Abstract
We consider some new identities and representations of solutions to the second-order partial differential equations connected with the Hopf-Cole transformation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Yu. E. Anikonov, “Constructive Methods of Studying the Inverse Problems of Evolution Equations,” Sibirsk. Zh. Industr. Mat. 11(2), 3–20 (2008) [J. Appl. Indust. Math. 3 (3), 301–317 (2009)].
Yu. E. Anikonov, “Inverse Problems and Classes of Solutions of Evolution Equations,” J. Inverse Ill-Posed Problems 11(1), 1–26 (2003).
Yu. E. Anikonov and M. V. Neshchadim, “Analytical Methods of the Theory of Inverse Problems for Parabolic Equations,” Vestnik Novosib. Gos. Univ. 11(3), 20–35 (2011) [J. Math. Sci. (New York) 195 (6), 754–770 (2013)].
Yu. E. Anikonov, N. B. Ayupova, V. G. Bardakov, V. P. Golubyatnikov, and M. V. Neshchadim, “Inversion of Mappings and Inverse Problems,” Siberian Electron. Math. Rep. 9, 382–432 (2012) [see URL: http://semr.math.nsc.ru/v9/p382-432.pdf].
Yu. E. Anikonov and M. V. Neshchadim, “Representations for the Solutions and Coefficients of Second-Order Differential Equations,” Sibirsk.Zh. Industr. Mat. 15(4), 17–23 (2012) [J. Appl. Indust. Math. 7 (1), 15–21 (2013)].
Yu. E. Anikonov and M. V. Neshchadim, “Representations for the Solutions and Coefficients of Evolution Equations,” Sibirsk. Zh. Industr. Mat. 16(2), 40–49 (2013) [J. Appl. Indust. Math. 7 (3), 326–334 (2013)].
Yu. E. Anikonov and M. V. Neshchadim, “On Analytical Methods in the Theory of Inverse Problems for Hyperbolic Equations. I,” Sibirsk. Zh. Industr. Mat. 14(1), 27–39 (2011) [J. Appl. Indust. Math. 5 (4), 506–518 (2011)].
E. Hopf, “The Partial Differential Equation u t + uu x = μu xx,” Comm. Pure Appl. Math. 3, 201–230 (1950).
J. D. Cole, “On a Quasi-Linear Parabolic Equation Occurring in Aerodynamics,” Quart. Appl. Math. 9(3), 225–236 (1951).
H. Bateman and A. Erdelyi, Higher Transcendential Functions (McGraW-Hill Book Comp., New York, 1953; Nauka, Moscow, 1973).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © Yu.E. Anikonov, N.B. Ayupova, 2014, published in Sibirskii Zhurnal Industrial’noi Matematiki, 2014, Vol. XVII, No. 4, pp. 31–37.
Rights and permissions
About this article
Cite this article
Anikonov, Y.E., Ayupova, N.B. The Hopf-Cole transformation and multidimensional representations of solutions to evolution equations. J. Appl. Ind. Math. 9, 11–17 (2015). https://doi.org/10.1134/S1990478915010020
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1990478915010020