MATH 175: NUMERICAL ANALYSIS IICHAPTER 3: Differential Equations

Lecturer: Jomar Fajardo Rabajante2nd Sem AY 2012-2013

IMSP, UPLB

HAPPY NEW YEAR!!!

NUMERICAL SOLUTIONS TO DIFFERENTIAL EQUATIONS

• Main Topic: Ordinary Differential Equations (Initial Value Problem)–Modeling–Numerical Solutions• Softwares: MS Excel, SciLab, Berkeley Madonna

• Optional Topics–Boundary Value Problem ODE–Partial Differential Equations– Stochastic Differential Equations

WHAT IS DE?

Differential equations – the major interface of mathematics with the real world – are the main tool with which scientists make mathematical models of real systems. That is, differential equations have a central role in connecting the power of mathematics with the description of real phenomena.

-John Hubbard, Cornell University

What is DE?Definition:A differential equation is an equation that contains

derivatives of one or more dependent variables with respect to one or more independent variables.

This is an extension of what you have learned in Math 30 series…

MATH 151: Ordinary Differential EquationsMath 152: Partial Differential EquationsIn Math 175: We will discuss how to solve problems that

are not solvable in Math 151 and Math 152.

Examples:

cx

y

xdx

dy

2 :Solution

:ODE .1

2

eby variabl separation use :Solution

:ODE .2 kydx

dy

Examples:

zyxz

xy

yxx

3'

'

23'

:ODEs of System .3

ODE and PDE

• An ordinary differential equation contains only ordinary derivatives. Example:

• A partial differential equation contains partial derivatives. Example:

210ydt

dy

x

yy

t

y

Some concepts

• Differential equations are for modeling continuous-time systems

• Those discrete-time systems are modeled using difference equations (iterative equations, similar to our fixed-point iterations). Example:

510 1 nn yy

Some concepts

• Differential equations (DE) and difference equations are applied to model DYNAMICAL SYSTEMS – systems that change over time.

• The order of a DE refers to the highest-order derivative that appears in the equation. Example:

tydt

dy

dt

yd 65

5

3

Some concepts

• Autonomous DEs – the variable t does not appear in the equation. Example:

• Non-autonomous DEs – the variable t appears in the equation. Example:

4kydt

dy

4 tkydt

dy

Some concepts• Linear DE does not have transcendental and

nonlinear dependent variables.

Example of Linear DE:

Example of Non-linear DE:

)sin(2'5"

22'5"

22'5"

tyyy

tyyy

yyy

)sin(2'5"

22'5)"(

22'5"2

2

yyyy

yyyy

yyy

Very few physical systems are purely linear. But many nonlinear physical systems can be approximated by linear systems.

ORDINARY DIFFERENTIAL EQUATIONS

Let’s focus on

SOLVING ODEsExample: The solution to the ODE dy/dt, also written

as y’, is y(t) or y(t,y).

• Obtaining an explicit formula: y(t) • Obtaining an implicit formula: y(t,y)• Obtaining a power series representation for y(t)• Numerically approximating the solution y(t) or y(t,y)• Sketching the geometric representation of y(t)

Geometric RepresentationFIRST-ORDER ODE

• Direction Field/Slope FieldConsider y’ = t – y Compute y’ (slopes) for every possible

equally-spaced points (t,y).

See http://www.math.psu.edu/cao/DFD/Dir.html

Qualitative AnalysisFIRST-ORDER ODE

• Equilibria – solution that does not change over time i.e. dy/dt=0.

• Stability of Equilibrium solution– Stable if solutions near it tend toward it as t∞– Unstable if solutions near it tend away from it as t∞– Saddle or semi-stable if stable on one side and unstable on the other.Note: A good practice when studying DEs is to investigate first its qualitative properties. Many physical systems modeled by DEs “spend most of their time at or near equilibrium states.”

y’=y2–4

INITIAL VALUE PROBLEM

The combination of a differential equation and an initial condition is called an initial-value problem.

Example:

(You do not need +c when dealing with IVP)

1)0(

3' 2

y

ty