How to draw Fig. 4

function main
Pr=1; L=-1;D=1;
R=0.1;Sc=1;
% D=input('D=');   %%D=0.5, 1, 1.5

xa=0;xb=6;
solinit=bvpinit(linspace(xa,xb,101),[0 1 0 1 0 1 0]);
sol=bvp4c(@ode,@bc,solinit);
xint=linspace(xa,xb,101);
sxint=deval(sol,xint);

figure(1)
plot(xint,sxint(2,:));    
xlabel('\eta'); 
ylabel('f^\prime');
hold on
function res=bc(ya,yb)
res=[ya(1); ya(2)-L-D*ya(3); ya(4)-1; ya(7)-1; yb(2)-1; yb(4);yb(6)];  
end
 function dydx=ode(x,y)    
dydx=[y(2);  y(3);  y(2)^2-y(3)*y(1)-1;  y(5);  -3*Pr*y(1)*y(5)/(3+4*R);  y(7); -Sc*y(1)*y(7)];              
end
end

Actually I dont know the concept of DUAL solution.

Answer 1

[Stephan]

Hi,

maybe this is useful:

https://de.mathworks.com/help/matlab/math/solve-bvp-with-two-solutions.html

As shown in the link, the results strongly depend on the initial guesses - here is an example:

[sol1, sol2] = main

function [sol1, sol2] = main
Pr=1;
L=-1;
D=[0.5, 1, 1.5];
R=0.1;
Sc=1;

for k = 1:numel(D)
    xa=0;xb=6;
    solinit1=bvpinit(linspace(xa,xb,101),[0 1 0 1 0 1 0]);
    solinit2=bvpinit(linspace(xa,xb,101),[-1 -1 -1 -1 -1 -1 0]);
    sol1=bvp4c(@ode,@bc,solinit1);
    sol2=bvp4c(@ode,@bc,solinit2);
        
    figure(1)
    plot(sol1.x,sol1.yp(1,:),'-r')
    xlabel('\eta');
    ylabel('f^\prime');
    hold on
    plot(sol2.x,sol2.yp(1,:),'-b')
end

hold off


    function res=bc(ya,yb)
        res=[ya(1); ya(2)-L-D(k)*ya(3); ya(4)-1; ya(7)-1; yb(2)-1; yb(4);yb(6)];
    end

    function dydx=ode(x,y)
        dydx=[y(2);  y(3);  y(2)^2-y(3)*y(1)-1;  y(5);  -3*Pr*y(1)*y(5)/(3+4*R);  y(7); -Sc*y(1)*y(7)];
    end

end

results in:

Best regards

Stephan

[MINATI]

@ Stephan

How to know the initial guesses.

Actually if we add some EXTRA parameters, previous guesses may not work.

OR if we change

solinit2=bvpinit(linspace(xa,xb,101),[0 1 0 1 0 1 0]); solution is different.

Also that guess doesn't work for other figures.

So what is the trick for new guesses?

[Stephan]

I would love to tell you, but i can not, because i have no in Sicht to your problem.

[madhan ravi]

Sicht => insight

[MINATI]

OK Stephan

Thanks

[MINATI]

@Stephan

Following a Very bad CYCLONE in our State, I didnt follow your code properly as we had no electricity, no internet.

Thats why came late to the forum.

You had considered

f=y(1,:), f'=yp(1,:)

What is f"

Is it yp(2,:), If yes not matching. And

I have taken

f=y(1), f'=y(2); f"= y(3);

Theta= y(4), Theta'=y(5);

phi=y(6);phi'=y(7)