Third Order Coupled ODE's

I have these coupled equations based off Falkner-Skan:

f′′′=−f⋅f′′−(2m/(m+1))⋅(1−(f′)2)−(2/(m+1))⋅Ri⋅θcos(m/(m+1))

&

θ′′=−Pr⋅f⋅θ'

With the BC's: f(0)=f′(0)=0 and θ(0)=1. f(5)=1 and θ(5)=0 (5 has been taken as the endpoint)

Where Pr, m and Ri are constants which can be assigned, f and θ are functions of η and the dash (') means derivative with respect to η.

I have a MATLAB file which solves this when Ri=0 by using the shooting method (as this is a BVP) and ODE45 but now it seems like that can no longer be used as easily because now they need to be solved simultaneously when Ri is not zero and I don't know where to start.

How can I solve this problem using MATLAB for the plots of f' and θ against η?

[ravinder khary]

Respected Andrew Amoateng-Agyemang, As I am a beginner in MATLAB programming, Can you please share your code of shooting method and ODE45 for the above mentioned coupled Falkner-Skan equation. I also have a coupled BVP to solve. I don't know how to use MATLAB inbuilt function so please share a full code if possible. Thank you

[Didarul Ahasan Redwan]

here,, it is

Answer 1

[Torsten]

function main

xa = 0; xb = 5;
solinit = bvpinit(linspace(xa,xb,10),[0 0 1 1 0]);
sol = bvp4c(@coupled_falkner_skan,@bc,solinit);

function res = bc(ya,yb)
res = [ ya(1); ya(2); ya(4)-1; yb(2)-1; yb(4)];

function dydx = coupled_falkner_skan(x,y)
m = 0.0909; Ri = 5; Pr = 0.71;

dydx = [y(2); y(3); -y(1)*y(3)-(2*m/(m+1))*(1-y(2)^2)-(2/(m+1))*Ri*y(4)*cos(m*pi/(m+1)); y(5); -Pr*y(1)*y(5)];

Best wishes

Torsten.

[Andrew Amoateng-Agyemang]

I have completed what I set out to do now, thanks for your help!

The final code:

function main

xa = 0; xb = 5;
solinit = bvpinit(linspace(xa,xb,1000),[0 0 0 1 0]);
sol = bvp4c(@coupled_falkner_skan,@bc,solinit);
xint = linspace(xa,xb,1000);
Sxint = deval(sol,xint);
plot(xint,Sxint([2 4],:));

function res = bc(ya,yb)
res = [ ya(1); ya(2); ya(4)-1; yb(2)-1; yb(4)];

function dydx = coupled_falkner_skan(x,y)
m = 0.0909; Ri = 5; Pr = 0.71;

dydx = [y(2); y(3); -y(1)*y(3)-(2*m/(m+1))*(1-y(2)^2)-(2/(m+1))*Ri*y(4)*cos(m*pi/(m+1)); y(5); -Pr*y(1)*y(5)];

[Taylor Nichols]

I know this is an old question, but shouldn't the 4th boudary condition be yb(1) - 1, not yb(2) - 1. The BC you have written at the beginning is not the 1st derivative.

[Torsten]

Yes, take a look at my answer from the 7th of April below. I subsequently copied the code from the comment to it (where the boundary condition is modified) and added the plotting.

Answer 2

[Torsten]

Rewrite your equations as a system of first-order ordinary differential equations and use bvp4c to solve.

Best wishes

Torsten.

[Andrew Amoateng-Agyemang]

I have rewritten the equations in first order form, thanks. But now I'm stuck on actually using bvp4c. I need more clarification on MATLAB's definition (typed "help bvpc4") which I have been trying to understand.

Can you please clarify?

Answer 3

[Torsten]

function dydx = ode(x,y)
m=...;
Ri=...;
Pr=...;
dydx = [ y(2); y(3); -y(1)*y(3)-2*m/(m+1)*(1-y(2)*2)-2/(m+1)*Ri*y(4)*cos(m/(m+1)); y(5); -Pr*y(1)*y(4)];

function res = bc(ya,yb)
res = [ ya(1); ya(2); yb(1)-1.0; ya(4)-1.0; yb(4)];

solinit = bvpinit(linspace(0,4,5),[0 0 0 0 0]);
sol = bvp4c(@ode,@bc,solinit);

Best wishes

Torsten.

[Andrew Amoateng-Agyemang]

Thank you, I now understand all this and corrected any mistakes leaving:

function dydx = coupled_falkner_skan(x,y)
m = 0.0909; Ri = 5; Pr = 0.71;

dydx = [y(2); y(3); -y(1)*y(3)-(2*m/(m+1))*(1-y(2)^2)-(2/(m+1))*Ri*y(4)*cos(m*pi/(m+1)); y(5); -Pr*y(1)*y(5)];

end

function res = bc(ya,yb)
res = [ ya(1); ya(2); ya(4)-1; yb(2)-1; yb(4)];
xa = 0; xb = 5;
solinit = bvpinit(linspace(xa,xb,10),[0 0 1 1 0]);
sol = bvp4c(@coupled_falkner_skan,@bc,solinit);

end

Now I want to plot the solution across the whole domain, I hit the run button but it's telling me "error not enough input arguments" ?

Thanks

Answer 4

[T K]

if true
  % code
end