Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Dual extremum principles relating to cooling fins

Authors: S. Bhargava and R. J. Duffin
Journal: Quart. Appl. Math. 31 (1973), 27-41
MSC: Primary 80.49
DOI: https://doi.org/10.1090/qam/416294
MathSciNet review: 416294
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Abstract: Under consideration is a differential equation $ {\left( {pu'} \right)'} = qu$ of the Sturm--Liouville type where the function $ q\left( x \right) > 0$ is given. The problem is to find a function $ p\left( x \right) > 0$ in $ 0 \le x < b$, a constant $ b$ and a solution $ u\left( x \right)$ of the corresponding differential equation such that the energy functional $ \int_0^b {\left[ {p{{\left( {u'} \right)}^2} + q{u^2}} \right]} dx$ is maximized when $ p\left( x \right)$ is subject to the constraint $ \int_0^b {{p^\rho }dx \le {K^\rho }} $ and $ u$ is subject to the boundary conditions $ u = 1$ at $ x = 0$ and $ p\left( {du/dx} \right) = 0$ at $ x = b$. Here $ K > 0$ and $ \rho \ge 1$ are constants. A key relation $ \left\vert {du/dx} \right\vert = \lambda {p^{\left( {\rho - 1} \right)/2}}$, where $ \lambda $ is a positive constant, is found. This criterion leads to explicit solution of the problem. A further consequence of this criterion together with a pair of dual extremum principles is a ``duality inequality'' giving sharp upper and lower estimates of the maximum value of the energy functional.

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DOI: https://doi.org/10.1090/qam/416294
Article copyright: © Copyright 1973 American Mathematical Society

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