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How can boundary formulas be manipulated using Taylor series and undetermined coefficients?

I'm solving a boundary value problem numerically where the boundary condition is non-standard. I need a systematic, algebraic approach to manipulate the condition itself, likely using a series expansion, to derive a computable form that can be coded into my solver.

 

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By Rani Answered 3 years ago

When facing tricky boundary conditions in my fluid dynamics work, I've often used this approach. First, expand your unknown solution as a Taylor series about the boundary point. Substitute this series into the boundary condition formula which might also be expanded if nonlinear. This yields an equation where powers of the spatial variable are matched. The method of undetermined coefficients is then applied: you solve the resulting system of algebraic equations for the series coefficients, thereby satisfying the boundary condition order-by-order. It transforms a differential constraint into an algebraic one.

 

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