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The material of this question is relevant to solving boundary value problems, Rayleigh-Benard convection and the Boussinesq equations.

How might I solve the boundary value problem ( BVP ),defined by

\begin{align} \left[-S+ \left( \frac{d^2}{dx^2} -k^2\right)\right]\circ \left( \frac{d^2}{dx^2} -k^2\right)f(x)&=0 \tag1\\ f(0)&=0\\ f(h)&=\Delta\theta \end{align} with $k$ constant.

NB: In context I like the use of the notation \begin{align} D&= \frac{d}{d x}\\ D^2&= \frac{d^2}{d x^2}\\ \end{align} Using this notation (1) can be written $$[-S+(D^2-k^2)]\circ(D^2-k^2)f(x)=0 \tag2$$

The ordinary differential equation (2) is of fourth order and is homogenous.

OTHER INFORMATION

I searched on this site and found a mention of the phrase ' undetermined coefficients ' , but I don't think that was in relation to the BVP idea, I can't remember.

I was hoping for some specific help with details.

user151522
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1 Answers1

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Without more information, $S$ is also a constant and the roots of the characteristic equation are

$$\pm k,\pm\sqrt{S+k^2}.$$

Hence the general solution

$$f(x)=C_0e^{kx}+C_1e^{-kx}+C_2e^{\sqrt{S+k^2}x}+C_3e^{-\sqrt{S+k^2}x}.$$

The given initial conditions yield

$$C_0+C_1+C_2+C_3=0.$$

$$C_0e^{kh}+C_1e^{-kh}+C_2e^{\sqrt{S+k^2}h}+C_3e^{-\sqrt{S+k^2}h}=\Delta\theta.$$

Two more conditions are necessary to determine all coefficients.

  • I realize now, I just needed to apply the general theory of how to solve 'nTH-Order Linear Homogenous Ordinary Differential Equations with Constant Coefficients', with n=4. – user151522 Oct 17 '18 at 11:28