# How to solve the eigenvalue problem that arises in stability of flows

I am an undergraduate student and I want to know how to solve this hydrodynamic stability problem. I tried but all my efforts have failed. Your suggestions will help me a lot. The equations are f''- a^2 f + a R h=0; h''- a^2 h + a f=0; And the conditions at the plates are z=0: f = 0, h = 0 and z=1: f = 0, h = 0. In my first attempt, I am fixing a = 3.1472 and then find R as an eigenvalue and print the eigenfunctions f and h. To do this I wrote the following code by introducing one more equation for R i.e R'[z]=0 with an artificial condition R[0]=n. The following code represents this.

Block[{a = 3.1479},
sol = ParametricNDSolve[{f''[z] - a^2 f[z] + a R[z] h[z] == 0,
h''[z] - a^2 h[z] + a f[z] == 0, R'[z] == 0, f[0] == f[1] == 0,
h[0] == h[1] == 0, R[0] == n}, {f[z], h[z], R[z]}, {z, 0, 1}, n]]


Can you please correct my code? I would be grateful to you for this help.

• Are you sure you have the equations set up correctly? I'm taking a Laplace transform of the equations and getting that f and h are both 0. I could be misunderstanding the problem. Can you add additional information as to the context of the problem? I'm not quite sure what you mean by "conditions at the plates" or what functions f and h represent. – Haff Sep 22 '17 at 18:33
• Dear Sir, Thank you for your reply. Please accept my sorry for the mistakes in my previous description. For the classical Benard convection in Porous media, the onset equations are precisely f''- a^2 f - a R h=0; h''- a^2 h - a f=0; f=h=0 on z=0 and z=1. If I wish to find the minimum of R in the neutral curve, I should have d(R)/da=0. This gives one more equation but one artificial boundary condition has to be supplied for R. I think in my previous description I made small mistake in my equations. The third term in each of those equations should have negative sign. – PAL Sep 25 '17 at 4:56
• Dear sir I just wanted to have a code like this For example, reference.wolfram.com/language/tutorial/NDSolveBVP.html#3518691 for the boundary value problem with parameter as given in last example of the above link. I tried in the same way but failed. Could you please assist me? – PAL Sep 25 '17 at 6:44