Dear Rinku Mishra,

I think it could be solved by ODE method. First, make a function for the systems equations, then use ode (ode45, ode23, ode15s, etc) to solve them.

Hope to be useful.

Best Regards

https://www.mathworks.com/help/matlab/ref/ode45.html

https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&cad=rja&uact=8&ved=0ahUKEwiG57_J4IPVAhUpKsAKHdJZAvgQFghFMAQ&url=http%3A%2F%2Fwww.eng.auburn.edu%2F~tplacek%2Fcourses%2F3600%2Fode45waterloo.pdf&usg=AFQjCNHNa1u7E9fEAhmcc27glFEKX0loMA

Here is a useful example.

Note that if you have boundary condition instead of initial condition you can use shooting method.

Dear Rinku Mishra,

I think it could be solved with MATLAB’s built-in solver bvp4c.

Hope to be useful.

kind Regards

@Ali Al-Fayadh, but how it can be applied for coupled boundary condition? please explain.

Ms Mishra

May be you need this

https://www.youtube.com/watch?v=u-b4fJGOxP0

This is not a boundary value problem. This is an initial value problem with three unknowns. x=yz is not a boundary condition. It should be treated a constraint.

You can use Lagrange multiplier to impose the constraint.

As suggested by the others, you have to use an ODE solver. However, I suspect if you can use MATLAB's ODE solvers directly. They don't have any features supporting constraints.

You may have to write your own code using some time integration technique.

Dear Rinku Mishra,

well, you can write the boundary conditions as the M-file bc.m, which records boudary residues.

function res=bc(y0,y1)

%BC: Evaluates the residue of the boundary condition

res=[y0(1);y1(1)-10]; % if, for example you have y(0)=1 and y(1)=10 with a given DEq.

By residue, I mean the left-hand side of the boundary condition once it has been set to 0. the second boundary condition is y(1) = 10, so its residue is y(1) − 10 (depends on the given DEq), which is recorded in the second component of the vector that bc.m returns.

Hope to be useful.

kind Regards

Dear Ali Al-Fayadh,

Given equations do not represent a BVP. Therefore, one can not solve the system as a BVP.

This misunderstanding is the result of Rinku Mishra mentioning the constraint as the boundary condition.

Dear Rinku,

If you do not describe your problem correctly, you will not get any useful answers to solve it.

Please post full details of your system of equations. Verify again about the equation "x=yz". It can not be a boundary condition for this system of equations. It must a be a constraint which the solution at every time instant has to satisfy (for example, conservation law). Calling it as a boundary condition makes no sense.

Regards,

Chenna.

Hi dear Mishra,

I think that the matlab command bvp4c may works well for this problem,

You can visit the Matlab for more detail: https://www.mathworks.com/help/matlab/ref/bvp4c.html

Sincerely, 

S.Cengizci

The index of your system of DAEs is greater than 1. You may not solve it using a directly available ODE solver in Matlab. You have to follow the technique explained here.

I have used the simple example of the motion of the rigid pendulum in 2D. It includes Matlab code as well.

https://ckadapa.wordpress.com/2017/07/20/numerical-solutions-of-daes/

This is an initial value problem in 3 unknowns, x, y an z that you want to obtain as function of t.  You have one: initial condition x(t_0)=y(t_0)z(t_0), but you need two more, so you can come up with x(t_0)=x_0, y(t_0)=y_0 z(t_0=z_0, with x_0, y_0 and z_0 values you have calculated from your BC's.  After that it will be plain sailing and you can use standard MATLAB ODE solvers. If you lack those other two initial conditions MATLAB isn't going to help you.

NDSolve[{x'[t] == y[t] - z[t], y'[t] == x[t] z[t], z'[t] == y[t]/z[t],

x[0] == y[0] z[0], x[1] == 2, z[1] == 4}, {x, y, z}, {t, 1, 3.9}]

gives you the solution in Mathematica. And a nice figure is obtained like this: Plot[{x[t], y[t], z[t]} /. nds, {t, 1, 3.9}]

Dear Rinku

On the numerical computation of Poincaré maps, M Hénon, Physica D, 1982

This paper might answer your question