3rd order non linear differential equation

2f'''+f.f''=0

The boundary conditions are given as

f(0)=r, f'(0)=0, f'(∞)=1 , where r is a constant .

How can I solve this differential equation using Matlab or Mathematica ??

Answer 1

[Torsten]

Set

y1=f, y2=f',y3=f'',

rewrite your system as

y1'=y2

y2'=y3

y3'=-0.5*y1*y3,

set oo to a sufficiently large value and use MATLAB's bvp4c to solve.

Best wishes

Torsten.

[arindam pandit]

Hey thanks a lot for giving me the initial thrust. I happen to be a mechanical engineer with near zero knowledge of MATLAB. This is what I have been able to come up with.. your inputs will be highly appreciated.

 if true
function dydx = ode(x,y)
  dydx = [ y(2)
           y(3)
          -0.5*y(1)*y(3)];
function res = obc(ya,yb)
  res = [ ya(1)-1 
          ya(2)
          yb(2)];
solinit = bvpinit()
SOL = bvp4c(ode,obc,SOLINIT)

Can you help me out with bvpint function?

[Torsten]

function dydx = ode(x,y)
  dydx = [ y(2)
           y(3)
          -0.5*y(1)*y(3)];
function res = obc(ya,yb)
  r=0.1;
  res = [ ya(1)-r 
          ya(2)
          yb(2)-1.0];

xend=10;
SOLINIT = bvpinit(linspace(0,xend),[0.1 1 0]);
SOL = bvp4c(@ode,@obc,SOLINIT)

Best wishes

Torsten.

[arindam pandit]

Thank you sir for your time and knowledge in helping me out, but I have been facing the following warning message.

function dydx = ode(x,y) | Error: Function definitions are not permitted in this context.

Do I need to write the code in separate windows ?

[Torsten]

Try out whether this code runs:

function matbvp4c

xend=10;
SOLINIT = bvpinit(linspace(0,xend),[0.1 1 0]);
SOL = bvp4c(@ode,@obc,SOLINIT);

function dydx = ode(x,y)
  dydx = [ y(2)
           y(3)
          -0.5*y(1)*y(3)];
function res = obc(ya,yb)
  r=0.1;
  res = [ ya(1)-r 
          ya(2)
          yb(2)-1.0];

Best wishes

Torsten.

[arindam pandit]

Thank you very much Sir for helping me out with this problem. The code did run and gave the values of f wrt x. Sir, what are the modifications I need to do to get the value of f' and f'' for this problem .

[Torsten]

function matbvp4c

xend=10;
SOLINIT = bvpinit(linspace(0,xend),[0.1 1 0]);
SOL = bvp4c(@ode,@obc,SOLINIT);

y=deval(SOL,SOL.x);
plot(SOL.x,y(1,:),SOL.x,y(2,:),SOL.x,y(3,:));

function dydx = ode(x,y)
  dydx = [ y(2)
           y(3)
          -0.5*y(1)*y(3)];
function res = obc(ya,yb)
  r=0.1;
  res = [ ya(1)-r 
          ya(2)
          yb(2)-1.0];

y(1,:) is f, y(2,:) is f' and y(3,:) is f''.

Best wishes

Torsten.

[arindam pandit]

Thank you a lot sir. Can the matbvp4c function be used to solve a 2nd order DE. Like I have this equation.

2y''+a.b.y'=0 .

y(0)=1

y(∞)=0

a and b can be assumed as constants with values 0.1 and 0.2 respectively .

Thank you.

[Torsten]

Solution is y(x)=exp(-a*b*x/2).

Best wishes

Torsten.

[arindam pandit]

thank you sir for the answer, but I am required to write a code for the same in Matlab and I am not getting the correct results. So I will be really obliged if you can help me out with the code.

Thank you.

[Torsten]

Please show what you have done so far. Maybe we can find the error together.

Best wishes

Torsten.

[assiya malik]

i know how to solve it by inbuilt methods but i want the coding of RK-4 step by step to solve this balsius equation. if somebody could help i will be grateful