solve second order non-linear partial differential equation with boundaries conditions

hello everyone!

I am struggling to find a way to solve this equation

=

with those boundaries condition

where a, b, pac, pev are known constants

hope someone can help me,

thank you!

[darova]

Its not partial equation. It's ordinary one. DId you try bvp4c?

Answer 1

[Shashank Gupta]

Hi Alessia,

There are many way to solve the above differential equation and some of them are well documented, refer to bvp4c function, it is really good way of solving boundary condition differential equation(@darova also recommend this function in the comments section). There are other ways to solve this, check out this example link.

I hope this references helps you.

Cheers.

[Alessia Nannetti]

thank you so much for your help!

i actually solved the problem with bvp4c function.

here there is my code

clear all

close all

clc

%stationary and rje uniform

global pac pev fi beta

rsco = 0.0024; %Reference SC resistance at h0 mmHg/(μl/min)

rje = 2.08; %mmHg/(μl/min)

rd = 0.694; %mmHg/(μl/min)

beta = rje/rsco;

pac = 15 %mmHg

pev = 9 %mmHg

n = 30; %number of CC

w = 300; %SC width μm

x0 = 1;

h0 = 1;

fi = rje/rd;

osc = (2*n*w*h0)/fi,

f = 1; %Hz frequency of ocular pulse

alpha = osc*rje*f;

pi = (pac-pev)/fi;

Hs = 1-pi*(fi/(1+fi));

options = bvpset('RelTol',1e-1,'Stats','on','NMax',10000) %needed this for tollerance errors

infinity = 1;

solinit = bvpinit(linspace(0,infinity,1000),[Hs^4 0]); %changed following equation (19)

sol = bvp4c(@fsode,@fsbc,solinit,options);

x = sol.x;

y = sol.y;

H=y(1,:).^(1/4);

dHdx=(1/4)*(y(1,:).^(-3/4).*y(2,:));

% plot of dH(x)/dx

figure(1)

plot(x,dHdx,x(end),dHdx(end),'o','linewidth',2);

grid on

title('dH/dx')

xlabel('x')

ylabel('dY^(1/4)/dx')

%plot of H(x)

figure(2)

plot(x,H,x(end),H(end),'o','linewidth',2);

grid on

title('H')

xlabel('x')

ylabel('Y^(1/4)')

Q = -(beta/alpha)*H.^3.*dHdx; %dimensionless flow rate

T = -(beta/alpha)*H.*dHdx; %dimensionless shear stress

%plot of Q(x) --> dimensionless flow rate

figure(3)

plot(x,Q,'linewidth',2)

grid on

title('dimensionless flow rate')

xlabel('x')

ylabel('Q(x)')

%plot of T(x)--> dimensionless shear stress

figure(4)

plot(x,T,'linewidth',2)

grid on

title('dimensionless shear stress')

xlabel('x')

ylabel('T(x)')

function dfdeta = fsode(x,y) %Function

global beta

dfdeta = [y(2)

(4/beta)*(y(1)^(0.25)-1)];

end

function res = fsbc(ya,yb) %Boundaries Conditions, [a,b]-->is the interval

global pac pev fi beta

res = [ya(2)

-(beta/(4*fi))*yb(2)-yb(1)^(1/4)+1+pev-pac];

end

I don't know why I got this error from Matlab:

Warning: Imaginary parts of complex X and/or Y arguments ignored.

I see that I got Imaginary parts on the solution ( sol.x and sol.y) but actually I don't want them, they need to be all real. I don't know what I am doing wrong.

Could you please help me find a solution for this problem?

Many Thanks