Solve a system of Partial Differential Equations using function BVP4C

Hello every one. I have a Boundary Value Problem to solve. When I press "run", an error appears: "Warning: Explicit solution could not be found. > In solve at 160". How can I fix my errors. Thank you in advance!

Below is my Boundary Value Problem and the code:

BVP problem:

S'''+3SS''-2*S'*S'+Y-109.963*F=0

Y''+3*5.076*S*Y'+0.0000007738*F'*Y'+0.00000255*Y'*Y'=0

F''+3*5.076*42770*S*F'+2.914*Y''=0

Boundary Conditions:

x=0 then S=S'=0, Y=1, F=1

x=infinitive then S'=Y=F=0

My code:

Editor window 1:

clear all; close all; clc;

init=bvpinit(linspace(0,4,100),[0 0 0 0 0 0 0 0 0])

sol=bvp4c(@rhs_bvp,@bc_bvp,init);

x=linspace(0,4,1000);

BS=deval(sol,x);

plot(x,BS);

Editor window 2:

function rhs=rhs_bvp(x,y)

rhs=[y(2);y(3);-3*y(1)*y(3)+2*y(2)*y(2)-y(5)+109.963*y(8);y(6);-3*5.076*y(1)*y(6)-0.0000007738*y(9)*y(6)-0.000002255*y(6)*y(6);y(9);-3*5.076*42770*y(1)*y(9)+2.914*3*5.076*y(1)*y(6)+2.914*0.0000007738*y(9)*y(6)+2.914*0.000002255*y(6)*y(6)];

Editor window 3:

function bc=bc_bvp(yl,yr)

bc=[yl(1);yl(2);yl(5)-1;yl(8)-1;yr(2);yr(5);yr(8)];

Answer 1

[Torsten]

rhs=[y(2);y(3);-3*y(1)*y(3)+2*y(2)*y(2)-y(4)+109.963*y(6);y(5);-3*5.076*y(1)*y(5)-0.0000007738*y(5)*y(7)-0.000002255*y(5)*y(5);y(7);-3*5.076*42770*y(1)*y(7)-2.914*(-3*5.076*y(1)*y(5)-0.0000007738*y(5)*y(7)-0.000002255*y(5)*y(5))];

bc=[yl(1);yl(2);yl(4)-1;yl(6)-1;yr(2);yr(4);yr(6)];

Best wishes

Torsten.

Answer 2

[Bill Greene]

It looks to me like you have 9 dependent variables and only 7 equations and boundary conditions. Clearly, that can't be solved.

Answer 3

[Hiep Hoang]

Thank Bill Greene and Torsten for your comments. I already got some results, but results are not so good. The residual is quite large as compared to tolerance.

I think the problem is the initial guess. The number of mesh points is 1000. I specify the initial guess as following:

init=bvpinit(linspace(0,10,1000),[0 0 1 1 -0.33 1 -1])

It means that at each of 1000 points, y(1)=0, y(2)=0, y(3)=1, y(4)=1, y(5)=-0.33, y (6)=1, y(7)=-1. But how can I specify the initial guess at only one point x=0, I think it can improve the results.

[Torsten]

Use a function for yinit:

https://de.mathworks.com/help/matlab/ref/bvpinit.html

Best wishes

Torsten.