STURM-LIOUVILLE DIFFERENCE EQUATIONS

(1.1.7) V(c„Ay„) =

anw2yn

, n = l , . . . , m - l .

1.2 Network Theory

1

Consider a cascade of LC circuits with inductance an, capacitance —, and

loop current un in the successive meshes (see Figure 1.2).

«-.

• • t,

J*

V^-d=r yCe-d=r v ^

V c

^ VcJ=

&

t • •

Figure 1.2: LONetwork

1 .

The current in the branch containing — is un — tin+i in the sense of un.

cn

Assume that un is of the form

ynctwtj

where yn is a complex constant, and, for

the nonhomogeneous problem, assume that a generator in the first mesh supplies

a voltage Eeiwt. Then, by KirchofTs rule, it follows that

(1.2.1) E + c-iyo(iw)~l + aoyoiw + c0(y0 - yi)(ttn)"1 = 0 ,

and, for n = 1,..., m — 1,

(1.2.2) cn-x(yn - yn-i)(iw)~l + anyniw + cn(yn - yn+i)(iw)~l = 0.

Note that y_i = ym = 0, so that multiplication by iw in (1.2.1) and (1.2.2) gives

V(c0Ay0) =

aow2yo

- #