Solving nonlinear boundary value problem

Question

I have an ODE of the form $$C_1 d_y u + C_2 (d_y u)^n = C_3 y + C_4$$ with boundary conditions $u(0) = 0, d_y u(L) = 0$, where $C_1 \to C_4$ are known constants, and where $0 < n \leq 1$ is a real number.

Special cases:

• If $C_2 = 0$, this is a simple linear ODE and is straightforward to solve.
• If $C_1 = 0$, I can take the $\sqrt[n]{}$ of both sides and proceed to solve the nonlinear first order ODE.

Question: Is there then an appropriate method for solving the general case ($C_1 \neq 0, C_2 \neq 0$)?

Answer 1

You simply have a nonlinear boundary value problem here. There is, in general, no technique where you can solve this in one shot, but there are many techniques where you can iteratively solve it via a sequence of solutions $u^{(k)}(y)$ that converge to $u(y)$ as $k\rightarrow \infty$. Examples of this are the fixed point method, pseudo-timestepping, or Newton's method.

You can find an overview of the various techniques in many books. There are also lectures 31.5 and following from my video lecture page: http://www.math.tamu.edu/~bangerth/videos.html

Answer 2

One specific method that you might find interesting is called the Shooting Method. For a nonlinear BVP, it will be an iterative approach, but it is kind of interesting because you treat the BVP like an initial value problem, so you can use normal integrators like Runge-Kutta, etc. Here's a link to give you an idea: Link.