Question

Problem 1 (Section 6.3) Starting with the finite difference expressions for the partial derivatives, re-derive the forward Euler method for the heat equation with an extra nonlinear term: u(0,t)- u(1t)-0 Then, find the solution over three time steps (i.e. find the twelve vawith 3 decimal digits of precision, assuming k = 1, γ=2, M = 0.01, L = 1 and N=5, with initial condition u a table to show your results. It is strongly recommended that you write a short program to do this, although you can do this by hand, of coursePros an n-0.6-(0, 1 2, 2.1 0). Make

Answer 1

% Code in Matlab

clc;
clear all;

L=1;
N=5;
k=1;
gamma=2;
del_t=0.01;
del_x=L/N;
alpha=k*del_t/del_x.^2;

for j=1:4
u(1,j)=0;
u(6,j)=0;
u(1,1)=0;
u(2,1)=1;
u(3,1)=2;
u(4,1)=2;
u(5,1)=1;
u(6,1)=0;
for i=2:N-1

u(i,j+1)=u(i,j)+alpha*(u(i+1,j)-2*u(i,j)+u(i-1,j))-gamma*u(i,j).^2;
end;
end;

%u(n=1,m=1)=u(2,2)=-1
%u(n=1,m=2)=u(2,3)=-4.0625
%u(n=1,m=3)=u(2,4)=-55.8047
%u(n=2,m=1)=u(3,2)=-6.25
%u(n=2,m=2)=u(3,3)=-83.0625
%u(n=2,m=3)=u(3,4)=-1.3862e4
%u(n=3,m=1)=u(4,2)=-6.25
%u(n=3,m=2)=u(4,3)=-82.8125
%u(n=3,m=3)=u(4,4)=-1.3778e4
%u(n=4,m=1)=u(5,2)=0
%u(n=4,m=2)=u(5,3)=0
%u(n=4,m=3)=u(5,4)=0