@misc{AncoAliWolf2011, author = {Anco, Stephen and Ali, Sajid and Wolf, Thomas}, title = {Exact solutions of nonlinear partial differential equations by the method of group foliation reduction}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-13171}, number = {11-25}, year = {2011}, abstract = {A novel symmetry method for finding exact solutions to nonlinear PDEs is illustrated by applying it to a semilinear reaction-diffusion equation in multi-dimensions. The method is based on group foliation reduction and employs a separation ansatz to solve an equivalent first-order group foliation system whose independent and dependent variables respectively consist of the invariants and differential invariants of a given one-dimensional group of point symmetries for the reaction-diffusion equation. With this method, solutions of the reaction-diffusion equation are obtained in an explicit form, including group-invariant similarity solutions and travelling-wave solutions, as well as dynamically interesting solutions that are not invariant under any of the point symmetries admitted by this equation.}, language = {en} } @misc{AncoAliWolf2011, author = {Anco, Stephen and Ali, Sajid and Wolf, Thomas}, title = {Symmetry analysis and exact solutions of semilinear heat flow in multi-dimensions}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-13213}, number = {11-26}, year = {2011}, abstract = {A symmetry group method is used to obtain exact solutions for a semilinear radial heat equation in \$n\>1\$ dimensions with a general power nonlinearity. The method involves an ansatz technique to solve an equivalent first-order PDE system of similarity variables given by group foliations of this heat equation, using its admitted group of scaling symmetries. This technique yields explicit similarity solutions as well as other explicit solutions of a more general (non-similarity) form having interesting analytical behavior connected with blow up and dispersion. In contrast, standard similarity reduction of this heat equation gives a semilinear ODE that cannot be explicitly solved by familiar integration techniques such as point symmetry reduction or integrating factors.}, language = {en} } @misc{Sullivan2016, author = {Sullivan, T. J.}, title = {Well-posed Bayesian inverse problems and heavy-tailed stable Banach space priors}, issn = {1438-0064}, doi = {10.3934/ipi.2017040}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-59422}, year = {2016}, abstract = {This article extends the framework of Bayesian inverse problems in infinite-dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451-559, 2010) and others, to the case of a heavy-tailed prior measure in the family of stable distributions, such as an infinite-dimensional Cauchy distribution, for which polynomial moments are infinite or undefined. It is shown that analogues of the Karhunen-Lo{\`e}ve expansion for square-integrable random variables can be used to sample such measures. Furthermore, under weaker regularity assumptions than those used to date, the Bayesian posterior measure is shown to depend Lipschitz continuously in the Hellinger metric upon perturbations of the misfit function and observed data.}, language = {en} } @misc{Wolf1999, author = {Wolf, Thomas}, title = {A comparison of four approaches to the calculation of conservation laws}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3896}, number = {SC-99-01}, year = {1999}, abstract = {The paper compares computational aspects of four approaches to compute conservation laws of single differential equations or systems of them, ODEs and PDEs. The only restriction, required by two of the four corresponding computer algebra programs, is that each DE has to be solvable for a leading derivative. Extra constraints may be given. Examples of new conservation laws include non-polynomial expressions, an explicit variable dependence and conservation laws involving arbitrary functions. Examples involve the following equations: Ito, Liouville, Burgers, Kadomtsev-Petviashvili, Karney-Sen-Chu-Verheest, Boussinesq, Tzetzeica, Benney.}, language = {en} } @misc{SokolovWolf1999, author = {Sokolov, Vladimir V. and Wolf, Thomas}, title = {A symmetry test for quasilinear coupled systems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3901}, number = {SC-99-02}, year = {1999}, abstract = {It is well known that the following class of systems of evolution equations \begin{eqnarray} \label{nsgen} \cases{ u_{t}=u_{xx}+F(u,v,u_x,v_x),\cr v_{t}=-v_{xx}+G(u,v,u_x,v_x),\cr} \end{eqnarray} is very rich in integrable cases. The complete classification problem is very difficult. Here we consider only the most interesting (from our opinion) subclass of systems (1). Namely, we consider equations linear in all derivatives of the form \begin{eqnarray} \label{kvazgen} \cases{ u_t = u_{xx} + A_{1}(u,v) u_x + A_{2}(u,v) v_x + A_{0}(u,v)\cr v_t = - v_{xx} + B_{1}(u,v) v_x + B_{2}(u,v) u_x + B_{0}(u,v). \cr} \end{eqnarray} without any restrictions on the functions \$A_{i}(u,v), B_{i}(u,v)\$.}, language = {en} } @misc{WolfBrandtMohammadzadeh1998, author = {Wolf, Thomas and Brandt, Andreas and Mohammadzadeh, Majid}, title = {Computer algebra algorithms and routines for the computation of conservation laws and fixing of gauge in differential expressions}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3482}, number = {SC-98-05}, year = {1998}, abstract = {Three different approaches for the determination of conservation laws of differential equations are presented. For three corresponding REDUCE computer algebra programs CONLAW1/2/3 the necessary subroutines are discribed. One of them simplifies general solutions of overdetermined PDE systems so that all remaining free functions and constants correspond to independent conservation laws. It determines redundant functions and constants in differential expressions and is equally useful for the determination of symmetries or the fixing of gauge freedom in differential expressions.}, language = {en} } @misc{EulerEulerWolf2012, author = {Euler, Marianna and Euler, Norbert and Wolf, Thomas}, title = {The Two-Component Camassa-Holm Equations CH(2,1) and CH(2,2): First-Order Integrating Factors and Conservation Laws}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-16087}, year = {2012}, abstract = {Recently, Holm and Ivanov, proposed and studied a class of multi-component generalisations of the Camassa-Holm equations [D D Holm and R I Ivanov, Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples, {\it J. Phys A: Math. Theor} {\bf 43}, 492001 (20pp), 2010]. We consider two of those systems, denoted by Holm and Ivanov by CH(2,1) and CH(2,2), and report a class of integrating factors and its corresponding conservation laws for these two systems. In particular, we obtain the complete sent of first-order integrating factors for the systems in Cauchy-Kovalevskaya form and evaluate the corresponding sets of conservation laws for CH(2,1) and CH(2,2).}, language = {en} } @misc{AncoMacNaughtonWolf2012, author = {Anco, Stephen and MacNaughton, Steven A. and Wolf, Thomas}, title = {Conservation laws and symmetries of quasilinear radial wave equations in multi-dimensions}, issn = {1438-0064}, arxiv = {http://arxiv.org/abs/1109.1719}, doi = {10.1063/1.4711814}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-16111}, year = {2012}, abstract = {Symmetries and conservation laws are studied for two classes of physically and analytically interesting radial wave equations with power nonlinearities in multi-dimensions. The results consist of two main classifications: all symmetries of point type and all conservation laws of a general energy-momentum type are explicitly determined, including those such as dilations, inversions, similarity energies and conformal energies that exist only for special powers or dimensions. In particular, all variational cases (when a Lagrangian formulation exists) and non-variational cases (when no Lagrangian exists) for these wave equations are considered. As main results, the classification yields generalized energies and radial momenta in certain non-variational cases, which are shown to arise from a new type of Morawetz dilation identity that produces conservation laws for each of the two classes of wave equations in a different way than Noether's theorem.}, language = {en} } @misc{AncoMohiuddinWolf2012, author = {Anco, Stephen and Mohiuddin, Mohammad and Wolf, Thomas}, title = {Travelling waves and conservation laws for complex mKdV-type equations}, issn = {1438-0064}, arxiv = {http://arxiv.org/abs/1110.2403}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-16124}, year = {2012}, abstract = {Travelling waves and conservation laws are studied for a wide class of \$U(1)\$-invariant complex mKdV equations containing the two known integrable generalizations of the ordinary (real) mKdV equation. The main results on travelling waves include deriving new complex solitary waves and kinks that generalize the well-known mKdV \$\sech\$ and \$\tanh\$ solutions. The main results on conservation laws consist of explicitly finding all 1st order conserved densities that yield phase-invariant counterparts of the well-known mKdV conserved densities for momentum, energy, and Galilean energy, and a new conserved density describing the angular twist of complex kink solutions.}, language = {en} } @misc{WeiserErdmannSchenkletal.2017, author = {Weiser, Martin and Erdmann, Bodo and Schenkl, Sebastian and Muggenthaler, Holger and Hubig, Michael and Mall, Gita and Zachow, Stefan}, title = {Uncertainty in Temperature-Based Determination of Time of Death}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-63818}, year = {2017}, abstract = {Temperature-based estimation of time of death (ToD) can be per- formed either with the help of simple phenomenological models of corpse cooling or with detailed mechanistic (thermodynamic) heat transfer mod- els. The latter are much more complex, but allow a higher accuracy of ToD estimation as in principle all relevant cooling mechanisms can be taken into account. The potentially higher accuracy depends on the accuracy of tissue and environmental parameters as well as on the geometric resolution. We in- vestigate the impact of parameter variations and geometry representation on the estimated ToD based on a highly detailed 3D corpse model, that has been segmented and geometrically reconstructed from a computed to- mography (CT) data set, differentiating various organs and tissue types. From that we identify the most crucial parameters to measure or estimate, and obtain a local uncertainty quantifcation for the ToD.}, language = {en} }