Plot a non homogeneous DPE -- Sum two plots in Matlab?

[Phys pilot]

Hello,

I want to plot this PDE which is non homogeneous:

ut=kuxx+cut=kuxx+c
u(x,0)=c0(1−cosπx)u(x,0)=c0(1−cosπx)
u(0,t)=0u(1,t)=2c0u(0,t)=0u(1,t)=2c0​

I have a code that can solve this problem and plot it with those boundary and initial conditions but not with the non homogeneous term.

function [u,er]=ftcs(t0,tf,nt,a,b,nx,ci,cca,ccb,dif,cada,sol)
% explicit method

%u solution
%er vector error
%t0 y tf time limits
%nt
%nx
%a b x value
%ci initial condition
%cca y ccb boundary conditions
%dif diffusion coefficient
%cada time lapse
%sol solution if it's known

x=linspace(a,b,nx);  x=x';  dx=x(2)-x(1); %h
t=linspace(t0,tf,nt); t=t';  dt=t(2)-t(1); %k
r=dt/dx^2*dif;
if r>.5
    clc
    disp('stability not satisfied')
    pause
end
di=1-2*r;
cca=inline(cca,'t');
ccb=inline(ccb,'t');
A=diag(di*ones(nx-2,1))+diag(r*ones(nx-3,1),1)+diag(r*ones(nx-3,1),-1);
A=[[r;zeros(nx-3,1)] A [zeros(nx-3,1);r]];
ci=vectorize(inline(ci,'x'));
u=ci(x);      % vectorwith a solution for t=0, initial condition
gra=plot(x,u);
axis([a b min(u) max(u)])
pause
z=u;
u(1)=cca(t(1));
u(end)=ccb(t(1));
for i=1:nt-1;
    u=A*u;
    u=[cca(t(i+1)); u ; ccb(t(i+1))];
    if mod(i,cada)==0
        set(gra,'ydata',u');
        pause(.1)
        z(:,end+1)=u;
    end
end
if nargin==12 & nargout==2
    sol=vectorize(inline(sol,'x','t'));
    er=u-sol(x,tf);
end

pause
figure
surfc(z);shading interp; colormap(hot);set(gca,'ydir','reverse');
rotate3d

But I don't know how to add that term, any ideas? I don't really understand the code very well. I just want to plot it. would it be very difficult?

The only idea I have is to divide the problem in two equations which sum would be the solution of the problem:

u(x,t)=v(x,t)+U(x)u(x,t)=v(x,t)+U(x)

kU(x)′′+c=0kU(x)″+c=0
U(x)=−cx22k+(2c0+c2k)xU(x)=−cx22k+(2c0+c2k)x​

and this other one

vt=kvxxvt=kvxx​

Which I can solve and plot with my code. So, is there any way to plot the U and sum it to the plot of the second one over the time? thanks

 

[mmwave]

Thank you for your question. It is indeed possible to include the non-homogeneous term in your code and plot the solution. Here are some steps you can follow:

1. Add the non-homogeneous term to your PDE equation in the code. It should look something like this:

ut = kuxx + cut + c

2. Modify the initial condition to include the non-homogeneous term. This can be done by adding the term to the existing initial condition function, like this:

ci = 'c0*(1-cos(pi*x)) + c'

3. Define the non-homogeneous boundary conditions in the code. This can be done by adding the term to the existing boundary condition functions, like this:

cca = 'c + 2*c0'
ccb = '2*c0'

4. Modify the code to incorporate the non-homogeneous term. This might involve changing the matrices and vectors used in the code, as well as the boundary conditions and initial conditions.

5. Run the code and plot the solution. You should now be able to see the solution with the non-homogeneous term included.

I hope this helps you plot the solution to your PDE. If you have any further questions, please let me know.