How can i code and solve this PDE on matlab?

Hi! I am looking to solve this PDE on matlab but I am unable to obtain any solution using pdepe function. Would appreciate any assistance!

[Torsten]

Please include your code.

[Torsten]

I don't see any similarity between your code and the equation you've written.

[Kevin Lee]

yes... I tried following the code on mathworks but I can't seems to be able to put my PDE into the code. May I know how should i start with?

Answer 1

[Torsten]

c = 1;
f = alpha2*sqrt(u(1))/(1-u(1))^1.5*DuDx;
s = alpha1*u(1)*(2-u(1))/(1-u(1))^2*DuDx;

u0=? 
(You didn't provide a profile at t=0 for epsilon)

pl=?
ql=?
pr=?
qr=?
(You didn't provide boundary conditions for epsilon)

[Kevin Lee]

The boundary condition is when z = z0, epsilon = 0.8 and when z = z1, epsilon = 0.2.

When t=0, epsilon(z,0)=0.261.

[Torsten]

So,

z0 = 0;
z1 = 1;
u0=0.8*(z==z0)+0.261*(z>z0 & z<z1)+0.2*(z==z1)

and

pl = ur-0.8;
ql = 0;
pr = ur-0.2;
qr = 0;

Does it work ?

[Kevin Lee]

matlab returned the following error:

[Torsten]

See corrected code. And you neither gave values to alpha1 nor to alpha2.

[Torsten]

You didn't define z0 and z1. Further, pdex1ic is a function of x, not z.

[Kevin Lee]

It returned the following error:

>> pdex11

Undefined function or variable 'alpha2'.

Error in pdex11>pdex1pde (line 36)

f = alpha2*sqrt(u(1))/(1-u(1))^1.5*DuDx;

Error in pdepe (line 246)

[c,f,s] = feval(pde,xi(1),t(1),U,Ux,varargin{:});

Error in pdex11 (line 17)

sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t);

Am i defining the terms wrongly?

[Torsten]

The variables must be accessible where you use them. So the easiest way is to define them in the respective functions. Or include the global statement from above in the function, too.

[Kevin Lee]

After i define them in the respective function, matlab returned the following error:

>> pdex11 Error using pdepe (line 293)

Spatial discretization has failed. Discretization supports only parabolic and elliptic equations, with flux term involving spatial derivative.

Error in pdex11 (line 7)

sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t);

[Torsten]

Please include your final code as plain text, not as a graphic file.

[Kevin Lee]

how do i do so?

[Torsten]

By copy-and-paste from the text file in which you develop your code.

[Kevin Lee]

I written the code on matlab script directly though

[Torsten]

Then save it to a text file.

[Kevin Lee]

I have saved the file to a text file but still getting the error on matlab.

>> pdex11 Error using pdepe (line 293)

Spatial discretization has failed. Discretization supports only parabolic and elliptic equations, with flux term involving spatial derivative.

Error in pdex11 (line 7)

sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t);

[Torsten]

I meant that you should use the text from the saved text file to include it here.

[Kevin Lee]

ah i see! the site say i have reached the limit of 10 uploads. so i have pasted it here below:

function pdex11

m = 0;

x = linspace(0,1,200);

t = linspace(0,600,5);

sol = pdepe(m,@pdex1pde,@pdex1ic,@pdex1bc,x,t);

% Extract the first solution component as u.

u = sol(:,:,1);

% A surface plot is often a good way to study a solution.

surf(x,t,u) 

title('Numerical solution computed with 20 mesh points.')

xlabel('Distance x')

ylabel('Time t')

% --------------------------------------------------------------
function [c,f,s] = pdex1pde(x,t,u,DuDx)

r = 0.01;        %radius

V = 4/3*pi*r^3;  %volume of sphere

rho = 1000;      %density

g = 9.81;        %gravity

l = 0.816*r;     %length

vis = 1.808*10^-3; %viscosity

ten = 4.189*10^-2; %surface tension

alpha1 = (V*g*rho)/(1999*l*vis);

alpha2 = (ten*sqrt(0.161*V))/(282.7*sqrt(10*l)*vis);

c = 1;

f = alpha2*sqrt(u(1))/(1-u(1))^1.5*DuDx;

s = alpha1*u(1)*(2-u(1))/(1-u(1))^2*DuDx;

% --------------------------------------------------------------

function u0 = pdex1ic(x)

x0 = 0;

x1 = 1;

u0=0.8*(x==x0)+0.261*(x>x0 & x<x1)+0.2*(x==x1);

% --------------------------------------------------------------
function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t)

pl = ur-0.8;

ql = 0;

pr = ur-0.2;

qr = 0;

[Torsten]

Two mistakes I can spot in your code:

1. Define r *before* you calculate V.
2. Use pl = ul-0.8 instead of pl = ur-0.8.

[Kevin Lee]

noted on point 2. However, i have already defined r in the line above V.

It returned the following error.

>> pdex11 Warning: Failure at t=1.883061e+00. Unable to meet integration tolerances without reducing the step size below the smallest value allowed (3.552714e-15) at time t. > In ode15s (line 668) In pdepe (line 289) In pdex11 (line 7) Warning: Time integration has failed. Solution is available at requested time points up to t=0.000000e+00. > In pdepe (line 303) In pdex11 (line 7) Error using surf (line 82) Z must be a matrix, not a scalar or vector.

Error in pdex11 (line 12) surf(x,t,u)

[Torsten]

The code is technically ok now.

Since I don't know about the background of your problem, I can't help you any longer.