MATLAB: Graphing the first PDE

graphing pde

Hello! I am new to Matlab and trying to graph my first PDE. I have no idea how it works, can any one help me with how to write the script for the following PDE

c(∂u∂x) + ∂u∂t = 0 with initial condition u(x, 0) = arctan(x). Plot the solution to this PDE for a few different values of c (small, large, positive, negative)

Thanks 🙂

Best Answer

how do find the correct boundary conditions when (initial condition) uo = atan(x) and the PDE equation is c(∂u∂x) + ∂u∂t = 0 ?

Related Solutions

MATLAB: PDEPE function

Yes it is possible in pdefun :)

Look at example:

function diffusion
m = 0;
x = linspace(0,1,20);
t = linspace(0,2,5);
sol = pdepe(m,@eqtn,@ic,@bc,x,t);
u = sol(:,:,1);
figure;
surf(x,t,u);    
xlabel('Distance x');
ylabel('Time t');
% --------------------------------------------------------------------------



function [c,f,s] = eqtn(x,t,u,DuDx)
c = 1;
f = Dfsn(u)*DuDx;
s = 0;
% --------------------------------------------------------------------------
function u0 = ic(x)
u0 = 1:length(x);
% --------------------------------------------------------------------------
function [pl,ql,pr,qr] = bc(xl,ul,xr,ur,t)
pl = ul;
ql = 2;
pr = ur;
qr = 2;
% --------------------------------------------------------------------------
function d = Dfsn(u)
d = sqrt(u+1);

Where Dfsn is your non linear diffusion coefficient function.

MATLAB: Solving second order PDE

The reason that pdepe imposes a boundary condition of the flux equal zero at the

center is that this is required for the problem to be mathematically well-posed.

Imposing a prescribed temperature at the center would require that the flux go to

infinity there.

An easy way to understand this is to solve the problem with the left end a small distance

from the center and with a fine mesh. I have attached a short script below that shows this.

function matlabAnswers_6_27_2020
r0=1e-6;
x = linspace(r0,1,1000);
tf=1;
t = linspace(0,tf,40);
pdeFunc = @(x,t,u,DuDx) heatpde(x,t,u,DuDx);
icFunc = @(x) heatic(x);
bcFunc = @(xl,ul,xr,ur,t) heatbcDirichlet(xl,ul,xr,ur,t);
m=1;
sol = pdepe(m, pdeFunc,icFunc,bcFunc,x,t);
figure; plot(t, sol(:,end)); grid on; title 'Temperature at outer surface'
figure; plot(t, sol(:,1)); grid on; title 'Temperature at center'
figure; plot(x, sol(end,:)); grid; title 'Temperature at final time'
end
function [c,f,s] = heatpde(x,t,u,DuDx)
c = 1;
f = DuDx;
s = 0;
end
function u0 = heatic(x)
u0 = 0;
end
function [pl,ql,pr,qr] = heatbcDirichlet(xl,ul,xr,ur,t)
pl = ul-1;
ql = 0;
pr = 0;
qr = 1;
end

Best Answer

Related Solutions

MATLAB: PDEPE function

MATLAB: Solving second order PDE

Related Question