Solution to Differential Equations Using Discrete Green's Function and Duhamel's Methods

ODE

first order:

dx

dt

+ a x = S(t)

parameter a

1

max time

1

method

pulses: Green's function

strips: Duhamel's

segments

4

source

harmonic: cos(p t)

parameter p

π

2

0.2

0.4

0.6

0.8

1.0

t

0.2

0.4

0.6

0.8

1.0

S(t)

source

0.2

0.4

0.6

0.8

1.0

t

0.1

0.2

0.3

0.4

x(t)

response

This Demonstration shows how to find approximate solutions to linear ordinary differential equations using two methods:

1. the discrete Green's function method, in which the source is approximated as a sequence of pulses;

2. the discrete Duhamel's method, in which the source is approximated by a sequence of strips.

The complete solution is approximated by a superposition of solutions for each individual pulse or strip. As the limit of the number of segments tends to infinity, the pulse and strip methods approach the continuous Green's method and Duhamel's method, respectively.

In the graphs:

• solid, filled, red lines represent the exact source and response;

• thin, black, dashed lines represent the individual sources and responses to each pulse or strip;

• thick, black, dashed lines represent the total approximate source and response.