yy3 is generating a straight line while it should generate a curve, i don't understand what is wrong!!!

function ABC

% Initialization of paramters

Pr=6.2;

M=4;

Bi=0.3;

beta=0.2;

ks1=0.5;

ks2=0.1;

rhos1=0.2;

rhos2=0.3;

n=3;

kbf=0.598;

phi1=0.1;

phi2=0.04;

rhocps1=0.3;

rhocps2=0.1;

rhof=997.1;

kf=0.613;

rhocpf=4179;

A1=((1-phi1)^(2.5)).*((1-phi2)^(2.5));

B1=(1-phi2).*((1-phi1).*rhof + rhos1.*phi1) + rhos2.*phi2;

A2=B1*(1/rhof); % rho_Hnf/rho_f

B2=ks2+(n-1)*kbf-(n-1)*(kbf-ks2).*phi2;

B3=ks2+(n-1)*kbf+(kbf-ks2).*phi2;

C5=B2/B3; % k_Hnf/k_bf Nanofluid Constant

B4=ks1+(n-1)*kf-(n-1)*(kf-ks1).*phi1;

B5=ks1+(n-1)*kf+(kf-ks1).*phi1;

C6=B4/B5; % k_bf/k_f

A4=C5*C6; % k_Hnf/k_f

B6=(1-phi2)*((1-phi1)*rhocpf+phi1*rhocps1)+phi2*rhocps2; % rhocp_Hnf

B7=1/rhocpf;

A3=B6*B7; % rhocp_Hnf/rhocp_f

% Initial Condition Input

sol = bvpinit(linspace(0,8,100), [1 0 0 0 0 0 0 0]);

% solution in structure form

sol1 = bvp4c(@bvpexam2, @bcexam2, sol);

x1 = sol1.x;

y1 = sol1.y;

%%% Plotting of the temperature

plot(x1, y1(7, :))

hold on

function res=bcexam2(y0, yinf)

res=[y0(2)-1;y0(5)-beta;y0(1)+y0(4);y0(8)+(Bi/A4)*(1-y0(7)); yinf(2);yinf(5);yinf(7); yinf(4)];

end

function dydx = bvpexam2(~,y)

yy1=A1*(A2*(y(2)^2-(y(1)+y(4))*y(3))+M*y(2));

yy2 = A1*(A2*(y(5)^2-(y(1)+y(4))*y(6))+M*y(5));

yy3 =-(A3/A4)*Pr*(y(1)+y(4))*y(8);

dydx= [y(2);y(3);yy1;y(5);y(6);yy2;y(7);yy3];

end

end

[Fareeha]

here is what i get from this code

[Fareeha]

this is what i need

[Shahid Hasnain]

@Fareeha

What the purpose of plotting y1(7,:)?

[Fareeha]

@Shahid Hasnain it denotes theta

Answer 1

[Torsten]

As initial condition for y(7), you set y(7) = 0, and you define the differential equation for y(7) as dy(7)/dx = y(7). The solution is y(7) = 0 for all x - and that's what is plotted.

The line

dydx= [y(2);y(3);yy1;y(5);y(6);yy2;y(7);yy3];

must read

dydx= [y(2);y(3);yy1;y(5);y(6);yy2;y(8);yy3];

[Fareeha]

@Torsten Thank you it helped

[Fareeha]

@Torsten i have another question

in function

function res=bcexam2(y0, yinf)

res=[y0(2)-1;y0(5)-beta;y0(1)+y0(4);y0(8)+(Bi/A4)*(1-y0(7)); yinf(2);yinf(5);yinf(7)];

end

I removed one BC, now i have 8 vectors in dydx and 7 BC, in such condition how could code work?

[Torsten]

You need as many boundary conditions as there are first-order differential equations. Otherwise, you get a one-dimensional solution manifold.

Consider

y '' = c with y(0) = 1

The solution is

y(x) = 1 + a1*x + c/2*x^2

with arbitrary parameter a1.