Example 7.10 1

Write the finite difference equation for the system

(7.0 1a)

(7.0 1b)

Examine the stability of the system.

The exact solution is

(7.0 2)

We note that y is a decreasing function of x.

• Using Equation (7.9 22b), the finite difference equation is

(7.0 3)

Instead of solving Equation (7.10 3) numerically, we seek an analytic solution. The solution is of the form

(7.0 4)

Substituting Equation (7.10 4) into Equation (7.10 3), we obtain

(7.0 5)

From Equation (7.10 5), we deduce that ? _1,2 are the roots of

(7.0 6)

The values of ? _{1, 2} are

(7.0 7)

Since h is assumed to be small, we expand in powers of h.

The roots are then approximately given by

(7.0 8a)

(7.0 8b)

The solution is given approximately by

(7.0 9)

Replacing j by x _j (x _j=jh), and noting that

(7.0 10a)

(7.0 10b)

The solution y _j can be written as

(7.0 11)

Applying boundary condition (7.10 1b), we find C ₂ to be zero and y _j is a decreasing function of x _j.

In an actual computation, there is round-off error and the value of C