In an electric transmission line of length l that carries alternating current of high frequency (called a "lossless" line), the voltage V and current i are described by
∂2V / ∂x2 = LC ∂2V / ∂t2, 0< x < l, 0 < t; ∂2i / ∂x2 = LC ∂2i / ∂t2, 0< x < l, 0 < t;
where L is the inductance per unit length, and C is the capacitance per unit length. Suppose the line is 200 ft long and the constants C and L are given by
C = 0.1 farads/ft and L = 0.3 henries/ft.
Suppose the voltage and current also satisfy
V(0, t) = V(200, t) = 0, 0 < t;
V(x, 0) = 110 sin πx / 200, 0≤ x ≤ 200;
∂V / ∂t (x, 0) = 0, 0 ≤ x ≤ 200;
i(0, t) = i(200, t) = 0, 0 < t;
i(x, 0) = 5.5 cos πx / 200, 0≤ x ≤ 200;
and
∂i / ∂t (x, 0) = 0, 0 ≤ x ≤ 200.
Approximate the voltage and current at t = 0.2 and t = 0.5 using Algorithm 12.4 with h = 10 and k = 0.1.