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- W2019436200 abstract "Two methods are given for treating a new type of potential problem arising in a recent study of the author on the encroachment of water into an oil sand. The problem is that of finding the potential distributions in two regions of different ``constants'' (``conductivities''), separated by a moveable surface of unknown shape, this motion and shape to be determined by the conditions that each point of the interface should have a velocity proportional to the vector gradient of the potential at the point, and that the area swept out by the moving surface shall assume the ``constant'' appropriate to that of the encroaching side of the interface. The first method is a perturbation method, valid when the difference between the ``constants'' is small, and reduces the problem of determining the shape and motion of the interface and the potential distributions simultaneously, to a series of problems in which potential distributions are to be found on the sides of known interfaces, and the motion of the surface is to be found in a known (time varying) potential field. This same type of reduction is effected also in the second method, which is one of direct successive approximations, but which may be used for any values of the potential ``constants.'' The perturbation theory is carried through explicitly to the third order for the case of a one dimensional system, and agrees to that order with the exact results which were derived for this case in a previous paper. The method of successive approximations is carried out for this same case through the second approximation for a ratio of the ``conductivity'' of the interior side of the interface to that on the exterior side equal to 5, and through the third approximation for the ratio 1/5. In the former, the calculations give a rapid monotonic convergence to the correct solution, and in the latter the convergence is oscillatory, and only slightly less rapid. These behaviors are shown to be generally true for the linear problem, in the course of the construction of an analytic convergence proof for the one dimensional problem. The convergence proof of the method of successive approximations is also given for the case of radial symmetry." @default.
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- W2019436200 date "1937-06-01" @default.
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- W2019436200 title "A Note on a Problem in Potential Theory" @default.
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- W2019436200 doi "https://doi.org/10.1063/1.1710318" @default.
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