Lecture 1.1: An Introduction to Ordinary Differential Equations - PowerPoint PPT Presentation

Dec 09, 2022 •283 likes •371 views

Lecture 1.1: An Introduction to Ordinary Differential Equations Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 2080, Differential Equations M. Macauley (Clemson) Lecture 1.1:

Lecture 1.1: An Introduction to Ordinary Differential Equations Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 2080, Differential Equations M. Macauley (Clemson) Lecture 1.1: Intro to ODEs Math 2080, ODEs 1 / 7
Introduction to ODEs What is a Differential Equation? It is an equation involving a function and its derivatives. Example (finance) The rate of growth of an investment is proportional to the amount of the investment. Equation: P ′ ( t ) = rP ( t ). (Often, we just write P ′ = rP .) For example, consider a mutual fund that grows at a 10% rate. Note: We assume that interest is compounded continuously , i.e., at any point in 1 time, the rate of change is 10 P ( t ). M. Macauley (Clemson) Lecture 1.1: Intro to ODEs Math 2080, ODEs 2 / 7
Modeling with ODEs Big idea If the rate of change of a function f is proportional to the function itself, then f ′ = rf . Example (biology) A colony of rabbits grows at a rate proportional to its size. M. Macauley (Clemson) Lecture 1.1: Intro to ODEs Math 2080, ODEs 3 / 7
Modeling with to ODEs Example (chemistry) A radioactive substance decays at a rate proportional to its size. Sample question : If there are 30 grams initially, and 20 grams after one year, what is the half-life? M. Macauley (Clemson) Lecture 1.1: Intro to ODEs Math 2080, ODEs 4 / 7
Modeling with ODEs Example (physics) The temperature of a cup of coffee cools at a rate proportional to the difference: “(temp. of coffee) – (ambient temp.)”. M. Macauley (Clemson) Lecture 1.1: Intro to ODEs Math 2080, ODEs 5 / 7
Exponential decay What else exhibits this “decay to a limiting value” behavior in nature (approximately)? Earth’s population. Velocity of a falling object with air resistance. M. Macauley (Clemson) Lecture 1.1: Intro to ODEs Math 2080, ODEs 6 / 7
Common theme: a family of solutions Some questions from calculus: What is the antiderivative of f ( t ) = 2 t ? The velocity of a car is x ′ ( t ) = 2 t . How far from home is it after t hours? An investment takes 5 years to double. How much is it worth after 8 years? M. Macauley (Clemson) Lecture 1.1: Intro to ODEs Math 2080, ODEs 7 / 7

Recommend

8. Ordinary Differential Equations Indispensable for many technical applications! 8. Ordinary

Introduction Approximation of IVP with FD Consistency and Convergence Multistep Methods Stiff Problems Applications 8. Ordinary Differential Equations Indispensable for many technical applications! 8. Ordinary Differential Equations

575 views • 30 slides

AM 205: lecture 13 Last time: Numerical solution of ordinary differential equations Today:

AM 205: lecture 13 Last time: Numerical solution of ordinary differential equations Today: Additional ODE methods, boundary value problems Thursdays lecture will be given by Thomas Fai Assignment 3 will be posted tonight

373 views • 19 slides

Parallel Numerical Algorithms Chapter 7 Differential Equations Section 7.1 Ordinary

Ordinary Differential Equations Parallel Numerical Algorithms Chapter 7 Differential Equations Section 7.1 Ordinary Differential Equations Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign CS

313 views • 12 slides

Partial Differential Equations Lecture Notes for Math 404 Rouben Rostamian Department of

Partial Differential Equations Lecture Notes for Math 404 Rouben Rostamian Department of Mathematics and Statistics UMBC Fall 2020 Introduction The heat Introduction to PDEs equation Instances of use You are already familiar with Ordinary

1.15k views • 104 slides

43 7 System Analysis 7.1 Introduction Models for chemical reactions: ordinary differential

43 7 System Analysis 7.1 Introduction Models for chemical reactions: ordinary differential equations a System parameters (reaction velocities) remain to be determined. Observe concentrations Y ( j ) for times t i . Statistics: Nonlinear

121 views • 8 slides

Notes about ordinary differential equations. Master BME, Math Level 2 October 10, 2019 1/33

Notes about ordinary differential equations. Master BME, Math Level 2 October 10, 2019 1/33 Differential equations and systems of differential equations Basic notions and examples Ordinary Differential Equation (ODE) : equation involving an

672 views • 33 slides

Numerical Methods for Ordinary Differential Equations (ODE) Introduction In this course, we focus

Numerical Methods for Ordinary Differential Equations (ODE) Introduction In this course, we focus on the following general initial-value problem (IVP) for a first order ODE: dx ( t ) x = f ( t, x ) = f ( t, x ( t )) or dt x ( a ) =

251 views • 4 slides

Modeling Physics with Differential-Algebraic Equations Lecture 1 General Introduction to

Modeling Physics with Differential-Algebraic Equations Lecture 1 General Introduction to Differential Equations COMASIC (M2) December 4th, 2019 Khalil Ghorbal k halil.ghorbal@inria.fr K. Ghorbal (INRIA) 1 COMASIC M2 1 / 21 Ordinary

600 views • 38 slides

JUST THE MATHS SLIDES NUMBER 15.5 ORDINARY DIFFERENTIAL EQUATIONS 5 (Second order

JUST THE MATHS SLIDES NUMBER 15.5 ORDINARY DIFFERENTIAL EQUATIONS 5 (Second order equations (B)) by A.J.Hobson 15.5.1 Non-homogeneous differential equations 15.5.2 Determination of simple particular integrals UNIT 15.5 - ORDINARY

97 views • 8 slides

AM 205: lecture 12 Last time: Numerical differentiation, numerical solution of ordinary

AM 205: lecture 12 Last time: Numerical differentiation, numerical solution of ordinary differential equations Today: Convergence and stability of ODE numerical methods The lecture on Tuesday October 16 will be given by Dr. Xiaolin

921 views • 34 slides

JUST THE MATHS SLIDES NUMBER 15.10 ORDINARY DIFFERENTIAL EQUATIONS 10 (Simultaneous

JUST THE MATHS SLIDES NUMBER 15.10 ORDINARY DIFFERENTIAL EQUATIONS 10 (Simultaneous equations (C)) by A.J.Hobson 15.10.1 Matrix methods for non-homogeneous systems UNIT 15.10 - ORDINARY DIFFERENTIAL EQUATIONS 10 SIMULTANEOUS

328 views • 13 slides

Lecture 4: Numerical solution of ordinary differential equations Habib Ammari Department of

Lecture 4: Numerical solution of ordinary differential equations Habib Ammari Department of Mathematics, ETH Z urich Numerical methods for ODEs Habib Ammari Numerical solution of ODEs General explicit one-step method: Consistency;

1.48k views • 119 slides

Matrix Formula of Differential Resultant for First Order Generic Ordinary Differential Polynomials

Matrix Formula of Differential Resultant for First Order Generic Ordinary Differential Polynomials Zhi-Yong Zhang Academy of Mathematics and Systems Science Chinese Academy of Sciences Oct, 2012 Joint work with C.M. Yuan and X.S. Gao Zhi-Yong

635 views • 42 slides

JUST THE MATHS SLIDES NUMBER 15.1 ORDINARY DIFFERENTIAL EQUATIONS 1 (First order

JUST THE MATHS SLIDES NUMBER 15.1 ORDINARY DIFFERENTIAL EQUATIONS 1 (First order equations (A)) by A.J.Hobson 15.1.1 Introduction and definitions 15.1.2 Exact equations 15.1.3 The method of separation of the variables UNIT 15.1 -

470 views • 11 slides

Dynamic Process Models Moritz Diehl Overview Ordinary Differential Equations (ODE)

Dynamic Process Models Moritz Diehl Overview Ordinary Differential Equations (ODE) Boundary Conditions, Objective Differential-Algebraic Equations (DAE) Multi Stage Processes Partial Differential Equations (PDE) and Method of

265 views • 22 slides

Population Modeling with Ordinary Differential Equations Michael J. Coleman November 6, 2006

Population Modeling with Ordinary Differential Equations Michael J. Coleman November 6, 2006 Abstract Population modeling is a common application of ordinary differential equations and can be studied even the linear case. We will investigate

168 views • 13 slides

Introduction to Ordinary Differential Equations Emily Weymier Department of Mathematics &

Introduction to Ordinary Differential Equations Emily Weymier Department of Mathematics & Statistics Stephen F. Austin State University, Nacogdoches, TX September 22, 2017 Emily Weymier (SFA) Perturbation Theory September 22, 2017 1 / 28

355 views • 31 slides

JUST THE MATHS SLIDES NUMBER 15.7 ORDINARY DIFFERENTIAL EQUATIONS 7 (Second order

JUST THE MATHS SLIDES NUMBER 15.7 ORDINARY DIFFERENTIAL EQUATIONS 7 (Second order equations (D)) by A.J.Hobson 15.7.1 Problematic cases of particular integrals UNIT 15.7 - ORDINARY DIFFERENTIAL EQUATIONS 7 SECOND ORDER EQUATIONS (D)

302 views • 9 slides

JUST THE MATHS SLIDES NUMBER 15.3 ORDINARY DIFFERENTIAL EQUATIONS 3 (First order

JUST THE MATHS SLIDES NUMBER 15.3 ORDINARY DIFFERENTIAL EQUATIONS 3 (First order equations (C)) by A.J.Hobson 15.3.1 Linear equations 15.3.2 Bernouillis equation UNIT 15.3 - ORDINARY DIFFERENTIAL EQUATIONS 3 FIRST ORDER

276 views • 11 slides

JUST THE MATHS SLIDES NUMBER 15.2 ORDINARY DIFFERENTIAL EQUATIONS 2 (First order

JUST THE MATHS SLIDES NUMBER 15.2 ORDINARY DIFFERENTIAL EQUATIONS 2 (First order equations (B)) by A.J.Hobson 15.2.1 Homogeneous equations 15.2.2 The standard method UNIT 15.2 - ORDINARY DIFFERENTIAL EQUATIONS 2 FIRST ORDER

264 views • 7 slides

Modelling Biochemical Reaction Networks Lecture 7: Numerical integration of ordinary

Modelling Biochemical Reaction Networks Lecture 7: Numerical integration of ordinary differential equations Marc R. Roussel Department of Chemistry and Biochemistry Recommended reading Fall, Marland, Wagner and Tyson, section 1.4.1

591 views • 12 slides

JUST THE MATHS SLIDES NUMBER 15.4 ORDINARY DIFFERENTIAL EQUATIONS 4 (Second order

JUST THE MATHS SLIDES NUMBER 15.4 ORDINARY DIFFERENTIAL EQUATIONS 4 (Second order equations (A)) by A.J.Hobson 15.4.1 Introduction 15.4.2 Second order homogeneous equations 15.4.3 Special cases of the auxiliary equation UNIT 15.4 -

322 views • 13 slides

Another look at estimating parameters in systems of ordinary differential equations via

Another look at estimating parameters in systems of ordinary differential equations via regularization c Ivan Vuja ci Seyed Mahdi Mahmoudi , Ernst Wit Department of Mathematics, Vrije Universiteit Amsterdam, The

757 views • 47 slides

JUST THE MATHS SLIDES NUMBER 15.8 ORDINARY DIFFERENTIAL EQUATIONS 8 (Simultaneous

JUST THE MATHS SLIDES NUMBER 15.8 ORDINARY DIFFERENTIAL EQUATIONS 8 (Simultaneous equations (A)) by A.J.Hobson 15.7.1 The substitution method UNIT 15.8 - ORDINARY DIFFERENTIAL EQUATIONS 8 SIMULTANEOUS EQUATIONS (A) 15.8.1 THE

388 views • 6 slides