Question

what is the fomula of the numerical solution of the heat eqation

Ut = a² Uxx ? (not the analytical solution the numerical solution)

Give example please

Answer 1

Simplest numerical formula for 1D heat equation is:

U_i^{new =} U_i^old + alpha*(U_i+1^old - 2U_i^old + U_i-1^old)

where alpha=a*dt/dx²

------ Here is a MATLAB code to generate the solution -------

% Solution of the Heat Equation Using a Forward Difference Scheme
% Initialize Data
% Length of Rod, Time Interval
% Number of Points in Space, Number of Time Steps

L=1;
T=0.1;
a=1;
N=10;
M=50;
dx=L/N;
dt=T/M;

alpha=a*dt/dx^2;

% Position

for i=1:N+1
    x(i)=(i-1)*dx;
end

% Initial Condition

for i=1:N+1
    u0(i)=sin(pi*x(i));
end

% Partial Difference Equation (Numerical Scheme)

for j=1:M
    for i=2:N
        u1(i)=u0(i)+alpha*(u0(i+1)-2*u0(i)+u0(i-1));
    end
    u1(1)=0;
    u1(N+1)=0;
    u0=u1;
end

% Plot solution

plot(x, u1);