Summary
A Cauchy problem for the Laplace equation is solved by analytic continuation of the space variables on the plane of the complex potential, thereby obtaining an explicit expression for the geometry of physical boundaries of interest. In an illustrative application to the inverse free boundary problem of electrochemical machining, the general solution comprises a closed-form description of a tool family which can be used to machine a prescribed workpiece. The method is extended to include the effects of variable electrolyte conductivity, and a general tool design procedure is suggested in which an analytic series with correct asymptotic behavior is used to represent the given workpiece geometry. Applications in other fields such as heat conduction and hydrodynamics are discussed. The inverted formulation described herein affords considerable advantage and generality in solving Cauchy problems which are encountered in engineering design.
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Nilson, R.H., Tsuei, Y.G. Inverted Cauchy problem for the Laplace equation in engineering design. J Eng Math 8, 329–337 (1974). https://doi.org/10.1007/BF02353499
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DOI: https://doi.org/10.1007/BF02353499