Self-similarity of nonlinear PDE

[Raisa]

Hi everyone,

I am a student in Mechanical Engineering and I am currently working on an assignment where I am exploiting the possibility of self-similarity for a PDE of a given problem. The PDE in my assignment consists of two independent variables (x for space and t for time), and one dependent variable h(x,t). The PDE is quite simple, however, my self-similar solution still contains the variable t. I have really spent a lot of time on trying to figure out where I went wrong in solving the problem but I cannot figure it out. I have summarized my solution method here: View attachment 6412

Does anyone have tips on how to solve this?

Thanks in advance!

 

[gtoguy97]

1. I assumed a self-similar solution of the form h(x,t) = t^αf(xt^β). 2. Substituting this into the PDE, I obtained a differential equation in f(xt^β). 3. Solving for f(xt^β), I got the solution as f(xt^β) = (xt^β)^γ, where γ is a constant. 4. Then, I substituted this back into the self-similar solution and got h(x,t) = t^α(xt^β)^γ. 5. Finally, I solved for α and β by comparing the exponents of both sides of the equation.It looks like you have done everything correctly, so there must be an error in your assumptions or calculations. One way to check this is to plug your solution into the original PDE and see if it satisfies the equation. If it does not, then you know something is wrong. If it does, then you should double check your calculations to make sure everything is correct. You could also try solving the PDE directly without making any assumptions. This could help to confirm that your self-similar solution is valid. You can use numerical methods to solve the PDE, such as the finite difference method or the finite element method. Finally, it may help to go over your notes and textbooks again to make sure you understand the concepts and methods you are using. Good luck!