Ordinary Differential Equations (ODEs). Objectives of Topic  Solve Ordinary Differential Equations (ODEs).  Appreciate the importance of numerical methods.

Slides:



Advertisements
Similar presentations
Ordinary Differential Equations
Advertisements

SE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture KFUPM Read , 26-2, 27-1 CISE301_Topic8L8&9 KFUPM.
Dr. Jie Zou PHY Chapter 9 Ordinary Differential Equations: Initial-Value Problems Lecture (I) 1 1 Besides the main textbook, also see Ref.: “Applied.
Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 Ordinary Differential Equations Equations which are.
ECE602 BME I Partial Differential Equations in Biomedical Engineering.
Part 6 Chapter 22 Boundary-Value Problems PowerPoints organized by Dr. Michael R. Gustafson II, Duke University All images copyright © The McGraw-Hill.
Mech300 Numerical Methods, Hong Kong University of Science and Technology. 1 Part Seven Ordinary Differential Equations.
Chapter 1 Introduction The solutions of engineering problems can be obtained using analytical methods or numerical methods. Analytical differentiation.
Ordinary Differential Equations S.-Y. Leu Sept. 21, 2005.
Chap 1 First-Order Differential Equations
Ordinary Differential Equations S.-Y. Leu Sept. 21,28, 2005.
“But, if this is true, and if a large stone moves with a speed of, say, eight while a smaller one moves with a speed of four, then when they are united,
Solve the differential equation. y'' - 10y' + 74y = 0
Solve the differential equation. y'' - 6y' + 45y = 0
Introduction to Differential Equations. Definition : A differential equation is an equation containing an unknown function and its derivatives. Examples:.
1 Chapter 9 Differential Equations: Classical Methods A differential equation (DE) may be defined as an equation involving one or more derivatives of an.
Differential Equations and Boundary Value Problems
CISE301_Topic8L1KFUPM1 CISE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture KFUPM Read , 26-2, 27-1.
MATH 685/ CSI 700/ OR 682 Lecture Notes Lecture 10. Ordinary differential equations. Initial value problems.
Fin500J Topic 7Fall 2010 Olin Business School 1 Fin500J Mathematical Foundations in Finance Topic 7: Numerical Methods for Solving Ordinary Differential.
PART 7 Ordinary Differential Equations ODEs
7.1 SOLVING SYSTEMS BY GRAPHING The students will be able to: Identify solutions of linear equations in two variables. Solve systems of linear equations.
Lecture 35 Numerical Analysis. Chapter 7 Ordinary Differential Equations.
EE3561_Unit 8Al-Dhaifallah14351 EE 3561 : Computational Methods Unit 8 Solution of Ordinary Differential Equations Lesson 3: Midpoint and Heun’s Predictor.
Fin500J Topic 6Fall 2010 Olin Business School 1 Fin500J: Mathematical Foundations in Finance Topic 6: Ordinary Differential Equations Philip H. Dybvig.
Ordinary Differential Equations
Computational Method in Chemical Engineering (TKK-2109)
Math 3120 Differential Equations with Boundary Value Problems
Differential Equations. Definition A differential equation is an equation involving derivatives of an unknown function and possibly the function itself.
Prepared by Mrs. Azduwin Binti Khasri
Scientific Computing Numerical Solution Of Ordinary Differential Equations - Euler’s Method.
Boundary-Value Problems Boundary-value problems are those where conditions are not known at a single point but rather are given at different values of.
Numerical Methods for Solving ODEs Euler Method. Ordinary Differential Equations  A differential equation is an equation in which includes derivatives.
The elements of higher mathematics Differential Equations
A Brief Introduction to Differential Equations Michael A. Karls.
MAT 1228 Series and Differential Equations Section 3.7 Nonlinear Equations
Ch 1.3: Classification of Differential Equations
Engineering Analysis – Computational Fluid Dynamics –
Differential Equations Also known as Engineering Analysis or ENGIANA.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 Chapter 27.
3.1 – Solve Linear Systems by Graphing A system of two linear equations in two variables x and y, also called a linear system, consists of two equations.
3.1 Solving Systems Using Tables and Graphs When you have two or more related unknowns, you may be able to represent their relationship with a system of.
1 Chapter 1 Introduction to Differential Equations 1.1 Introduction The mathematical formulation problems in engineering and science usually leads to equations.
Dr. Mujahed AlDhaifallah ( Term 342)
Differential Equations Linear Equations with Variable Coefficients.
Today’s class Ordinary Differential Equations Runge-Kutta Methods
Ch 1.1: Basic Mathematical Models; Direction Fields Differential equations are equations containing derivatives. The following are examples of physical.
9.1 Solving Differential Equations Mon Jan 04 Do Now Find the original function if F’(x) = 3x + 1 and f(0) = 2.
Section 1.1 Basic Definitions and Terminology. DIFFERENTIAL EQUATIONS Definition: A differential equation (DE) is an equation containing the derivatives.
Boyce/DiPrima 9 th ed, Ch1.3: Classification of Differential Equations Elementary Differential Equations and Boundary Value Problems, 9 th edition, by.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 Part 6 - Chapters 22 and 23.
OBJECTIVES Students will able to Students will able to 1. define differential equation 1. define differential equation 2. identify types, order & degree.
Introduction to Differential Equations
DIFFERENTIAL EQUATIONS
Introduction to Differential Equations
SLOPE FIELDS & EULER’S METHOD
SLOPE FIELDS & EULER’S METHOD
Basic Definitions and Terminology
Differential Equations
Business Mathematics MTH-367
525602:Advanced Numerical Methods for ME
SE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture KFUPM (Term 101) Section 04 Read , 26-2, 27-1 CISE301_Topic8L4&5.
Ch 1.3: Classification of Differential Equations
Ch 1.3: Classification of Differential Equations
SE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture KFUPM (Term 101) Section 04 Read , 26-2, 27-1 CISE301_Topic8L2.
Introduction to Ordinary Differential Equations
Chapter 1: Introduction to Differential Equations
Differential Equations
CISE301: Numerical Methods Topic 8 Ordinary Differential Equations (ODEs) Lecture KFUPM Read , 26-2, 27-1 CISE301_Topic8L1 KFUPM.
Chapter 1: First-Order Differential Equations
Presentation transcript:

Ordinary Differential Equations (ODEs)

Objectives of Topic  Solve Ordinary Differential Equations (ODEs).  Appreciate the importance of numerical methods in solving ODEs.  Assess the reliability of the different techniques.  Select the appropriate method for any particular problem.

Introduction to ODEs

Learning Objectives  Recall basic definitions of ODEs: Order Linearity Initial conditions Solution  Classify ODEs based on: Order, linearity, and conditions.  Classify the solution methods.

Partial Derivatives u is a function of more than one independent variable Ordinary Derivatives v is a function of one independent variable Derivatives

Partial Differential Equations involve one or more partial derivatives of unknown functions Ordinary Differential Equations involve one or more Ordinary derivatives of unknown functions Differential Equations Differential Equations

Ordinary Differential Equations Ordinary Differential Equations (ODEs) involve one or more ordinary derivatives of unknown functions with respect to one independent variable x(t): unknown function t: independent variable

Example of ODE: Model of Falling Parachutist The velocity of a falling parachutist is given by:

Definitions Ordinary differential equation

Definitions (Cont.) (Dependent variable) unknown function to be determined

Definitions (Cont.) (independent variable) the variable with respect to which other variables are differentiated

Order of a Differential Equation The order of an ordinary differential equation is the order of the highest order derivative. Second order ODE First order ODE Second order ODE

A solution to a differential equation is a function that satisfies the equation. Solution of a Differential Equation

Linear ODE An ODE is linear if The unknown function and its derivatives appear to power one No product of the unknown function and/or its derivatives Linear ODE Non-linear ODE

Nonlinear ODE An ODE is linear if The unknown function and its derivatives appear to power one No product of the unknown function and/or its derivatives

Solutions of Ordinary Differential Equations Is it unique?

Uniqueness of a Solution In order to uniquely specify a solution to an n th order differential equation we need n conditions. Second order ODE Two conditions are needed to uniquely specify the solution

Auxiliary Conditions Boundary Conditions  The conditions are not at one point of the independent variable Initial Conditions  All conditions are at one point of the independent variable Auxiliary Conditions

Boundary-Value and Initial value Problems Boundary-Value Problems  The auxiliary conditions are not at one point of the independent variable  More difficult to solve than initial value problems Initial-Value Problems  The auxiliary conditions are at one point of the independent variable same different

Classification of ODEs ODEs can be classified in different ways:  Order First order ODE Second order ODE N th order ODE  Linearity Linear ODE Nonlinear ODE  Auxiliary conditions Initial value problems Boundary value problems

Analytical Solutions  Analytical Solutions to ODEs are available for linear ODEs and special classes of nonlinear differential equations.

Numerical Solutions  Numerical methods are used to obtain a graph or a table of the unknown function.  Most of the Numerical methods used to solve ODEs are based directly (or indirectly) on the truncated Taylor series expansion.

Classification of the Methods Numerical Methods for Solving ODE Multiple-Step Methods Estimates of the solution at a particular step are based on information on more than one step Single-Step Methods Estimates of the solution at a particular step are entirely based on information on the previous step