How to solve nonlinear coupled ode by Shooting method .

Respected sir ,

I am getting problem to solve non linear coupled BVP by shooting method . Can you help me to solve that problem?

My problem is

F'''=(1/(1+epsilon1*(1-G)))*(A*F'+.5*t*A*F''+(F')^2-F*F''+epsilon1*G'*F''-lambda*G-delta*H+M*F'-(-1+epsilon1*G-epsilon1)*bita*F');

G''=(1/(1+epsilon2*G+Nr))*(Pr*(2*A*G+.5*A*t*G'+F'*G-F*G'-M*Ec*(F')^2-Ec*(1-epsilon1*G'+epsilon1)*(F'')^2)-epsilon2*(G')^2);

H''= Sc*(2*A*H+.5*A*t*H'+F'*H-F*H'+Rex*Zai*H);

where A=0 ; epsilon1= 0 ; epsilon2=1; lambda= 1; delta=1; bita=0; Nr=.1; Pr=5; M=.5;

Ec=.1 ; Sc=1; Rex= .3 ; Zai= .1 ;

and F(0)=0 , F'(0)=1 F'(infity)=0 G(0)=1 G(infity)=0 H(0)= 1 H(infity)=0

and F''(0) , G'(0) & H'(0) we have to guess

so tell me how to solve by shooting method with using rk -4 method .

[darova]

What have you tried? What ideas do you have? Did you see examples in MATLAB help?

[Chandan Kumawat]

I general do it with general shooting method but i'm getting problem how to do initial guess.

Here i'm attaching code which i have tried .

Answer 1

[Chandan Kumawat]

It is okey but my problem still remain same that how i do appropriate guess for according to your code

F2(0) & G1(0) & H1(0)

[darova]

Why did you accept your answer instead of mine?

[Chandan Kumawat]

here I'm not getting any option for accept your answer .

Thanks for solve my problem .

[darova]

Whta is this?

Answer 2

[darova]

Try bvp4c

Suggestion:

F0 = y(1);
    %% ...
H1 = y(7);
    % and use these variables to make your code more redable
dy(1) = F1;
    %% ..

You can also use temporary variables to make your code simpler

dy(3) = 1/(1+e1*(1-G0))* ...
    (A*F1 + 0.5*t*A*F2 + F1^2 - F0*F2 + e1*G1*F2 - lambda*G0 - delta*H0 + M*F1 - (-1+e1*G0-e1)*bita*F1);
%%
TEMP0 = 1/(1+e1*(1-G0));
TEMP1 = 0.5*t*A*F2;
TEMP2 = e1*G1*F2;
TEMP3 = (-1+e1*G0-e1)*bita*F1;
dy(3) = TEMP0 * (A*F1 + TEMP1 + F1^2 - F0*F2 + TEMP2 - lambda*G0 - delta*H0 + M*F1 - TEMP3);

See attached scripts

[Chandan Kumawat]

It is okey but my problem still remain same that how i do appropriate guess for according to your code

F2(0) & G1(0) & H1(0)

[darova]

How do you think those points mean?

[Chandan Kumawat]

Please can you attach total code which you made for this problem?

[darova]

Please see the picture

[Chandan Kumawat]

Thank you sir give attention on my problem and this code working but again i'm getting one is that it is give answer but also give some error notification in command window which i'm write below is ..........

Warning: Unable to meet the tolerance without using more than 1428 mesh

points.

The last mesh of 1000 points and the solution are available in the output

argument.

The maximum residual is 0.159818, while requested accuracy is 0.001.

> In bvp4c (line 323)

In Untitled5 (line 6)

[darova]

Impossible. What is Untitled6 ? Why don't you run the script i attached Untitled21?

[Chandan Kumawat]

Here i use untitled 5 is same with which you attached untitled 21 .This error come when change constant value of M Ec Pr ...... and one more thing solution and plot are coming which i need .

Does is there any code for solve this problem by using shooting method and Runge kutta method of 4th order .

[darova]

constant value of M Ec Pr

Are you sure those values are correct? Maybe for something values diving by zero occurs?

[Chandan Kumawat]

Thank you sir for give attention on my problem and now code is working well .

Can we solve it by ODE45 if yes then what values for F2(0) G1(0) & H1(0) should we take and how we will find that values .

[darova]

I just changed main code

init = [0 1 -0.58 1 -1.52 1 -1.12];
% solinit = bvpinit([0 2],zeros(1,7));
% sol = bvp4c(@new,@bvpf,solinit);
[t,y] = ode45(@new,[0 2], init);

% plot(sol.x,sol.y)
plot(t,y)
legend('F','dF','d2F','G','dG','H','dH')

I took initial conditions from last calc. There is no rule for F2(0) G1(0) & H1(0) values, only guessing or something like bvp4c