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Example 4.8
Find the first-order iterative scheme to solve the first-order differential equa-
tion given by:

dy
at() + bt y() = u t() (4.31)
dt
with the initial condition y(t ) specified.
1
Solution: Substituting Eq. (4.16) for the numerical differentiator in the dif-
ferential equation, we deduce the following first-order difference equation
for y(k):

−1 
yk() = 2 ak() + bk()  2 ak y k()( − 1 ) + ak D k() ( − 1 ) + u k()   (4.32)
 

t ∆
t ∆

 
to which we should add, in the numerical subroutine, the expression for the
first-order differentiator D(k) as given by Eq. (4.16). The initial condition for
the function at the origin of time, specify the first elements of the y and D
arrays:
y()1 = y t( = t )
1
D()1 = ( /1 a())[1 u()1 − b()1 y()]1

Application
To illustrate the use of the above algorithm, let us solve, over the interval 0 ≤
t ≤ 6, for the potential across the capacitor in an RC circuit with an ac source;
that is,

dy
a + y = sin(2π t) (4.33)
dt
where a = RC and y(t = 0) = 0.

Solution: Edit and execute the following script M-file, for a = 1/(2π):

tin=0;
tfin=6;
t=linspace(tin,tfin,3000);
N=length(t);
y=zeros(1,N);

© 2001 by CRC Press LLC