Mathematical Models with Maple
|
|
- Maximilian Kelly
- 5 years ago
- Views:
Transcription
1 Algebraic Biology Mathematical Models with Maple Tetsu YAMAGUCHI Applied System nd Division, Cybernet Systems Co., Ltd., Otsuka -9-3, Bunkyo-ku, Tokyo , Japan Abstract In this paper, we introduce the fundamental capability of well-known computer algebra system called Maple. Using with this Maple system, we investigated on how computer algebra system can work for research and development study in the biology. In particular, we present a review of Maple s functional role in the development of mathematical models. 1. Introduction The Maple system has been originally developed on 1980, by SCG (Symbolic Computation Group) in University of Waterloo, Canada. The basic aspect of the system was compact, but powerful computer algebra system with fundamental algebraic computation capabilities []. Throughout many significant contributions from researchers in computer algebra world for the past 5 years, it has recently released the latest version called Maple 10 on 005 from Maplesoft, a division of Waterloo Maple Inc, which has established for the purpose of marketing, supporting and distributing the software. Recent research and development work in designing and modeling requires the stable and the flexible mathematical models which can describe the system. In this purpose, researchers and engineers also requires high-level mathematical knowledge such as transformation of various differential equations and its related analytical operation so that numerical simulation can t handle the problem with unified and coordinated methodology. The computer algebra system can handle to state the system by exact and can convert its expression into an equivalent another form with analytical transformation and simplification by implementing thousands of mathematical formulae. Using with computer algebra system, we can develop the mathematical model which has less amount of calculation, for example. In Section, we introduce the fundamental capability of Maple system. In particular, we reviewed about basic functionality for ordinary and partial differential equations (ODE, PDE) so that Maple is well-known for solving and evaluating of various differential equations. Section 3 discusses a typical model development and its classic flow on Maple system. As a specific understanding of Maple system, we show steps to solve the differential equation for epidemics. Finally, the paper ends with a brief summary referencing some applications of Maple in various area. pp c 005 by Universal Academy Press, Inc. / Tokyo, Japan
2 15 Fig 1. Embedded interactive components in Maple Worksheet. Fundamental Capability of Maple system Recent computer algebra system including Maple has following common capabilities; Symbolic Computation: Polynomial Operations (GCD, Factorization, Root-Finding, Gröbner Base), Symbolic-Differentiation/Integration, Solving Algebraic Equation, Exact Solution for ODEs/PDEs, Formula Conversion, Series Expansion. Numerical Computation: Approximation, Arbitrary Precision Computation, Numerical Integration, Interval Arithmetic, Statistical Computation (Probability Computation, Descriptive Statistics, Tabulation, Data Manipulation/Smoothing, Hypothesis Testing, Regression/ Estimation), Interpolation/Fitting, Integral/Discrete Transformation. Visualization: D/3D Graphics, Animation, Density Plot, Contour Plot, D/3D Implicit Plot, Statistical Plot, Interactive Operation for Graphics, Exporting to other formats. Miscellaneous: Code Conversion (L A TEX, C, Fortran, Visual Basic, Java, MATLAB), API (C, VB, Java), Educational Contents, GUI, Unit Arithmetic, Objective Programming. In addition, Maple has an intuitive Graphical User Interface (GUI) called Worksheet for entering mathematical expression, and also enables us to author a mathematical document and interactive operation with embedded component such as slider, text field, plot area and command button (see Fig. 1). Internal expressions in the worksheet are stored as Maple-based XML format, so we can easily convert any kind of mathematical expression to an appropriate MathML expression or other structured expression. As listed above, Maple also has various functionalities based on symbolic and numeric algorithms for solving of differential equation. Basically, we can use two typical commands implemented in Maple as dsolve for ODE and pdsolve for PDE, respectively. We can select a lot of algorithm for solving ODE(s) symbolically such that Lie symmetries, classification, integration factors, integral transformations and series expansion at given point. Furthermore, optional functions for applying enhanced operation to given ODE as following can be available by loading the DEtools package; Commands for simplifying ODE(s) with integrability condition.
3 153 Commands for constructing closed-form solution. Commands for classifying ODE(s). Commands for Visualization. Commands for Conversion. Using with this DE solver, we discuss a typical mathematical model development and its general flow on Maple system in the next section. On the other hand, Maple can also handle the PDE(s) with both of symbolic and numeric for finding solution. For the symbolic PDE solving, Maple takes a phased procedure automatically for rewriting the system which can simply express in form of polynomial by using an approach with differential algebraic equation. With some verifications such as integrability conditions in the system, it also produces to divide subsystems of given PDE, and then tries to obtain a solution on one variable. After iteration of finding all possible solutions, Maple performs a set of solutions for given PDE system. In addition, we can compute, if exists, a Traveling Wave Solution (TWS) as a power series in tanh function or several (special) mathematical functions. For the numeric computation to solve PDE system, we can take 11 classical methods such as Euler, Backward-Euler, Crank-Nicholson, and so on. The numerically computed result from PDE solver is a form of intermediate expression of the approximate solution. Hence, for example, we can indicate some of step size to construct an approximate value as free and can estimate of errors. For instance, let us consider a simple one-way wave equation as u (x, t) = u (x, t) t x with a boundary condition of {u (x, 0) = sin ( π x), u (0, t) = sin ( π t)}. After constructing an intermediate expression of the solution and its error function, we can see the band difference between each step size in time and space: 1/16 for the left side and 1/3 for right side, respectively, in Fig.. Fig. Error estimations for PDE solution As we introduced the fundamental functionalities in this section, we can analyze the mathematical model described by differential equation using with Maple.
4 Typical Model development on Maple system In this section, we review the typical strategy with respect to mathematical model development using with Maple system by solving differential equation. Let us consider the case in biology with famous mathematical model of epidemics [3]. We use a following first-order ordinary differential equation system which describes on infection model for the number of infected patients denoted by P (t), carrier patients denoted by C(t) and removed patients denoted by R(t). S(t), C(t) and R(t) satisfy Ṡ = r S P, (1) P = r S P γ P, Ṙ = γp. where r and γ are positive constants for infection rate and cull rate, respectively. Note that d(s + P + R)/dt = 0. That is, S(t) + P (t) + R(t) = N () where N is a constant which means a total population in that area. Since equation () and (), we can derive a single differential equation for the number of removed patients as follows; Ṙ = γ(n R S 0 e R/ρ ) (3) where ρ = γ/r and S 0 = S(t 0 ). After this formulation, we can get the following implicit solution by applying dsolve command for given differential equation. R(t) ( ) t + γ 1 N + a + S 0 e a 1 ρ d a + C1 = 0 (4) where a and C1 are generated by Maple automatically as internal variables. To obtain an explicit solution of the differential equation (3), we need to replace e R/ρ to its 3-rd order taylor series in case that R(t) 1. ρ In Maple, we can get taylor series expansion of e x by taylor command with specified order, and can substitute x = R(t)/ρ by subs command. Thus, equation (3) can be expanded as following expression; Ṙ = γ N γ R (t) γ S 0 + γ S 0R (t) ρ 1 γ S 0 (R (t)) (5) ρ Again, we can get the solution of differential equation (5) by dsolve command with an initial condition R(0) = N. ( ( ( ρ 1 S 0 ρ + tan tγ ( )) ) ) ρ φ ρ 1 arctan ρ + 1 S φ ρ ρ φ ρ R (t) = φ (6) S 0 where φ = ρ S 0 ρ + S 0 N S 0. However, we have to mention here that expression of Maple s solution (6) is not automatically collected terms s.t. rhs of φ. User sometimes needs to indicate explicitly each rules to abbreviation or term replacement.
5 155 Theoretically, according to the paper [3], it can be a form of [ S0 ρ 1 + α tanh R(t) = ρ S 0 ( 1 αγ t φ )] (7) where [ (S0 ) ] 1 α = ρ 1 + S 0(N S 0 ), ρ ( ) φ = tanh 1 1 S0 α ρ 1. To summarize of section, we must point out that computer algebra system, however, can specify a determinate transformation which is based on user own strategy and enables us to measure what kind of expression is the most efficient in that context by applying algebraic operations. After obtaining an explicit form of the solution for given differential equation, we can then process to identify specific values in unknown coefficients by taking the linear and/or nonlinear least square fitting or other symbolic-numeric methods. 4. Summary We have reviewed the fundamentals of Maple system in the viewpoint of evaluation for the mathematical model which is expressed by differential equation. Recently, Maple and other computer algebra system are required to identify more comprehensive model so that dynamic simulation by only numerical computation can t handle instability and uncertainty of the system. Meanwhile, it still exists a lot of computation cost in almost computer algebraic algorithms. Hence, we need to develop symbolic-numeric mixed type of algorithm in the application area too. References [1] M. Braun, Differential Equation and Their Applications (1983) Springer-Verlag, New York. [] B.W. Char, K.O. Geddes, W.M. Gentleman and G.H. Gonnet, The design of Maple: A compact, portable, and powerful computer algebra system. (1983) Computer Algebra (Proc. of EUROCAL 83), No.16, Springer-Verlag, Berlin, 101. [3] W.O. Kermack and A.G. McKendrick, Contributions to the mathematical theory of epidemics, Proc. Roy. Stat. Soc. A, 115 (197) 700. [4] L. Ljung, System Identification: Theory for the User (1987) Prentice-Hall. [5] M.B. Monagan, K.O. Geddes, K.M. Heal, G. Labahn et al., Maple Advanced Programming Guide (005) [6] T. Yabe, F. Xiao and T. Utsumi, The Constrained Interpolation Profile Method for Multiphase Analysis, Journal of Comp. Physics 169 (001) 556.
A Symbolic Numeric Environment for Analyzing Measurement Data in Multi-Model Settings (Extended Abstract)
A Symbolic Numeric Environment for Analyzing Measurement Data in Multi-Model Settings (Extended Abstract) Christoph Richard 1 and Andreas Weber 2? 1 Institut für Theoretische Physik, Universität Tübingen,
More informationLecture 17: Ordinary Differential Equation II. First Order (continued)
Lecture 17: Ordinary Differential Equation II. First Order (continued) 1. Key points Maple commands dsolve dsolve[interactive] dsolve(numeric) 2. Linear first order ODE: y' = q(x) - p(x) y In general,
More informationMathematics with Maple
Mathematics with Maple A Comprehensive E-Book Harald Pleym Preface The main objective of these Maple worksheets, organized for use with all Maple versions from Maple 14, is to show how the computer algebra
More informationSparse Polynomial Multiplication and Division in Maple 14
Sparse Polynomial Multiplication and Division in Maple 4 Michael Monagan and Roman Pearce Department of Mathematics, Simon Fraser University Burnaby B.C. V5A S6, Canada October 5, 9 Abstract We report
More informationClassroom Tips and Techniques: Electric Field from Distributed Charge
Classroom Tips and Techniques: Electric Field from Distributed Charge Introduction Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft This past summer I was asked if Maple could
More informationHistorical Perspective on Numerical Problem Solving
Historical Perspective on Numerical Problem Solving Mordechai Shacham Department of Chemical Engineering Ben Gurion University of the Negev Beer-Sheva, Israel Michael B. Cutlip Department of Chemical Engineering
More informationExcel for Scientists and Engineers Numerical Method s. E. Joseph Billo
Excel for Scientists and Engineers Numerical Method s E. Joseph Billo Detailed Table of Contents Preface Acknowledgments About the Author Chapter 1 Introducing Visual Basic for Applications 1 Chapter
More informationClassroom Tips and Techniques: Series Expansions Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft
Introduction Classroom Tips and Techniques: Series Expansions Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft Maple has the ability to provide various series expansions and
More informationODEs and Redefining the Concept of Elementary Functions
ODEs and Redefining the Concept of Elementary Functions Alexander Gofen The Smith-Kettlewell Eye Research Institute, 2318 Fillmore St., San Francisco, CA 94102, USA galex@ski.org, www.ski.org/gofen Abstract.
More informationThe Maple Computer Algebra System
Computer Science Journal of Moldova, vol.1, no.2(2), 1993 The Maple Computer Algebra System Michael Monagan Maple is a comprehensive general purpose computer algebra system. It is used primarily in education
More informationThe Tangent Parabola The AMATYC Review, Vol. 23, No. 1, Fall 2001, pp
The Tangent Parabola The AMATYC Review, Vol. 3, No., Fall, pp. 5-3. John H. Mathews California State University Fullerton Fullerton, CA 9834 By Russell W. Howell Westmont College Santa Barbara, CA 938
More informationReactor Design within Excel Enabled by Rigorous Physical Properties and an Advanced Numerical Computation Package
Reactor Design within Excel Enabled by Rigorous Physical Properties and an Advanced Numerical Computation Package Mordechai Shacham Department of Chemical Engineering Ben Gurion University of the Negev
More informationFour Point Gauss Quadrature Runge Kuta Method Of Order 8 For Ordinary Differential Equations
International journal of scientific and technical research in engineering (IJSTRE) www.ijstre.com Volume Issue ǁ July 206. Four Point Gauss Quadrature Runge Kuta Method Of Order 8 For Ordinary Differential
More informationLeading Flying Objects
Leading Flying Objects Stephen M. Watt Western University J. W. Graham Medal Seminar 13 June 2012, University of Waterloo Leading adj. 1. guiding, directing or influencing 2. of greatest importance or
More informationMathematical Visualization Tool GCLC/WinGCLC
Mathematical Visualization Predrag Janičić URL: www.matf.bg.ac.rs/ janicic Faculty of Mathematics, University of Belgrade, The Third School in Astronomy: Astroinformatics Virtual Observatory University
More informationA Brief Introduction To. GRTensor. On MAPLE Platform. A write-up for the presentation delivered on the same topic as a part of the course PHYS 601
A Brief Introduction To GRTensor On MAPLE Platform A write-up for the presentation delivered on the same topic as a part of the course PHYS 601 March 2012 BY: ARSHDEEP SINGH BHATIA arshdeepsb@gmail.com
More informationPartial Differential Equations
Next: Using Matlab Up: Numerical Analysis for Chemical Previous: Ordinary Differential Equations Subsections Finite Difference: Elliptic Equations The Laplace Equations Solution Techniques Boundary Conditions
More informationShijun Liao. Homotopy Analysis Method in Nonlinear Differential Equations
Shijun Liao Homotopy Analysis Method in Nonlinear Differential Equations Shijun Liao Homotopy Analysis Method in Nonlinear Differential Equations With 127 figures Author Shijun Liao Shanghai Jiao Tong
More informationComputer Aided Design of Thermal Systems (ME648)
Computer Aided Design of Thermal Systems (ME648) PG/Open Elective Credits: 3-0-0-9 Updated Syallabus: Introduction. Basic Considerations in Design. Modelling of Thermal Systems. Numerical Modelling and
More informationNUMERICAL METHODS. lor CHEMICAL ENGINEERS. Using Excel', VBA, and MATLAB* VICTOR J. LAW. CRC Press. Taylor & Francis Group
NUMERICAL METHODS lor CHEMICAL ENGINEERS Using Excel', VBA, and MATLAB* VICTOR J. LAW CRC Press Taylor & Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Croup,
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 informationChabot College Fall Course Outline for Mathematics 25 COMPUTATIONAL METHODS FOR ENGINEERS AND SCIENTISTS
Chabot College Fall 2010 Course Outline for Mathematics 25 COMPUTATIONAL METHODS FOR ENGINEERS AND SCIENTISTS Catalog Description: MTH 25 - Computational Methods for Engineers and Scientists 3.00 units
More informationComputer algebra systems (CAS) have been around for a number of years,
Problem solving in calculus with symbolic geometry and CAS Philip Todd Saltire Software James Wiechmann Tualatin High School, Tualatin Oregon, USA Computer
More informationNUMERICAL METHODS FOR ENGINEERING APPLICATION
NUMERICAL METHODS FOR ENGINEERING APPLICATION Second Edition JOEL H. FERZIGER A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York / Chichester / Weinheim / Brisbane / Singapore / Toronto
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 informationSection 2.1 Differential Equation and Solutions
Section 2.1 Differential Equation and Solutions Key Terms: Ordinary Differential Equation (ODE) Independent Variable Order of a DE Partial Differential Equation (PDE) Normal Form Solution General Solution
More informationRelations in epidemiology-- the need for models
Plant Disease Epidemiology REVIEW: Terminology & history Monitoring epidemics: Disease measurement Disease intensity: severity, incidence,... Types of variables, etc. Measurement (assessment) of severity
More informationA Mathematica Companion for Differential Equations
iii A Mathematica Companion for Differential Equations Selwyn Hollis PRENTICE HALL, Upper Saddle River, NJ 07458 iv v Contents Preface viii 0. An Introduction to Mathematica 0.1 Getting Started 1 0.2 Functions
More information1 One-Dimensional, Steady-State Conduction
1 One-Dimensional, Steady-State Conduction 1.1 Conduction Heat Transfer 1.1.1 Introduction Thermodynamics defines heat as a transfer of energy across the boundary of a system as a result of a temperature
More informationAppendix F. + 1 Ma 1. 2 Ma Ma Ma ln + K = 0 (4-173)
5:39p.m. Page:949 Trimsize:8.5in 11in Appendix F F.1 MICROSOFT EXCEL SOLVER FOR NON-LINEAR EQUATIONS The Solver is an optimization package that finds a maximum, minimum, or specified value of a target
More informationA Note on the Spread of Infectious Diseases. in a Large Susceptible Population
International Mathematical Forum, Vol. 7, 2012, no. 50, 2481-2492 A Note on the Spread of Infectious Diseases in a Large Susceptible Population B. Barnes Department of Mathematics Kwame Nkrumah University
More informationSPATIAL-TEMPORAL TECHNIQUES FOR PREDICTION AND COMPRESSION OF SOIL FERTILITY DATA
SPATIAL-TEMPORAL TECHNIQUES FOR PREDICTION AND COMPRESSION OF SOIL FERTILITY DATA D. Pokrajac Center for Information Science and Technology Temple University Philadelphia, Pennsylvania A. Lazarevic Computer
More informationNumerical Methods for Engineers
Numerical Methods for Engineers SEVENTH EDITION Steven C Chopra Berger Chair in Computing and Engineering Tufts University Raymond P. Canal Professor Emeritus of Civil Engineering of Michiaan University
More informationExact Solutions for the Nonlinear (2+1)-Dimensional Davey-Stewartson Equation Using the Generalized ( G. )-Expansion Method
Journal of Mathematics Research; Vol. 6, No. ; 04 ISSN 96-9795 E-ISSN 96-9809 Published by Canadian Center of Science and Education Exact Solutions for the Nonlinear +-Dimensional Davey-Stewartson Equation
More informationÜbungsaufgaben zur VU Computermathematik Serie 9
Winfried Auzinger Sommersemester 214 Dirk Praetorius 4. 6. Juni 214 Übungsaufgaben zur VU Computermathematik Serie 9 Eine Sammlung verschiedener Problemstellungen. Die Angaben zu manchen der Aufgaben sind
More informationEnabling Advanced Problem Solving in Spreadsheets with Access to Physical Property Data
Enabling Advanced Problem Solving in Spreadsheets with Access to Physical Property Data Michael B. Cutlip, Professor of Chemical Engineering, University of Connecticut (speaker) michael.cutlip@uconn.edu
More informationMATHEMATICS (MATH) Calendar
MATHEMATICS (MATH) This is a list of the Mathematics (MATH) courses available at KPU. For information about transfer of credit amongst institutions in B.C. and to see how individual courses transfer, go
More information1.1 Motivation: Why study differential equations?
Chapter 1 Introduction Contents 1.1 Motivation: Why stu differential equations?....... 1 1.2 Basics............................... 2 1.3 Growth and decay........................ 3 1.4 Introduction to Ordinary
More informationAN EVALUATION SCHEME FOR THE UNCERTAINTY ANALYSIS OF A CAPTIVE TRAJECTORY SYSTEM
24th INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES AN EVALUATION SCHEME FOR THE UNCERTAINTY ANALYSIS OF A CAPTIVE TRAJECTORY SYSTEM Sean Tuling Defencetek, CSIR Keywords: Uncertainty Analysis, CTS
More informationMATHia Unit MATHia Workspace Overview TEKS
1 Function Overview Searching for Patterns Exploring and Analyzing Patterns Comparing Familiar Function Representations Students watch a video about a well-known mathematician creating an expression for
More informationScientific Computing II
Scientific Computing II Molecular Dynamics Numerics Michael Bader SCCS Technical University of Munich Summer 018 Recall: Molecular Dynamics System of ODEs resulting force acting on a molecule: F i = j
More informationSolutions Preliminary Examination in Numerical Analysis January, 2017
Solutions Preliminary Examination in Numerical Analysis January, 07 Root Finding The roots are -,0, a) First consider x 0 > Let x n+ = + ε and x n = + δ with δ > 0 The iteration gives 0 < ε δ < 3, which
More informationWhat are Numerical Methods? (1/3)
What are Numerical Methods? (1/3) Numerical methods are techniques by which mathematical problems are formulated so that they can be solved by arithmetic and logic operations Because computers excel at
More information1.6 Computing and Existence
1.6 Computing and Existence 57 1.6 Computing and Existence The initial value problem (1) y = f(x,y), y(x 0 ) = y 0 is studied here from a computational viewpoint. Answered are some basic questions about
More informationNumerical methods for the Navier- Stokes equations
Numerical methods for the Navier- Stokes equations Hans Petter Langtangen 1,2 1 Center for Biomedical Computing, Simula Research Laboratory 2 Department of Informatics, University of Oslo Dec 6, 2012 Note:
More informationOn Newton-type methods with cubic convergence
Journal of Computational and Applied Mathematics 176 (2005) 425 432 www.elsevier.com/locate/cam On Newton-type methods with cubic convergence H.H.H. Homeier a,b, a Science + Computing Ag, IT Services Muenchen,
More informationIn this work is used the version of the Mathematica software for Windows environment. The examples which are treated are "generic" and should ex
How to Use Mathematica to Solve Typical Problems in Celestial Mechanics D.C. Lob~ao y Λ y Instituto de Pesquisa e Desenvolvimento IPD Universidade do Vale do Para ba UNIVAP S~ao José dos Campos, SP Brasil
More informationMODELING OF ELASTO-PLASTIC MATERIALS IN FINITE ELEMENT METHOD
MODELING OF ELASTO-PLASTIC MATERIALS IN FINITE ELEMENT METHOD Andrzej Skrzat, Rzeszow University of Technology, Powst. Warszawy 8, Rzeszow, Poland Abstract: User-defined material models which can be used
More informationNumerical Methods for Engineers and Scientists
Numerical Methods for Engineers and Scientists Second Edition Revised and Expanded Joe D. Hoffman Department of Mechanical Engineering Purdue University West Lafayette, Indiana m MARCEL D E К К E R MARCEL
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 informationA Glimpse at Scipy FOSSEE. June Abstract This document shows a glimpse of the features of Scipy that will be explored during this course.
A Glimpse at Scipy FOSSEE June 010 Abstract This document shows a glimpse of the features of Scipy that will be explored during this course. 1 Introduction SciPy is open-source software for mathematics,
More informationPredici 11 Quick Overview
Predici 11 Quick Overview PREDICI is the leading simulation package for kinetic, process and property modeling with a major emphasis on macromolecular systems. It has been successfully utilized to model
More informationAPPLICATIONS OF FD APPROXIMATIONS FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS
LECTURE 10 APPLICATIONS OF FD APPROXIMATIONS FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS Ordinary Differential Equations Initial Value Problems For Initial Value problems (IVP s), conditions are specified
More informationPartial Differential Equations
Partial Differential Equations Introduction Deng Li Discretization Methods Chunfang Chen, Danny Thorne, Adam Zornes CS521 Feb.,7, 2006 What do You Stand For? A PDE is a Partial Differential Equation This
More informationWhile using the input and output data fu(t)g and fy(t)g, by the methods in system identification, we can get a black-box model like (In the case where
ESTIMATE PHYSICAL PARAMETERS BY BLACK-BOX MODELING Liang-Liang Xie Λ;1 and Lennart Ljung ΛΛ Λ Institute of Systems Science, Chinese Academy of Sciences, 100080, Beijing, China ΛΛ Department of Electrical
More informationGraphical User Interface (GUI) for Torsional Vibration Analysis of Rotor Systems Using Holzer and MatLab Techniques
Basrah Journal for Engineering Sciences, vol. 14, no. 2, 2014 255 Graphical User Interface (GUI) for Torsional Vibration Analysis of Rotor Systems Using Holzer and MatLab Techniques Dr. Ameen Ahmed Nassar
More informationIntegration Using Tables and Summary of Techniques
Integration Using Tables and Summary of Techniques Philippe B. Laval KSU Today Philippe B. Laval (KSU) Summary Today 1 / 13 Introduction We wrap up integration techniques by discussing the following topics:
More informationBusiness Calculus
Business Calculus 978-1-63545-025-5 To learn more about all our offerings Visit Knewtonalta.com Source Author(s) (Text or Video) Title(s) Link (where applicable) OpenStax Senior Contributing Authors: Gilbert
More informationComputational Modeling for Physical Sciences
Computational Modeling for Physical Sciences Since the invention of computers the use of computational modeling and simulations have revolutionized the way we study physical systems. Their applications
More informationCHAPTER 10: Numerical Methods for DAEs
CHAPTER 10: Numerical Methods for DAEs Numerical approaches for the solution of DAEs divide roughly into two classes: 1. direct discretization 2. reformulation (index reduction) plus discretization Direct
More informationSYMBOLIC AND NUMERICAL COMPUTING FOR CHEMICAL KINETIC REACTION SCHEMES
SYMBOLIC AND NUMERICAL COMPUTING FOR CHEMICAL KINETIC REACTION SCHEMES by Mark H. Holmes Yuklun Au J. W. Stayman Department of Mathematical Sciences Rensselaer Polytechnic Institute, Troy, NY, 12180 Abstract
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 informationPART II: Research Proposal Algorithms for the Simplification of Algebraic Formulae
Form 101 Part II 6 Monagan, 195283 PART II: Research Proposal Algorithms for the Simplification of Algebraic Formulae 1 Research Area Computer algebra (CA) or symbolic computation, as my field is known
More informationAthena Visual Software, Inc. 1
Athena Visual Studio Visual Kinetics Tutorial VisualKinetics is an integrated tool within the Athena Visual Studio software environment, which allows scientists and engineers to simulate the dynamic behavior
More informationComputational Biology Course Descriptions 12-14
Computational Biology Course Descriptions 12-14 Course Number and Title INTRODUCTORY COURSES BIO 311C: Introductory Biology I BIO 311D: Introductory Biology II BIO 325: Genetics CH 301: Principles of Chemistry
More informationFin System, Inc. Company Report. Temperature Profile Calculators. Team 1 J. C. Stewards, Lead A. B. Williams, Documentation M. D.
Fin System, Inc. Company Report Temperature Profile Calculators Team 1 J. C. Stewards, Lead A. B. Williams, Documentation M. D. Daily, Programmer Submitted in Fulfillment of Management Requirements August
More informationSyllabus for Applied Mathematics Graduate Student Qualifying Exams, Dartmouth Mathematics Department
Syllabus for Applied Mathematics Graduate Student Qualifying Exams, Dartmouth Mathematics Department Alex Barnett, Scott Pauls, Dan Rockmore August 12, 2011 We aim to touch upon many topics that a professional
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 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 informationThe Solution of Weakly Nonlinear Oscillatory Problems with No Damping Using MAPLE
World Applied Sciences Journal (): 64-69, 0 ISSN 88-495 IDOSI Publications, 0 DOI: 0.589/idosi.wasj.0..0.09 The Solution of Weakly Nonlinear Oscillatory Problems with No Damping Using MAPLE N. Hashim,
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 informationDifferential Equations DIRECT INTEGRATION. Graham S McDonald
Differential Equations DIRECT INTEGRATION Graham S McDonald A Tutorial Module introducing ordinary differential equations and the method of direct integration Table of contents Begin Tutorial c 2004 g.s.mcdonald@salford.ac.uk
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 informationA NEW SOLUTION OF SIR MODEL BY USING THE DIFFERENTIAL FRACTIONAL TRANSFORMATION METHOD
April, 4. Vol. 4, No. - 4 EAAS & ARF. All rights reserved ISSN35-869 A NEW SOLUTION OF SIR MODEL BY USING THE DIFFERENTIAL FRACTIONAL TRANSFORMATION METHOD Ahmed A. M. Hassan, S. H. Hoda Ibrahim, Amr M.
More informationMATLAB for Chemical Engineering
MATLAB for Chemical Engineering Dr. M. Subramanian Associate Professor Department of Chemical Engineering Sri Sivasubramaniya Nadar College of Engineering OMR, Chennai 603110 msubbu.in[at]gmail.com 16
More informationANIMATION OF LAGRANGE MULTIPLIER METHOD AND MAPLE Pavel Prazak University of Hradec Kralove, Czech Republic
ANIMATION OF LAGRANGE MULTIPLIER METHOD AND MAPLE Pavel Prazak University of Hradec Kralove, Czech Republic Experience shows that IT is a powerful tool for visualization of hidden mathematical concepts.
More informationSeymour Public Schools Curriculum
Algebra II continues the study of functions. Students learn operations with functions and graphing of different types of functions. Students solve equations and perform operations with exponents, matrices
More informationMATLAB TOOL FOR IDENTIFICATION OF NONLINEAR SYSTEMS
MATLAB TOOL FOR IDENTIFICATION OF NONLINEAR SYSTEMS M. Kalúz, Ľ. Čirka, M. Fikar Institute of Information Engineering, Automation, and Mathematics, FCFT STU in Bratislava Abstract This contribution describes
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 informationArches Part 1: Introduction to the Uintah Computational Framework. Charles Reid Scientific Computing Summer Workshop July 14, 2010
Arches Part 1: Introduction to the Uintah Computational Framework Charles Reid Scientific Computing Summer Workshop July 14, 2010 Arches Uintah Computational Framework Cluster Node Node Node Node Node
More informationThe Modified (G /G)-Expansion Method for Nonlinear Evolution Equations
The Modified ( /-Expansion Method for Nonlinear Evolution Equations Sheng Zhang, Ying-Na Sun, Jin-Mei Ba, and Ling Dong Department of Mathematics, Bohai University, Jinzhou 11000, P. R. China Reprint requests
More informationDifferential Equations Dynamical Systems And An Introduction To Chaos Solutions Manual Pdf
Differential Equations Dynamical Systems And An Introduction To Chaos Solutions Manual Pdf Math 134: Ordinary Differential Equations and Dynamical Systems and an Introduction to Chaos, third edition, Academic
More informationExact Solutions for a Fifth-Order Two-Mode KdV Equation with Variable Coefficients
Contemporary Engineering Sciences, Vol. 11, 2018, no. 16, 779-784 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ces.2018.8262 Exact Solutions for a Fifth-Order Two-Mode KdV Equation with Variable
More informationTheory of Equations. Lesson 5. Barry H. Dayton Northeastern Illinois University Chicago, IL 60625, USA. bhdayton/theq/
Theory of Equations Lesson 5 by Barry H. Dayton Northeastern Illinois University Chicago, IL 60625, USA www.neiu.edu/ bhdayton/theq/ These notes are copyrighted by Barry Dayton, 2002. The PDF files are
More information5.2 - Euler s Method
5. - Euler s Method Consider solving the initial-value problem for ordinary differential equation: (*) y t f t, y, a t b, y a. Let y t be the unique solution of the initial-value problem. In the previous
More informationCURRICULUM GUIDE Algebra II College-Prep Level
CURRICULUM GUIDE Algebra II College-Prep Level Revised Sept 2011 Course Description: This course builds upon the foundational concepts of Algebra I and Geometry, connects them with practical real world
More informationLet's begin by evaluating several limits. sqrt x 2 C4 K2. limit. x 2, x = 0 ; O limit. 1K cos t t. , t = 0 ; 1C 1 n. limit. 6 x 3 C5.
Dr. Straight's Maple Examples Example II: Calculus perations and Functions Functions Covered: diff, dsolve, implicitdiff, implicitplot, int, limit, odeplot, piecewise, plot, solve, subs, sum, taylor Let's
More informationComputational Methods for Engineers Programming in Engineering Problem Solving
Computational Methods for Engineers Programming in Engineering Problem Solving Abu Hasan Abdullah January 6, 2009 Abu Hasan Abdullah 2009 An Engineering Problem Problem Statement: The length of a belt
More informationSolving Nonlinear Wave Equations and Lattices with Mathematica. Willy Hereman
Solving Nonlinear Wave Equations and Lattices with Mathematica Willy Hereman Department of Mathematical and Computer Sciences Colorado School of Mines Golden, Colorado, USA http://www.mines.edu/fs home/whereman/
More informationENGI 3424 First Order ODEs Page 1-01
ENGI 344 First Order ODEs Page 1-01 1. Ordinary Differential Equations Equations involving only one independent variable and one or more dependent variables, together with their derivatives with respect
More informationTransactions on Modelling and Simulation vol 12, 1996 WIT Press, ISSN X
Simplifying integration for logarithmic singularities R.N.L. Smith Department ofapplied Mathematics & OR, Cranfield University, RMCS, Shrivenham, Swindon, Wiltshire SN6 SLA, UK Introduction Any implementation
More informationMathematical Modelling Using Simulink
Experiment Two Mathematical Modelling Using Simulink Control Systems Laboratory Dr. Zaer Abo Hammour Dr. Zaer Abo Hammour Control Systems Laboratory 1. Mathematical Model Definition A mathematical model
More informationApplication demonstration. BifTools. Maple Package for Bifurcation Analysis in Dynamical Systems
Application demonstration BifTools Maple Package for Bifurcation Analysis in Dynamical Systems Introduction Milen Borisov, Neli Dimitrova Department of Biomathematics Institute of Mathematics and Informatics
More informationMoments of Inertia. Maplesoft, a division of Waterloo Maple Inc., 2008
Introduction Moments of Inertia Maplesoft, a division of Waterloo Maple Inc., 2008 This application is one of a collection of educational engineering eamples using Maple. These ever stage of the solution,
More information1 Introduction to MATLAB
L3 - December 015 Solving PDEs numerically (Reports due Thursday Dec 3rd, carolinemuller13@gmail.com) In this project, we will see various methods for solving Partial Differential Equations (PDEs) using
More informationSolving Parametric Polynomial Systems by RealComprehensiveTriangularize
Solving Parametric Polynomial Systems by RealComprehensiveTriangularize Changbo Chen 1 and Marc Moreno Maza 2 1 Chongqing Key Laboratory of Automated Reasoning and Cognition, Chongqing Institute of Green
More informationSTEAMEST: A Software Tool for Estimation of Physical Properties of Water and Steam
226 JOURNAL OF SOFTWARE, VOL. 4, NO. 3, MAY 2009 STEAMEST: A Software Tool for Estimation of Physical Properties of Water and Steam Muhammad Faheem Department of Chemical Engineering, University of Engineering
More informationCompute the behavior of reality even if it is impossible to observe the processes (for example a black hole in astrophysics).
1 Introduction Read sections 1.1, 1.2.1 1.2.4, 1.2.6, 1.3.8, 1.3.9, 1.4. Review questions 1.1 1.6, 1.12 1.21, 1.37. The subject of Scientific Computing is to simulate the reality. Simulation is the representation
More informationExact Solutions of Kuramoto-Sivashinsky Equation
I.J. Education and Management Engineering 01, 6, 61-66 Published Online July 01 in MECS (http://www.mecs-press.ne DOI: 10.5815/ijeme.01.06.11 Available online at http://www.mecs-press.net/ijeme Exact Solutions
More informationComputational Fluid Dynamics-1(CFDI)
بسمه تعالی درس دینامیک سیالات محاسباتی 1 دوره کارشناسی ارشد دانشکده مهندسی مکانیک دانشگاه صنعتی خواجه نصیر الدین طوسی Computational Fluid Dynamics-1(CFDI) Course outlines: Part I A brief introduction to
More information