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I have a homogeneous second-order differential equation [in function $y(x)$] with variable coefficients - with no given specific functional forms for $a(x)$ or $b(x)$ - given as follows:

$$y\hspace{0.01in}''+a(x)\hspace{0.01in}y\hspace{0.01in}'+b(x)\hspace{0.01in}y=0$$

given the boundary conditions,

$$y(1)=1,\text{ and $\,\,$}y(0)=\alpha\hspace{0.01in}y\hspace{0.01in}'(0)$$

where, $\alpha$ is a constant. Clearly, the second boundary value/condition is in form of a constraint; from what I understand, this type of boundary value problem is called the Mixed Boundary Value Problem (or alternatively, the Robin Boundary Value Problem). Now, what is the most optimal method (in terms of computational cost) to solve this problem numerically?

Thanks for the help folks!

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  • $\begingroup$ How many boundary conditions, two or three? Are numerical values $y'(0), y(0), y(1) $ given? $\endgroup$ – Narasimham Apr 7 '15 at 18:00
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I cannot say it is most optimal.

Either separately solve the differential equation and input the boundary conditions,

$$ Y(x)=\alpha\hspace{0.01in}Y\hspace{0.01in}'(x), Y(0) = Y_i = y(0), y(1) = 1 $$

or, solve them together numerically as a coupled differential equations,like say:

$$y(1)=1,\text{ and $\,\,$}y(0)=\alpha\hspace{0.01in}y\hspace{0.01in}'(0) = 0.2 $$

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  • $\begingroup$ In your case, $Y_i$ is unknown. Also, the numerical values for $y\hspace{0.01in}'(0)$ or $y(0)$ are not given. $\endgroup$ – Avneet Singh Apr 8 '15 at 11:50
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I think herein the correct way is given. In principle, it is a method to solve the system analytically. One can find an optimal numerical method easily since the overall method simply includes solving two differential equations independently, and then linearly combining the results with an appropriate pre-factor.

Link: http://nptel.ac.in/courses/111107063/module4/lecture4/lecture4.pdf

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