Question 12.3.1: Approximate the solution to the hyperbolic problem ∂²u/∂t² (...
Approximate the solution to the hyperbolic problem
\frac{\partial^{2} u}{\partial t^{2}}(x, t)-4 \frac{\partial^{2} u}{\partial x^{2}}(x, t)=0, \quad 0<x<1, \quad 0<t,
with boundary conditions
u(0, t) = u(1, t) = 0, for 0 < t,
and initial conditions
u(x, 0)=\sin (\pi x), \quad 0 \leq x \leq 1, \quad \text { and } \quad \frac{\partial u}{\partial t}(x, 0)=0, \quad 0 \leq x \leq 1 ,
using h = 0.1 and k = 0.05. Compare the results with the exact solution
u(x, t) = sin πx cos 2πt.
Choosing h = 0.1 and k = 0.05 gives λ = 1, m = 10, and N = 20. We will choose a maximum time T = 1 and apply the Finite-Difference Algorithm 12.4. This produces the approximations w_{i, N} \text { to } u(0.1 i, 1) for i = 0, 1, … , 10. These results are shown in Table 12.6 and are correct to the places given.
Table 12.6
\begin{array}{cc}\hline x_{i} & w_{i, 20} \\\hline 0.0 & 0.0000000000 \\0.1 & 0.3090169944 \\0.2 & 0.5877852523 \\0.3 & 0.8090169944 \\0.4 & 0.9510565163 \\0.5 & 1.0000000000 \\0.6 & 0.9510565163 \\0.7 & 0.8090169944 \\0.8 & 0.5877852523 \\0.9 & 0.3090169944 \\1.0 & 0.0000000000 \\\hline\end{array}