TY - JOUR A1 - Makhmudov, K. O. A1 - Makhmudov, O. I. A1 - Tarkhanov, Nikolai Nikolaevich T1 - A nonstandard Cauchy problem for the heat equation JF - Mathematical Notes N2 - We consider the Cauchy problem for the heat equation in a cylinder C (T) = X x (0, T) over a domain X in R (n) , with data on a strip lying on the lateral surface. The strip is of the form S x (0, T), where S is an open subset of the boundary of X. The problem is ill-posed. Under natural restrictions on the configuration of S, we derive an explicit formula for solutions of this problem. Y1 - 2017 U6 - https://doi.org/10.1134/S0001434617070264 SN - 0001-4346 SN - 1573-8876 VL - 102 SP - 250 EP - 260 PB - Pleiades Publ. CY - New York ER - TY - JOUR A1 - Shlapunov, Alexander A1 - Tarkhanov, Nikolai Nikolaevich T1 - Golusin-Krylov formulas in complex analysis JF - Complex variables and elliptic equations N2 - This is a brief survey of a constructive technique of analytic continuation related to an explicit integral formula of Golusin and Krylov (1933). It goes far beyond complex analysis and applies to the Cauchy problem for elliptic partial differential equations as well. As started in the classical papers, the technique is elaborated in generalised Hardy spaces also called Hardy-Smirnov spaces. KW - Analytic continuation KW - inegral formulas KW - Cauchy problem Y1 - 2017 U6 - https://doi.org/10.1080/17476933.2017.1395872 SN - 1747-6933 SN - 1747-6941 VL - 63 IS - 7-8 SP - 1142 EP - 1167 PB - Routledge CY - Abingdon ER - TY - JOUR A1 - Mera, Azal A1 - Stepanenko, Vitaly A. A1 - Tarkhanov, Nikolai Nikolaevich T1 - Successive approximation for the inhomogeneous burgers equation JF - Journal of Siberian Federal University : Mathematics & Physics N2 - The inhomogeneous Burgers equation is a simple form of the Navier-Stokes equations. From the analytical point of view, the inhomogeneous form is poorly studied, the complete analytical solution depending closely on the form of the nonhomogeneous term. KW - Navier-Stokes equations KW - classical solution Y1 - 2018 U6 - https://doi.org/10.17516/1997-1397-2018-11-4-519-531 SN - 1997-1397 SN - 2313-6022 VL - 11 IS - 4 SP - 519 EP - 531 PB - Siberian Federal University CY - Krasnoyarsk ER - TY - JOUR A1 - Elin, Mark A1 - Shoikhet, David A1 - Tarkhanov, Nikolai Nikolaevich T1 - Analytic Semigroups of Holomorphic Mappings and Composition Operators JF - Computational Methods and Function Theory N2 - In this manuscript we provide a review on the classical and resent results related to the problem of analytic extension in parameter for a semigroup of holomorphic self-mappings of the unit ball in a complex Banach space and its relation to the linear continuous semigroup of composition operators. KW - Non-linear semigroups KW - Composition operators KW - Analytic extension KW - Holomorphic mappings Y1 - 2017 U6 - https://doi.org/10.1007/s40315-017-0227-x SN - 1617-9447 SN - 2195-3724 VL - 18 IS - 2 SP - 269 EP - 294 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Fedchenko, Dmitry A1 - Tarkhanov, Nikolai Nikolaevich T1 - A Rado theorem for the porous medium equation JF - Boletin de la Sociedad Matemática Mexicana N2 - We prove that if u is a locally Lipschitz continuous function on an open set chi subset of Rn + 1 satisfying the nonlinear heat equation partial derivative(t)u = Delta(vertical bar u vertical bar(p-1) u), p > 1, weakly away from the zero set u(-1) (0) in chi, then u is a weak solution to this equation in all of chi. KW - Quasilinear equations KW - Removable sets KW - Porous medium equation Y1 - 2017 U6 - https://doi.org/10.1007/s40590-017-0169-3 SN - 1405-213X SN - 2296-4495 VL - 24 IS - 2 SP - 427 EP - 437 PB - Springer CY - Cham ER - TY - JOUR A1 - Mera, Azal A1 - Shlapunov, Alexander A. A1 - Tarkhanov, Nikolai Nikolaevich T1 - Navier-Stokes Equations for Elliptic Complexes JF - Journal of Siberian Federal University. Mathematics & Physics N2 - We continue our study of invariant forms of the classical equations of mathematical physics, such as the Maxwell equations or the Lam´e system, on manifold with boundary. To this end we interpret them in terms of the de Rham complex at a certain step. On using the structure of the complex we get an insight to predict a degeneracy deeply encoded in the equations. In the present paper we develop an invariant approach to the classical Navier-Stokes equations. KW - Navier-Stokes equations KW - classical solution Y1 - 2019 U6 - https://doi.org/10.17516/1997-1397-2019-12-1-3-27 SN - 1997-1397 SN - 2313-6022 VL - 12 IS - 1 SP - 3 EP - 27 PB - Sibirskij Federalʹnyj Universitet CY - Krasnojarsk ER - TY - JOUR A1 - Malass, Ihsane A1 - Tarkhanov, Nikolai Nikolaevich T1 - The de Rham Cohomology through Hilbert Space Methods JF - Journal of Siberian Federal University. Mathematics & physics N2 - We discuss canonical representations of the de Rham cohomology on a compact manifold with boundary. They are obtained by minimising the energy integral in a Hilbert space of differential forms that belong along with the exterior derivative to the domain of the adjoint operator. The corresponding Euler-Lagrange equations reduce to an elliptic boundary value problem on the manifold, which is usually referred to as the Neumann problem after Spencer. KW - De Rham complex KW - cohomology KW - Hodge theory KW - Neumann problem Y1 - 2019 U6 - https://doi.org/10.17516/1997-1397-2019-12-4-455-465 SN - 1997-1397 SN - 2313-6022 VL - 12 IS - 4 SP - 455 EP - 465 PB - Sibirskij Federalʹnyj Universitet CY - Krasnoyarsk ER - TY - JOUR A1 - Alsaedy, Ammar A1 - Tarkhanov, Nikolai Nikolaevich T1 - A Hilbert Boundary Value Problem for Generalised Cauchy-Riemann Equations JF - Advances in applied Clifford algebras N2 - We elaborate a boundary Fourier method for studying an analogue of the Hilbert problem for analytic functions within the framework of generalised Cauchy-Riemann equations. The boundary value problem need not satisfy the Shapiro-Lopatinskij condition and so it fails to be Fredholm in Sobolev spaces. We show a solvability condition of the Hilbert problem, which looks like those for ill-posed problems, and construct an explicit formula for approximate solutions. KW - Dirac operator KW - Clifford algebra KW - Riemann-Hilbert problem KW - Fredholm operators Y1 - 2017 U6 - https://doi.org/10.1007/s00006-016-0676-8 SN - 0188-7009 SN - 1661-4909 VL - 27 SP - 931 EP - 953 PB - Springer CY - Basel ER - TY - JOUR A1 - Tarkhanov, Nikolai Nikolaevich T1 - Deformation quantization and boundary value problems JF - International journal of geometric methods in modern physics : differential geometery, algebraic geometery, global analysis & topology N2 - We describe a natural construction of deformation quantization on a compact symplectic manifold with boundary. On the algebra of quantum observables a trace functional is defined which as usual annihilates the commutators. This gives rise to an index as the trace of the unity element. We formulate the index theorem as a conjecture and examine it by the classical harmonic oscillator. KW - Symplectic manifold KW - star product KW - trace KW - index Y1 - 2016 U6 - https://doi.org/10.1142/S0219887816500079 SN - 0219-8878 SN - 1793-6977 VL - 13 SP - 176 EP - 195 PB - World Scientific CY - Singapore ER - TY - JOUR A1 - Ly, Ibrahim A1 - Tarkhanov, Nikolai Nikolaevich T1 - A Rado theorem for p-harmonic functions JF - Boletin de la Sociedad Matemática Mexicana N2 - Let A be a nonlinear differential operator on an open set X subset of R-n and S a closed subset of X. Given a class F of functions in X, the set S is said to be removable for F relative to A if any weak solution of A(u) = 0 in XS of class F satisfies this equation weakly in all of X. For the most extensively studied classes F, we show conditions on S which guarantee that S is removable for F relative to A. KW - Quasilinear equations KW - Removable sets KW - p-Laplace equation Y1 - 2016 U6 - https://doi.org/10.1007/s40590-016-0109-7 SN - 1405-213X SN - 2296-4495 VL - 22 SP - 461 EP - 472 PB - Springer CY - Basel ER -