
Numerische Mathematik 6, 35 -- 39 (964) Exactess Coditios i Numerical Quadrature* By HERBERT S. WILl* The itegral b () I--f t(x)dx is usually computed umerically by meas of a formula of the type (2) I
MULTIPLE TIME SCALES SOLUTION OF AN EQUATION WITH QUADRATIC AND CUBIC NONLINEARITIES HAVING FRAC- TIONAL-ORDER DERIVATIVE
Mathematical ad Computatioal Applicatios, Vol. 6, No., pp. 3-38,. Associatio for Scietific Research MULIPLE IME SCALES SOLUION OF AN EQUAION WIH QUADRAIC AND CUBIC NONLINEARIIES HAVING FRAC- IONAL-ORDER
MAXIMALLY FLAT FIR FILTERS
MAXIMALLY FLAT FIR FILTERS This sectio describes a family of maximally flat symmetric FIR filters first itroduced by Herrma [2]. The desig of these filters is particularly simple due to the availability
Numerical Methods for Ordinary Differential Equations
Numerical Methods for Ordiary Differetial Equatios Braislav K. Nikolić Departmet of Physics ad Astroomy, Uiversity of Delaware, U.S.A. PHYS 460/660: Computatioal Methods of Physics http://www.physics.udel.edu/~bikolic/teachig/phys660/phys660.html
Bangi 43600, Selangor Darul Ehsan, Malaysia (Received 12 February 2010, accepted 21 April 2010)
O Cesáro Meas of Order μ for Outer Fuctios ISSN 1749-3889 (prit), 1749-3897 (olie) Iteratioal Joural of Noliear Sciece Vol9(2010) No4,pp455-460 Maslia Darus 1, Rabha W Ibrahim 2 1,2 School of Mathematical
Special Modeling Techniques
Colorado School of Mies CHEN43 Secial Modelig Techiques Secial Modelig Techiques Summary of Toics Deviatio Variables No-Liear Differetial Equatios 3 Liearizatio of ODEs for Aroximate Solutios 4 Coversio
Mathematics 3 Outcome 1. Vectors (9/10 pers) Lesson, Outline, Approach etc. This is page number 13. produced for TeeJay Publishers by Tom Strang
Vectors (9/0 pers) Mathematics 3 Outcome / Revise positio vector, PQ = q p, commuicative, associative, zero vector, multiplicatio by a scalar k, compoets, magitude, uit vector, (i, j, ad k) as well as
CS537. Numerical Analysis and Computing
CS57 Numerical Aalysis ad Computig Lecture Locatig Roots o Equatios Proessor Ju Zhag Departmet o Computer Sciece Uiversity o Ketucky Leigto KY 456-6 Jauary 9 9 What is the Root May physical system ca be
IN many scientific and engineering applications, one often
INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS, VOL 3, NO, FEBRUARY 07 5 Secod Degree Refiemet Jacobi Iteratio Method for Solvig System of Liear Equatio Tesfaye Kebede Abstract Several
Signals and Systems. Problem Set: From Continuous-Time to Discrete-Time
Sigals ad Systems Problem Set: From Cotiuous-Time to Discrete-Time Updated: October 5, 2017 Problem Set Problem 1 - Liearity ad Time-Ivariace Cosider the followig systems ad determie whether liearity ad
SOLUTION SET VI FOR FALL [(n + 2)(n + 1)a n+2 a n 1 ]x n = 0,
![SOLUTION SET VI FOR FALL [(n + 2)(n + 1)a n+2 a n 1 ]x n = 0,](/thumbs/94/119566625.jpg "SOLUTION SET VI FOR FALL [(n + 2)(n + 1)a n+2 a n 1 ]x n = 0,")
4. Series Solutios of Differetial Equatios:Special Fuctios 4.. Illustrative examples.. 5. Obtai the geeral solutio of each of the followig differetial equatios i terms of Maclauri series: d y (a dx = xy,
Some New Iterative Methods for Solving Nonlinear Equations
World Applied Scieces Joural 0 (6): 870-874, 01 ISSN 1818-495 IDOSI Publicatios, 01 DOI: 10.589/idosi.wasj.01.0.06.830 Some New Iterative Methods for Solvig Noliear Equatios Muhammad Aslam Noor, Khalida
3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,
![3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,](/thumbs/82/85448051.jpg "3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,")
3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [
The Numerical Solution of Singular Fredholm Integral Equations of the Second Kind
WDS' Proceedigs of Cotributed Papers, Part I, 57 64, 2. ISBN 978-8-7378-39-2 MATFYZPRESS The Numerical Solutio of Sigular Fredholm Itegral Equatios of the Secod Kid J. Rak Charles Uiversity, Faculty of
Chapter 9: Numerical Differentiation
178 Chapter 9: Numerical Differetiatio Numerical Differetiatio Formulatio of equatios for physical problems ofte ivolve derivatives (rate-of-chage quatities, such as velocity ad acceleratio). Numerical
LOWER BOUNDS FOR THE BLOW-UP TIME OF NONLINEAR PARABOLIC PROBLEMS WITH ROBIN BOUNDARY CONDITIONS
Electroic Joural of Differetial Equatios, Vol. 214 214), No. 113, pp. 1 5. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.ut.edu ftp ejde.math.txstate.edu LOWER BOUNDS FOR THE BLOW-UP
L 5 & 6: RelHydro/Basel. f(x)= ( ) f( ) ( ) ( ) ( ) n! 1! 2! 3! If the TE of f(x)= sin(x) around x 0 is: sin(x) = x - 3! 5!
aylor epasio: Let ƒ() be a ifiitely differetiable real fuctio. At ay poit i the eighbourhood of =0, the fuctio ca be represeted as a power series of the followig form: X 0 f(a) f() ƒ() f()= ( ) f( ) (
Analysis of a Numerical Scheme An Example
http://www.d.edu/~gtryggva/cfd-course/ Computatioal Fluid Dyamics Lecture 3 Jauary 5, 7 Aalysis of a Numerical Scheme A Example Grétar Tryggvaso Numerical Aalysis Example Use the leap-frog method (cetered
1. Linear advection equations Let u R be a scalar unknown function of x R and t R + satisfying the Cauchy problem. u t + a(x, t)u x = 0, x R, t > 0,
1 Liear advectio equatios Let u be a scalar ukow fuctio of x ad t + satisfyig the Cauchy problem u t + au x =, x, t >, 1 ux, = u x, where a is a give positive costat, ad u x is a kow fuctio To solve this
AN OPEN-PLUS-CLOSED-LOOP APPROACH TO SYNCHRONIZATION OF CHAOTIC AND HYPERCHAOTIC MAPS
http://www.paper.edu.c Iteratioal Joural of Bifurcatio ad Chaos, Vol. 1, No. 5 () 119 15 c World Scietific Publishig Compay AN OPEN-PLUS-CLOSED-LOOP APPROACH TO SYNCHRONIZATION OF CHAOTIC AND HYPERCHAOTIC
Benaissa Bernoussi Université Abdelmalek Essaadi, ENSAT de Tanger, B.P. 416, Tanger, Morocco
EXTENDING THE BERNOULLI-EULER METHOD FOR FINDING ZEROS OF HOLOMORPHIC FUNCTIONS Beaissa Beroussi Uiversité Abdelmalek Essaadi, ENSAT de Tager, B.P. 416, Tager, Morocco e-mail: Beaissa@fstt.ac.ma Mustapha
Adaptation and Assessment of a High Resolution
Adaptatio ad Assessmet of a High Resolutio Semi-Discrete Numerical Scheme for Hyperbolic Systems with Source Terms ad Stiffess R. Naidoo, ad S. Baboolal Departmet of Mathematics ad Physics, M.L. Sulta
MATHEMATICS. 61. The differential equation representing the family of curves where c is a positive parameter, is of
MATHEMATICS 6 The differetial equatio represetig the family of curves where c is a positive parameter, is of Order Order Degree (d) Degree (a,c) Give curve is y c ( c) Differetiate wrt, y c c y Hece differetial
Appendix: The Laplace Transform
Appedix: The Laplace Trasform The Laplace trasform is a powerful method that ca be used to solve differetial equatio, ad other mathematical problems. Its stregth lies i the fact that it allows the trasformatio
Mathematical Series (You Should Know)
Mathematical Series You Should Kow Mathematical series represetatios are very useful tools for describig images or for solvig/approimatig the solutios to imagig problems. The may be used to epad a fuctio
HOUSEHOLDER S APPROXIMANTS AND CONTINUED FRACTION EXPANSION OF QUADRATIC IRRATIONALS. Vinko Petričević University of Zagreb, Croatia
HOUSEHOLDER S APPROXIMANTS AND CONTINUED FRACTION EXPANSION OF QUADRATIC IRRATIONALS Viko Petričević Uiversity of Zagre, Croatia Astract There are umerous methods for ratioal approximatio of real umers
A comparative study of a system of Lotka-Voltera type of PDEs through perturbation methods
Computatioal Ecology ad Software, 13, 3(4): 11-15 Article A comparative study of a system of Lotka-Voltera type of PDEs through perturbatio methods H. A. Wahab 1, M. Shakil 1, T. Kha 1, S. Bhatti, M. Naeem
September 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1
September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright
Median and IQR The median is the value which divides the ordered data values in half.
STA 666 Fall 2007 Web-based Course Notes 4: Describig Distributios Numerically Numerical summaries for quatitative variables media ad iterquartile rage (IQR) 5-umber summary mea ad stadard deviatio Media
Introduction to Optimization Techniques. How to Solve Equations
Itroductio to Optimizatio Techiques How to Solve Equatios Iterative Methods of Optimizatio Iterative methods of optimizatio Solutio of the oliear equatios resultig form a optimizatio problem is usually
Inverse Matrix. A meaning that matrix B is an inverse of matrix A.
Iverse Matrix Two square matrices A ad B of dimesios are called iverses to oe aother if the followig holds, AB BA I (11) The otio is dual but we ofte write 1 B A meaig that matrix B is a iverse of matrix
INTRODUCTORY MATHEMATICAL ANALYSIS
INTRODUCTORY MATHEMATICAL ANALYSIS For Busiess, Ecoomics, ad the Life ad Social Scieces Chapter 4 Itegratio 0 Pearso Educatio, Ic. Chapter 4: Itegratio Chapter Objectives To defie the differetial. To defie
Numerical Solution of the First-Order Hyperbolic Partial Differential Equation with Point-Wise Advance
Iteratioal oural of Sciece ad Research (ISR) ISSN (Olie): 39-74 Ide Copericus Value (3): 4 Impact Factor (3): 4438 Numerical Solutio of the First-Order Hyperbolic Partial Differetial Equatio with Poit-Wise
Numerical Solution of the Two Point Boundary Value Problems By Using Wavelet Bases of Hermite Cubic Spline Wavelets
Australia Joural of Basic ad Applied Scieces, 5(): 98-5, ISSN 99-878 Numerical Solutio of the Two Poit Boudary Value Problems By Usig Wavelet Bases of Hermite Cubic Splie Wavelets Mehdi Yousefi, Hesam-Aldie