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To the problem of the recovery of nonlinearities in equations of mathematical physics

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Abstract

The general construction for finding all “essentially different” nonlinearities in equations of mathematical physics is exemplified by the inverse problem of the Grad–Shafranov equation. We present an algorithm that allows one to recover relatively fast all essentially different sought-for nonlinear right-hand sides of the Grad–Shafranov equation. We present the first example of a domain with smooth boundary for which the inverse problem has at most one solution in the class of affine functions, and also in the class of exponential functions. We select some subset of simply connected domains that model, in some sense, the so-called doublet configurations, for which the inverse problem has at least two essentially different solutions in the class of analytic functions. In the concluding subsection of the paper, we indicate a method of recovery, from boundary data, of all essentially different nonlinearities in equations of mathematical physics of considerably general kind, which includes a system of equations that describes combustion and detonation processes.

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Authors and Affiliations

  1. Moscow State University, Moscow, Russia

    A. S. Demidov, A. S. Kochurov & A. Yu. Popov

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  1. A. S. Demidov
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  2. A. S. Kochurov
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  3. A. Yu. Popov
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Correspondence to A. S. Demidov.

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Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 27, Part I, pp. 75–125, 2009.

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Demidov, A.S., Kochurov, A.S. & Popov, A.Y. To the problem of the recovery of nonlinearities in equations of mathematical physics. J Math Sci 163, 46–77 (2009). https://doi.org/10.1007/s10958-009-9658-x

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  • DOI: https://doi.org/10.1007/s10958-009-9658-x

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