@article{MakhmudoMakhmudovTarkhanov2011,
  author    = {Makhmudo, K. O. and Makhmudov, O. I. and Tarkhanov, Nikolai Nikolaevich},
  title     = {Equations of Maxwell type},
  series = {Journal of mathematical analysis and applications},
  volume    = {378},
  journal   = {Journal of mathematical analysis and applications},
  number    = {1},
  publisher = {Elsevier},
  address   = {San Diego},
  issn      = {0022-247X},
  doi       = {10.1016/j.jmaa.2011.01.012},
  pages     = {64 -- 75},
  year      = {2011},
  abstract  = {For an elliptic complex of first order differential operators on a smooth manifold X, we define a system of two equations which can be thought of as abstract Maxwell equations. The formal theory of this system proves to be very similar to that of classical Maxwell's equations. The paper focuses on boundary value problems for the abstract Maxwell equations, especially on the Cauchy problem.},
  language  = {en}
}
@article{WeissHuisinga2011,
  author    = {Weiss, Andrea Y. and Huisinga, Wilhelm},
  title     = {Error-controlled global sensitivity analysis of ordinary differential equations},
  series = {Journal of computational physics},
  volume    = {230},
  journal   = {Journal of computational physics},
  number    = {17},
  publisher = {Elsevier},
  address   = {San Diego},
  issn      = {0021-9991},
  doi       = {10.1016/j.jcp.2011.05.011},
  pages     = {6824 -- 6842},
  year      = {2011},
  abstract  = {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.},
  language  = {en}
}
@article{BaerWafo2015,
  author    = {B{\"a}r, Christian and Wafo, Roger Tagne},
  title     = {Initial value problems for wave equations on manifolds},
  series = {Mathematical physics, analysis and geometry : an international journal devoted to the theory and applications of analysis and geometry to physics},
  volume    = {18},
  journal   = {Mathematical physics, analysis and geometry : an international journal devoted to the theory and applications of analysis and geometry to physics},
  number    = {1},
  publisher = {Springer},
  address   = {Dordrecht},
  issn      = {1385-0172},
  doi       = {10.1007/s11040-015-9176-7},
  pages     = {29},
  year      = {2015},
  abstract  = {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.},
  language  = {en}
}
@article{Ly2020,
  author    = {Ly, Ibrahim},
  title     = {A Cauchy problem for the Cauchy-Riemann operator},
  series = {Afrika Matematika},
  volume    = {32},
  journal   = {Afrika Matematika},
  number    = {1-2},
  publisher = {Springer},
  address   = {Heidelberg},
  issn      = {1012-9405},
  doi       = {10.1007/s13370-020-00810-4},
  pages     = {69 -- 76},
  year      = {2020},
  abstract  = {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.},
  language  = {en}
}