How to Solve this “First-order wave equation” using Euler’s time-stepping series to obtain a graph similar to above. Solve this using Runga Kutta time-stepping series also.

% This program describes a moving 1-D wave

% using the finite difference method

clc

close all;

%%

% Initialization

Nx= 480; % x-Grid

dx= 3/480; % Step size

x(:,1)= (0:Nx-1)*dx;

f= 125.7; % Wave number

U(:,1)= -16*(x(:,1)-1/2).^2

% U(:,1) = exp(-16*(x(:,1)-1/2).^2);

U(:,1)= exp(U(:,1));

U(:,1)= U(:,1).*sin(f*x(:,1))

%U(:,1)= exp(-16*(x(:,1)-1/2).^2)%sin*(f*x(:,1))

%U(:,1)= U(:,1).*sin(f*x(:,1))

mpx= (Nx+1)/2; % Mid point of x-axis

% (Mid point of 1 to 3= 2 here)

T= 10; % Total no. of time step

f= 125.7; % Wave number

dt= 0.000625; % time-step

t(:,1)= (0:T-1)*dt;

v= 1; % wave velocity

c=v*(dt/dx); % CFL condition

s1= floor(T/f);

%%

% Initial condition

x=0:.001:3;

u=@(t)exp(-16*(x-(.75*t+.5)).^2).*sin(125.7*x);

plot(x,u(0))

figure

plot(x,u(2))