Question 16.18: Solve the wave equation ∂²u/∂t² = 4∂²u/∂x² with initial cond...

It is given that step size for t is 0.1 (k = 0.1) and step size for x is 0.25 (h = 0.25). So, we have x = 0, 0.25, 0.5, 0.75, 1 and t = 0, 0.1, 0.2, 0.3.

Using the initial condition

u(x, 0)= \begin{cases}0.1(x) & 0 \leq x \leq 1 / 2 \\ 0.1(1-x) & 1 / 2 \leq x \leq 1\end{cases}

and boundary conditions

u(0, t) = u(1, t) = 0            t ≥ 0,

we have following table for values of u_{i, j}=u\left(x_i, t_j\right)

The explicit scheme (16.88) for the solution of wave equation is as follows

u_{i, j+1}=r\left(u_{i+1, j}+u_{i-1, j}\right)+2(1-r) u_{i, j}-u_{i, j-1}            (16.88)

Using r=\frac{c k^2}{h^2}=4 \frac{(0.1)^2}{(0.25)^2}=0.64 , we have

u_{i, j+1}=0.64\left(u_{i+1, j}+u_{i-1, j}\right)+0.72 u_{i, j}-u_{i, j-1}         (16.91)

Equation (16.91) can be used to compute various nodal values for different j.

j = 1.

At time t = 0, the initial condition is \left.\frac{\partial u}{\partial t}\right|_{t=0}=0 . The central difference formula provides the following equation

\left.\frac{\partial u}{\partial t}\right|_{t=0}=\left.\frac{u_{i, j+1}-u_{i, j-1}}{2 k}\right|_{j=0}=\left.\frac{u_{i, 1}-u_{i,-1}}{2 k}\right|_{j=0}=0

\Rightarrow u_{i, 1}=u_{i,-1}          (16.92)

The scheme (16.91) for j = 0 is given by

u_{i, 1}=0.64\left(u_{i+1,0}+u_{i-1,0}\right)+0.72 u_{i, 0}-u_{i,-1}

Using Eq. (16.92), we have

u_{i, 1}=0.32\left(u_{i+1,0}+u_{i-1,0}\right)+0.36 u_{i, 0}

For i = 1, 2, 3, we can easily obtain the following values

The following table shows these values in the second row.

j = 2,
The explicit scheme (16.91) for j = 1 gives following equation

u_{i, 2}=0.64\left(u_{i+1,1}+u_{i-1,1}\right)+0.72 u_{i, 1}-u_{i, 0}

For i = 1, 2, 3 , we can easily obtain the following values

j = 3,

For j = 2, the explicit scheme (16.91) becomes

u_{i, 3}=0.64\left(u_{i+1,2}+u_{i-1,2}\right)+0.72 u_{i, 2}-u_{i, 1}

For i = 1, 2, 3 , the following values are obtained