Question

Need help solving it using matlab with for loop

Objective: Solve the wave equation numerically using finite difference methods with both dirichlet and neumann conditions. Consider the wave equation for a string with fixed ends, L=1. lu lu Initial conditions. To make the string behave like a plucked guitar string, use a triangual initial condition. For x less than or equal to 0.5, set u(x, t 0) = 2HX and for x greater than 0.5, use u(x, t = 0) = 2H (1-x) , where H =1 is the initial height of the string. Boundary conditions 11(x 0.1) = 0 11(x = L, t) = 0 The string is L=1 with spacing of Ax = 0.2 from time 0 to 1 seconds with increments of 0.2s. Solve using the finite difference method and name your solution "U".

Answer 1

close all

clear
clc

m = 0;
x = 0:0.2:1;
t = 0:0.01:0.5;

sol = pdepe(m,@pdefun,@icfun,@bcfun,x,t);
u = sol(:,:,1);

surf(x,t,u)
title('Solution of PDE')
xlabel('x'), ylabel('t'), zlabel('u')

% Error
[X,T] = meshgrid(x,t);
ua = exp(-(pi^2) T) . sin(pi*X);
error = abs(u - ua);
En = norm(error, Inf);
disp('En ='), disp(En)

function [c,f,s] = pdefun(x,t,u,DuDx)
c = 1;
f = DuDx;
s = 0;
end

function u0 = icfun(x)
u0 = sin(pi*x);
end

function [pl,ql,pr,qr] = bcfun(xl,ul,xr,ur,t)
pl = ul;
ql = 0;
pr = 0;
qr = 1;
end

Output:

En =
1.434450078921155

Plot:

Solution of PDE 0.8 0.6 0.4 0.2 0.6 0.4 0.8 0.6 0.2 0.4 0.2