TY - JOUR A1 - Ly, Ibrahim T1 - A Cauchy problem for the Cauchy-Riemann operator JF - Afrika Matematika N2 - We study the Cauchy problem for a nonlinear elliptic equation with data on a piece S of the boundary surface partial derivative X. By the Cauchy problem is meant any boundary value problem for an unknown function u in a domain X with the property that the data on S, if combined with the differential equations in X, allows one to determine all derivatives of u on S by means of functional equations. In the case of real analytic data of the Cauchy problem, the existence of a local solution near S is guaranteed by the Cauchy-Kovalevskaya theorem. We discuss a variational setting of the Cauchy problem which always possesses a generalized solution. KW - nonlinear PDI KW - Cauchy problem KW - Zaremba problem Y1 - 2020 U6 - https://doi.org/10.1007/s13370-020-00810-4 SN - 1012-9405 SN - 2190-7668 VL - 32 IS - 1-2 SP - 69 EP - 76 PB - Springer CY - Heidelberg 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 - Bär, Christian A1 - Wafo, Roger Tagne T1 - Initial value problems for wave equations on manifolds JF - Mathematical physics, analysis and geometry : an international journal devoted to the theory and applications of analysis and geometry to physics N2 - We study the global theory of linear wave equations for sections of vector bundles over globally hyperbolic Lorentz manifolds. We introduce spaces of finite energy sections and show well-posedness of the Cauchy problem in those spaces. These spaces depend in general on the choice of a time function but it turns out that certain spaces of finite energy solutions are independent of this choice and hence invariantly defined. We also show existence and uniqueness of solutions for the Goursat problem where one prescribes initial data on a characteristic partial Cauchy hypersurface. This extends classical results due to Hormander. KW - Wave equation KW - Globally hyperbolic Lorentz manifold KW - Cauchy problem KW - Goursat problem KW - Finite energy sections Y1 - 2015 U6 - https://doi.org/10.1007/s11040-015-9176-7 SN - 1385-0172 SN - 1572-9656 VL - 18 IS - 1 PB - Springer CY - Dordrecht ER - TY - JOUR A1 - Weiss, Andrea Y. A1 - Huisinga, Wilhelm T1 - Error-controlled global sensitivity analysis of ordinary differential equations JF - Journal of computational physics N2 - We propose a novel strategy for global sensitivity analysis of ordinary differential equations. It is based on an error-controlled solution of the partial differential equation (PDE) that describes the evolution of the probability density function associated with the input uncertainty/variability. The density yields a more accurate estimate of the output uncertainty/variability, where not only some observables (such as mean and variance) but also structural properties (e.g., skewness, heavy tails, bi-modality) can be resolved up to a selected accuracy. For the adaptive solution of the PDE Cauchy problem we use the Rothe method with multiplicative error correction, which was originally developed for the solution of parabolic PDEs. We show that, unlike in parabolic problems, conservation properties necessitate a coupling of temporal and spatial accuracy to avoid accumulation of spatial approximation errors over time. We provide convergence conditions for the numerical scheme and suggest an implementation using approximate approximations for spatial discretization to efficiently resolve the coupling of temporal and spatial accuracy. The performance of the method is studied by means of low-dimensional case studies. The favorable properties of the spatial discretization technique suggest that this may be the starting point for an error-controlled sensitivity analysis in higher dimensions. KW - ODE with random initial conditions KW - Global sensitivity analysis KW - Cauchy problem KW - Error control/adaptivity KW - Rothe method KW - Approximate approximations Y1 - 2011 U6 - https://doi.org/10.1016/j.jcp.2011.05.011 SN - 0021-9991 VL - 230 IS - 17 SP - 6824 EP - 6842 PB - Elsevier CY - San Diego ER - TY - INPR A1 - Shlapunov, Alexander A1 - Tarkhanov, Nikolai Nikolaevich T1 - Golusin-Krylov Formulas in Complex Analysis T2 - Preprints des Instituts für Mathematik der Universität Potsdam 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. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 6 (2017) 2 KW - analytic continuation KW - integral formulas KW - Cauchy problem Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-102774 VL - 6 IS - 2 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Alsaedy, Ammar T1 - Variational primitive of a differential form N2 - In this paper we specify the Dirichlet to Neumann operator related to the Cauchy problem for the gradient operator with data on a part of the boundary. To this end, we consider a nonlinear relaxation of this problem which is a mixed boundary problem of Zaremba type for the p-Laplace equation. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 4 KW - Dirichlet-to-Neumann operator KW - Cauchy problem KW - p-Laplace operator KW - calculus of variations Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-89223 SN - 2193-6943 VL - 5 IS - 4 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Makhmudov, K. O. A1 - Makhmudov, O. I. A1 - Tarkhanov, Nikolai Nikolaevich T1 - A nonstandard Cauchy problem for the heat equation N2 - We consider a Cauchy problem for the heat equation in a cylinder X x (0,T) over a domain X in the n-dimensional space 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. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015)11 KW - heat equation KW - Cauchy problem KW - Carleman formulas Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-83830 SN - 2193-6943 VL - 4 IS - 11 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - De-Xing, Kong A1 - Hui, Yao T1 - Global exact boundary controllability of a class of quasilinear hyperbolic systems of conservation laws II N2 - In this paper, by a new constructive method, the authors reprove the global exact boundary controllability of a class of quasilinear hyperbolic systems of conservation laws with linearly degenerate fields. It is shown that the system with nonlinear boundary conditions is globally exactly boundary controllable in the class of piecewise C¹ functions. In particular, the authors give the optimal control time of the system. Finally, a new application is also given. T3 - Preprint - (2003) 08 KW - Quasilinear hyperbolic system KW - conservation laws KW - global exact boundary controllability KW - Cauchy problem KW - Goursat problem Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26565 ER -