Solving a third order non-linear ode using ode45

Question

Abhinav 2012-10-28

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评论： David After 2020-9-4

I need to solve

F''' + 2FF" + 1 - F'^2 = 0

with the boundary conditions

F(0)=F'(0)=0 and F'(infinity)=1.

I am new to using the ode solver in matlab and am not sure how to make it solve a non-linear third order equation. Any suggestion would be appreciated.

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Shiv Patel 2017-11-17

Thank you for this comment, it helped with my homework!

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Answer 1

Star Strider 2012-10-28

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https://ww2.mathworks.cn/matlabcentral/answers/52075-solving-a-third-order-non-linear-ode-using-ode45#answer_63539

在 MATLAB Online 中打开

The ode45 function is not applicable to two-point boundary value problems such as yours. You need to use bvp4c for example, as I did here:

dYdX = @(X,Y) [Y(2);  Y(3);  Y(2).^2-Y(1).*Y(2).*2.0-1.0];  % Differential equation
res = @(ya,yb) [ya(1);  ya(2);  yb(2)-1];                   % Boundary conditions
SolYinit = bvpinit([0 1E+1], [1; 1; 1]);
Fsol = bvp4c(dYdX, res, SolYinit);
X = Fsol.x;
F = Fsol.y;
figure(1)
plot(X, F)
legend('F_1', 'F_2', 'F_3', 3) 
grid

You will not be able to set your endpoint to Inf or other number larger than about 10 (at least I could not) because that generates a singular Jacobian (or so MATLAB tells me). There are links to bvpinit and the other functions throughout and at the end of the bvp4c page. You might also want to experiment with bvp5c.

This was an interesting problem. I am glad you asked the question.

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Abhinav 2012-11-1

编辑：Abhinav 2012-11-1

在 MATLAB Online 中打开

Thanks for taking the time to go through my question. The plots being generated by thus code do not match the results that I was expecting to get, and I am not able to figure out why, because everything in the code seems good.

I finally did solve this with ODE45 and here is the code :

 clc
 clear all
 xmin=0;
 xmax=2;
 while((xmax-xmin)>10e-6)
    x=(xmin+xmax)/2;
    init = [0 0 x];
    eta = [0 5];
    [t,y] = ode45('eqn',eta,init)
    %max(y(:,2))
     if max(y(:,2))>1
         xmax=x;
     else
         xmin=x;
     end
 end
plot(t,y(:,3),'r+-',t,y(:,2),'b*-',t,y(:,1),'g+-')
axis equal
axis([0 3 0 2])
legend('d2F','dF','F')
grid on

The function was defined in another file as so....

    function f = eqn(t,y)
    f = zeros(size(y));
    f(1) = y(2);
    f(2) = y(3);
    f(3) = -1 -2*y(1)*y(3) + y(2)*y(2);

The plots that I am getting from this are exactly what I wanted.

It mainly involved figuring the initial conditions for the given boundary condition, i.e., finding a value of F"(0) so that F'(infinity)=1, which this piece of code calculates as 1.3119 approximately. Here, I tried different values for 'infinity' and finally settled on 5 as a good number.

David After 2020-9-4

How can i solve this eqaution?

F''' (eta)+ F(eta)F"(eta) + F'^2(eta) = 0

with the boundary conditions

F(0)=F''(0)=0 and F'(infinity)=0 and eta is 0:0.1:6

i want to plot it and create a table for it (eta-f-f'-f'')

I am new to using the ode solver in matlab and am not sure how to make it solve a equation. Any suggestion would be appreciated.

thank you

my email : ff223325@gmail.com

David After 2020-9-4

How can i solve this eqaution?

F''' (eta)+ F(eta)F"(eta) + F'^2(eta) = 0

with the boundary conditions

F(0)=F''(0)=0 and F'(infinity)=0 and eta is 0:0.1:6

i want to plot it and create a table for it (eta-f-f'-f'')

I am new to using the ode solver in matlab and am not sure how to make it solve a equation. Any suggestion would be appreciated.

thank you

my email : ff223325@gmail.com

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Answer 2

R Vishnu 2014-1-30

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Hi I hope to correct a minor error. it is f(3) = -1 -2*y(1)*y(2) + y(2)*y(2); and not f(3) = -1 -2*y(1)*y(3) + y(2)*y(2); as per your equation F''' + 2FF' + 1 - (F'^2) = 0.

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Abhinav 2014-1-30

Hi Vishnu, My bad.. the second term was supposed to be 2FF". I have edited the original question too.

Thanks for pointing that out!

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Answer 3

Hemanth kumar C 2017-11-8

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shootlinear2.m

I need to solve

(D^2-a^2)fi+a*[R*thetha+(Ra*Phi/Le)] = 0,

(D^2-a^2)thetha-(a*(dt/dy)*fi)-Pe*D(thetha)+gamma*Phi=0,

(D^2-a^2)Phi-Le*Pe*D(Phi)-a*le*(dc/dy)*fi=0

with the boundary conditions y=0;fi=Phi=thetha=0

I am new to using the ode solver in matlab and am not sure how to make it solve a non-linear second order three equation and i have written the program but i am not getting proper output . if you want to see my practise paper i will attach here . Any suggestion would be appreciated.

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Hemanth kumar C 2017-12-7

dc/dy ,dt/dy are the temperature and concentration gradients.i derived equations for that. i attached the program sir you can see the both equations

Hemanth kumar C 2017-12-7

deepika & Narayan.pdf

i am solving this above attached paper .i derived everything only programming part i am not getting few things .please help me to solve this problem. you can see what is dc/dy and dt/dy in the paper.

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