Solve wave equation  with
 with , using explicit method upto 4 time levels.

Let's solve the wave equation using the explicit finite difference method up to 4 time levels.

Given:
- \( k = h = 0.25 \)
- \( c = \frac{h}{k} = \frac{0.25}{0.25} = 1 \)
- \( Nx = 4 \) (number of spatial grid points)
- \( Nt = 4 \) (number of time steps)
- \( \Delta x = \frac{1}{Nx - 1} = \frac{1}{4 - 1} = \frac{1}{3} \)
- \( \Delta t = \frac{1}{Nt - 1} = \frac{1}{4 - 1} = \frac{1}{3} \)

Now, let's initialize the grid and apply the initial and boundary conditions:

1. Initialize the grid:
\[ \begin{align*} u &= \begin{bmatrix} u_{1,1} & u_{2,1} & u_{3,1} & u_{4,1} \\ u_{1,2} & u_{2,2} & u_{3,2} & u_{4,2} \\ u_{1,3} & u_{2,3} & u_{3,3} & u_{4,3} \\ u_{1,4} & u_{2,4} & u_{3,4} & u_{4,4} \end{bmatrix} \end{align*} \]

2. Apply initial and boundary conditions:
- \( u(x,0) = 0 \implies \) first row = \( [0, 0, 0, 0] \)
- \( u_t(x,0) = 0 \implies \) second row = \( [0, 0, 0, 0] \)
- \( u(0,t) = 0 \implies \) first column = \( [0, ?, ?, ?] \)
- \( u(1,t) = 100 \sin(\pi t) \implies \) last column = \( [100 \sin(0), ?, ?, ?] = [0, ?, ?, ?] \)

Now, let's calculate the missing values in the grid using the explicit finite difference scheme:

\[ u_{i,j+1} = 2u_{i,j} - u_{i,j-1} + \frac{c^2 \Delta t^2}{\Delta x^2} (u_{i+1,j} - 2u_{i,j} + u_{i-1,j}) _________ _________ _____ _____ _____ _________ ___ _______ ___ ___ ______.
_________ ___ __________ _________ _________ __________ __________ _______ _____.
_______ _________ ____ __________ ____.
_________ __________ ____ ____ _____ _________ _________ _____.
_________ _____ _______ _____ _______ _________ ______ _________ ___ _________ ________.
________ ___ ____ ___ ______ _______ _____ __________ _______.
____ ______ ________ __________ ______ _________.
________ ______ _______ _____ __________ ___ _______.
_____ _______ ____ ____ ____ ________ ___ _________ _________ ___ _____.
____ _________ ___ ________ _________ ______ _______ ________ ______ ________.
______ ___ ________ ______ ___ _____ ______ ____.
______ ____ _______ __________ __________ ________.
_________ ___ ____ ________ ___ ___ ___.
____ _______ __________ ___ __________ ________ __________ _____.
_________ _______ _______ ________ _____ ____ _____.
____ _____ _______ _______ _________ ___ ________ _________ _________ _________.
____ _______ ______ _______ _____ ___ _____ ___.
________ _______ ________ ________ ____ _____.
_____ __________ ________ _________ _________ ____ ____ ________ _____ __________.
__________ ____ _________ ___ ____ _________ _____ ________ ______ ___ _______ _________.
_________ ______ _________ ____ ___ __________.
______ _________ _____ _________ _____ _____ _____ _______ ___ __________.
______ ____ _________ ________ __________ _______ ____.
__________ _____ ____ ________ ________ __________ _________ _________ ____ _____ _______ _________.
_________ ___ _________ __________ _______ __________ ________ _____.
_________ ______ _______ _____ _______ _____ ______ ________ _______ _________.
___ _________ ____ _____ ______ ______.
________ _____ __________ ________ _________ ____ __________ ___ ________.
___ ______ _____ _______ ____ ________ ________.
_____ _____ __________ _______ __________.
___ ______ _____ ________ _____ _____.
___ ____ ________ _______ ______.
_______ ______ ___ ______ _________ ________.
_______ ___ ______ _________ _______ ____ _________ ________ ___.
__________ _________ ____ ______ ___ __________ ________ _______.
______ _________ ___ ________ __________ ________ _________ ________ _____.
__________ ____ __________ _________ __________ ______.
____ _______ ________ _______ ________ __________ ___.
____ __________ _________ ______ ______ _________ ____.
____ __________ _________ ______ __________ ___ _______ _________ _______ ____.
_________ ______ ______ ___ __________ __________ ____ ____.
___ ________ ______ ____ _________ _____ _______.
__________ ________ _________ __________ _____ ____ __________.
___ ____ ______ _________ _________ _____ ____ _____ ________.
___ _____ __________ _________ ________ _____ _______ ____ ______ ______ ______ _________.
_____ __________ ___ ___ _____ _______ _________.
__________ __________ ____ _______ _________ __________ ___ __________.
_________ _________ ___ __________ ___ _________ __________ ______.
_______ ____ ______ _______ __________ ________ ___ _________ _____.
____ ____ ___ _______ __________.
_______ ______ __________ _______ _____ ____ ___ _____.
_________ _____ _______ _____ _______ ______ ________.
_________ ______ _______ ______ ________ _________.
_____ _______ ________ ___ _____ _________ _______ ______ ___ ___ ___ _________.
________ _____ ___ _________ _________.
_______ ______ ____ ______ __________ __________ ____ _____ ___ ____ ____.
_____ _______ __________ ______ ___ ____ ________ ________ ____ ________ ______.
________ __________ ______ _________ __________ _____.
___ _____ _____ ________ _____.
________ ___ _____ ________ ____ ___ _______.
________ ____ _________ _____ _______.
___ ___ _________ ____ ___ ___ ______ __________.
____ _____ _______ __________ ________ __________ ________ _______ _______.
_______ _____ ______ ______ _____ ___.
____ _________ _________ _______ _____ ___ _________ _____ _________ ________ ___.
__________ _______ ______ _____ ________.
_______ ______ ________ _________ ____ ___ ___.
_________ ______ _______ ___ ________ _________ ______ _________.
______ _______ ___ ________ ______.
__________ ________ _______ _________ _____ ____ ____ __________ ___ _____.
__________ _________ ___ _______ ________.
_______ ______ __________ _________ ____ __________ _________ _____ __________ ______ ________.
______ _________ _______ _________ ____ _________ ________ ____ _________ ____ _______.
_________ __________ ____ _________ ___ ________ _____ __________ ____ ___ _______ ___.
____ _______ ______ __________ ____ _____ _____ ________.
__________ ________ ______ _______ _________ ________ ___ ____.
_____ _______ __________ ____ ____ _________.
_____ ___ __________ ________ _________ ____ __________ ____.
_______ ______ _________ _______ ______ _____ ________ __________.
__________ ______ _______ _________ ___ ____ __________.
________ _____ ____ ________ ________ ______ __________ _________ ___ ____.
______ _____ _____ _________ _______ _____.
_______ _________ __________ _______ ______ ______ ________ ___ ________ _____ _____ _____.
__________ _________ _______ ________ _________ ______ _______ _____.
_____ ________ __________ ___ __________ __________ ______ ___ ______.
______ _________ __________ _______ ____ ___ ___.
___ ___ _______ ____ ____ ___ ______ ______ _____.
_________ _________ _________ ______ ________ ____ _________ _______.
_______ _______ _________ ___ _____ _________ __________ ______.
___ _________ ____ ______ __________ ___ ________ _______.
______ _____ ___ ___ ____ ____ __________ ________.
_________ _________ ________ __________ _____ _______ ___ _____ __________ ___ __________ __________.
___ _____ ______ __________ _______.
___ _______ ________ _______ ________.
____ ______ _______ ___ __________ _____ ___ ______ _______ _______ ___ ____.
______ _____ _________ ______ ____.
___ ____ ______ _____ ________ ____.
___ __________ _____ __________ _______ ___ _________ ___.
____ _________ ____ ___ ____ ____ __________ ____ _________.
_____ ______ _____ ____ ____.
________ __________ __________ _______ ____ ___ ___ __________.
_________ ______ ___ _______ __________ _______ ____ ______ _______.
____ __________ ___ ___ ____ ________ _____.
__________ ________ ____ _____ ____ ___ ____.
_______ __________ ___ _________ __________ ___ ______ _____ ____.
__________ ______ ___ ___ ________ _________ ___ ____ __________ ____ _____.
______ __________ ________ ____ ________ ________.
_____ ____ ______ __________ ___ _____ ____ ____ ______ ____ ___ ______.
___ ___ _________ ______ ______ ______ _________ _____ ___ ____ ______ _________.
_____ _____ _______ __________ ___.
________ __________ ____ _________ ______ ______ __________ ____ ______ ______ ________ ______.
________ ________ _______ __________ ___ _______ _______ ____ ___ _________.
_____ ___ _______ ______ ___ ________.
____ _________ _____ __________ _____ _________ _____ ___.
________ _____ _________ _____ ____ _______ __________.
________ ________ _______ _______ _____ _________ __________ ___ _________ ________.
_________ ________ ______ _______ _______.
_________ ______ __________ __________ __________ _________ _________ _____ _____ ______ ______ ________.
________ ____ _________ ______ ___ ________ __________ ____.
___ ______ _____ _____ ___ ________ _______ __________ ___ ____.
_______ ______ ________ _______ ________ ____ ____ ______.
_____ ____ ___ _____ __________ _________ ____ ___ ___.
___ ___ _____ __________ ___ __________ _________ _____ ___ __________ ___.
_______ ______ _____ ___ _________.
_____ _____ ________ _________ ____ ___ ______ ______.
_________ ____ __________ _______ _______ ________ ______ ____ _____.
__________ _________ ________ _____ _________ __________ _______ ________ ________ ______.
______ ________ _________ _______ _____ ________ ________ ___ ________ _______ _______ __________.
____ ____ _____ ________ _______ _____ ______ _________ ______ _____.
_____ ___ _________ _________ ____ __________ ___ ______ ______ ___.
____ _________ _____ __________ _____.
_________ ____ ________ _____ _____ __________ __________ _______ _____.
______ ___ _____ ________ _________.
___ _________ ______ _________ ______ _________.
___ ____ _________ ____ __________ _____ ________.
________ ____ ___ ________ ________.
__________ __________ __________ ____ ______ _________ ________ ________ ________.
__________ __________ ________ ________ ______ ________ ___ _________ ____.
________ ___ _______ __________ _______ _______ ________ __________.
__________ ____ _______ _________ _____ ____ _____ ___ ________.
___ _________ ____ ________ _____ _________.
____ __________ _______ ___ __________ _________ ________ ______ _________ ________ __________ ___.
___ __________ _______ _____ ______ __________ ___.
___ ______ _____ _____ ____ ______ ______ ________ _________ _________ ___ _________.
__________ ________ _____ ____ _________ _______ ____ ____ ____ ______ _______ _________.
__________ ______ ______ ___ _______.
_______ _______ ______ _________ ________ _____ ________ ____ ________ ______ ___ ______.
____ ______ ________ _______ _____ _________.
____ _____ __________ __________ _________ __________ _____.
______ ___ ________ ____ ____ ___ ______.
________ __________ _____ _________ _____.
________ _____ ____ __________ _______ _________ _____.
_____ _______ __________ ______ ___ ________ ___ ______ ____ __________ ___ ________.
_______ _____ __________ ________ ________ _____ ___ _________ _________ ___ _____.
_______ ____ _______ _____ ___ ______ ____.
__________ __________ ____ __________ __________ ___ ________ _____ _____ _________ ___.
______ _________ ______ ___ ___ _____ _______.
__________ _______ ______ ________ ____.
_______ __________ ______ ________ __________ __________ ___ ______ ___ _____ ______ _______.
_____ ___ ________ ___ ________ __________ _________.
Get Full Answer on WhatsApp