# Solving a fourth-order linear non-homogeneous PDE

I'm trying to solve a fourth-order linear non-homogeneous PDE

$$\frac{\partial u}{\partial t} + \frac{\partial^2 u}{\partial \theta^2} + \frac{\partial^4 u}{\partial \theta^4} = - \sin{\theta}.$$

This PDE is periodic in $$\theta$$ and has domain $$\theta \in [0, 2 \pi)$$, $$t\geq 0$$, with boundary conditions

$$\frac{\partial u}{\partial \theta} = 0$$

on $$\theta = \pi/2$$ and $$\theta = 3 \pi/2$$, with initial condition $$u(\theta , t=0) = 0$$.

I'm not sure of the best way to approach solving this PDE, so any help would be much appreciated!

I know that in the non-homogeneous case with a constant initial condition, then there is a constant solution, but I'm not sure how to proceed in the non-homogeneous case with the $$-\sin{\theta}$$ term on the RHS.

• Welcome to Mathematics SE. Take a tour. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. Jul 15, 2023 at 11:54
• What's the domain? Jul 15, 2023 at 12:55
• @tommy1996q I've edited the question to add the domain. Jul 15, 2023 at 13:08
• Since it's fourth order the solution is gonna be non unique, if you're ok with stationary solutions you basically reduce to an ODE Jul 15, 2023 at 13:13
• By simple inspection one see that $u(t,\theta)=-t\,\sin(\theta)$ satisfies the PDE and the conditions. Jul 15, 2023 at 13:37

You can, for each time $$t$$, project the solution $$u$$ on the $$\sin$$ and $$\cos$$ basis: $$u(t,\theta) = \sum_{n=1}^\infty a_n(t) \sin(n \theta) + \sum_{n=0}^\infty b_n(t) \cos(n \theta)$$ You can project the PDE to get: $$a_n' + (n^4-n^2) a_n = - \delta_{1,n}$$ $$b_n' + (n^4-n^2) b_n = 0$$ and solve these simple ODE The $$a_n(t)$$, $$n\neq 1$$ and $$b_n(t)$$ will be exponentials, and I beleieve we can find equations on their coefficients with the boundary conditions. I Hope this approach helps.