initial guessing - for mixed boundary condition

f'''+(1/2)*f*f''+lamda*theta

theta''+pr*(1/2*f*theta'+f'*theta

with B.C f(eta) = 0,f'(eta) = delta*f''(eta) , theata(eta)=1+beta*theta'(eta) at eta = zero

f'(eta) = 1 ,theta(eta) = 0 as eta = infinity

close all

solinit = bvpinit(linspace(0,2),@exlinit );

options = bvpset('RelTol',1e-6);

sol = bvp4c(@convec,@bvp,solinit,options);

figure(1)

plot(sol.x,sol.y(4,:),'linewidth',1.5);

%plot(sol.x,sol.y(5,:),'--','linewidth',1.5);

hold on

function dy = convec(t,y)

lamda = 1;

Pr = 6.2;

dy(1) = y(2);

dy(2) = y(3);

dy(3) = -(1/2)*y(1)*y(3)+lamda*y(4);

dy(4) = y(5);

dy(5) = -(Pr)*(1/2*y(1)*y(5)+y(2)*y(4));

dy =[dy(1);dy(2);dy(3);dy(4);dy(5)];

end

function res = bvp(ya,yb)

delta = 0.5;

beta = 1;

res = [ya(1)-0

ya(2)-delta*ya(3)

yb(2)-1;

ya(4)-1-beta*ya(5);

yb(4)-0];

end

function v = exlinit(t)

v = [0

1

1-exp(-t)

1

exp(-t)];

end

the program is running but not getting correct one for example temperature profile ie y(4)

kindly give a suggestion for guessing a initial condition for the problem

[Torsten]

From your equations, it should be

dy(3) = -(1/2)*y(1)*y(3)-lamda*y(4);

instead of

dy(3) = -(1/2)*y(1)*y(3)+lamda*y(4);

Answer 1

[Sarthak]

Hi N Nithya,

I understand you are looking for a way of initial guessing, but I found something else in the code which might help.

The equation for dy(3) should be:

dy(3) = -(1/2)*y(1)*y(3) - lamda*y(4)

The original equation, dy(3) = -(1/2)y(1)y(3) + lamday(4), is incorrect because the term lamday(4) should be negative.

The negative sign in front of the term lamda*y(4) is important because it ensures that the derivative of dy(3) is always negative. This is necessary because dy(3) represents the rate of change of y(3), and the rate of change of y(3) should always be negative when y(3) is increasing.

Hope this solves your query.