Shooting method for PDE

Hello all, There is a paper which I was going through (attached), where there are two equations which I would like to solve. Eq 16 and 17 based on 18 and 19 using shooting methods. Problem is the boundary conditions (infinity) and first order). I would be very thankful if you could help me solve the problem.

Thanking you.

Regards, Judah

Answer 1

[Zoltán Csáti]

During a shooting method you guess initial values from where you start solving the boundary value problem (BVP) as an initial value problem (IVP). You expect the result to be accurate when the right boundary condition (BC at infinity) is fulfilled. Since you solve an IVP, you can set the interval of integration. Choose it sufficiently large. If you use L1<L2 and there is little difference between them, then the truncated interval describes the problem of infinity well.

I solved similar boundary-layer-type equations several times. So I suggest you to try one or more of the followings:

• Solve the BVP with bvp4c or bvp5c
• Use Chebfun
• Perform group analysis on them so that you can use a transformation method that can transform the BVP to an IVP (see e.g. Töpfer-transformation)
• Use a series of transformations (see e.g. Iterative Transformation Method)
• Code the shooting method (but be aware, this can be quite susceptible to the initial conditions)

[Judah S]

Thanks Zoltan, I am able to solve this using using euler's method or RK if boundary conditions were not given in derivative terms. If so, I would solve for two initial guesses and do interpolation to find answer at boundary.

I will be thankful if you could give me one of your similar solutions which will help me proceed in correct direction.

Thanks.

Regards, Judah

[Judah S]

I worked on the code just now and seem to approach in the right direction.

Here is my code (attached) Figure 2 seems to give similar result compared to the paper but Y axis values are not correct (due to boundary condition F'(0) = f(0) and F(infinity) = 0). I don't know how to implement these boundary conditions correctly. I would be very thankful if you could look into it.

Thank you.

Regards, Judah

[Judah S]

Any help will be very appreciated.

Thanks,