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Humboldt University of Berlin (Germany)
Hyperbolic operators with finite time propagation of singularities
Thursday, 25 January 2024 at 12:00
On line: https://meet.google.com/nif-zfwd-mrg
Abstract
We will discuss a class of (nonlinear) initial-boundary value problems for first order 1D (integro)-differential strictly hyperbolic systems which are distinguished by finite time propagation of singularities. We characterize the last property in terms of boundary and/or equation coefficients. This class of problems exhibits a number of interesting mathematical phenomena such as the smoothing effect for solutions, the superstability property, robustness of exponential dichotomy, the Fredholm property and the absence of small divisors in periodic problems. Our analysis has applications to solving inverse problems and finding global classical solutions to quasilinear problems.
University of Split, Croatia
Fractional integral inequalities for convex functions
Thursday, 30 November 2023 at 17:10
On line: https://meet.google.com/nif-zfwd-mrg
Abstract
In recent years, considerable interest in the theory of fractional calculus has been stimulated due to its many applications in almost all applied sciences, especially in numerical analysis and various fields of physics and engineering. Fractional calculus has enabled the adoption of a theoretical model based on experimental data. Inequalities which involve integrals of functions and their derivatives, whose study has a history of about a century, are of great importance in mathematics, with far-reaching applications in the theory of differential equations, approximations, and probability, among others. Fractional differentiation inequalities have applications to fractional differential equations; the most important ones are in establishing uniqueness of the solution of initial problems and giving upper bounds to their solutions. These applications have motivated many researchers in the field of integral inequalities to investigate certain extensions and generalizations using different fractional differential and integral operators. Our recently introduced new class of (h,g.m)-convex functions unifies a certain range of convexity, thus allowing the generalizations of know results. The goal is to apply inequalities of classical types to this class and to give them in more general settings using fractional calculus, for the purpose of their extension, generalization and improvement of boundary evaluations.
Seminar video report
Technical University of Braunschweig, Germany
d.langemann@tu-bs.de
Reaction-Diffusion Equations for Modeling Locally Resolved Life-Science Applications
Thursday, 26 October 2023 at 14:15
On line: https://meet.google.com/nif-zfwd-mrg
Abstract
Life-science applications like the course of liver infections, the development of resistant plants or bacteria or the spreading of wildfires are described by local reaction terms and non-local terms describing diffusive or convective migration of the effects. That leads to the interesting class of reaction-diffusion equations which provides surprising phenomena. After the introduction of the life-science applications, we will discuss the typical solution behaviour like travelling waves and Turing instabilities, which might occur when leveling effects are combined. Finally, we demonstrate the consequences for the modeled applications.
Seminar video report
Sergei Solodky
Institute of Mathematics of NASU, Ukraine,
University of Giessen, Germany
solodky@imath.kiev.ua
Optimization of Numerical Differentiation Methods.
Approximation and Information Aspects
Friday, 20 October 2023 at 12:00
On line: https://meet.google.com/afw-vnwo-dfb
Abstract
In the talk, I will incorporate the so-called self-regularization into the numerical differentiation of multivariable functions. The proposed approach is a combination of the truncation method and a discretization scheme using the idea of a hyperbolic cross. It will be shown that numerical differentiation methods constructed in this way not only have a simple implementation but are also optimal in terms of the accuracy and volume of discrete information used.
Seminar video report
Universität Osnabrück, Germany
i.m.kovalyov@gmail.com
Indefinite Stieltjes Moment Problem and Jacobi Matrices
Friday, 29 September 2023 at 13:00
On line: https://meet.google.com/nif-zfwd-mrg
Abstract
Let \(\mathbf{J}\) be a monic generalized Jacobi matrix, i.e. a three-diagonal block matrix of a special form, introduced by M. Derevyagin and V. Derkach in 2004. We find conditions for a monic generalized Jacobi matrix \(\mathbf{J}\) to admit a factorization \(\mathbf{J} = \mathbf{L}\mathbf{U}\) with \(\mathbf{L}\) and \(\mathbf{U}\) being lower and upper triangular two-diagonal block matrices of a special form. In this case the Darboux transformation of \(\mathbf{J}\) defined by \(\mathbf{J}^p = \mathbf{L}\mathbf{U}\) it is shown to be also a monic generalized Jacobi matrix. Analogues of Christoffel formulas for polynomials of the first and the second kind, corresponding to the Darboux transformation \(\mathbf{J}^p\) are found. As was known, the generalized Jacobi matrix is associated with the indefinite Stieltjes moment problem. Hence, we consider one in the class \(N_{\kappa}^k\) of generalized Stieltjes functions. The set of the solution can be described by Schur step-by-step algorithm, which is based on the expansion of the solutions in the generalized Stieltjes continued fraction. The resolvent matrix is represented in terms of generalized Stieltjes polynomials.
Seminar video report
Jaan Janno
Tallinn University of Technology, Estonia
jaan.janno@taltech.ee
On some inverse problems that use nonlocality of fractional derivatives
Tuesday, 20 June 2023 at 13:00
On line: https://meet.google.com/nif-zfwd-mrg
Abstract
We consider some inverse problems for time-fractional diffusion equation of the order between 0 and 1. In the first class of problems, time-dependent source factors are reconstructed from final data. The proof of uniqueness uses power-type asymptotics of Mittag-Leffler functions involved in solution formulas of corresponding direct problems. This is related to the nonlocality of the fractional derivative included in the diffusion equation. In the second class of problems, the data are given in an arbitrarily small time neighborhood of the final value. Then, due to the nonlocality of a fractional derivative, a function given in a such neighborhood can be extended backward in time to the initial time value provided the fractional derivative of this function is also given in the mentioned neighborhood. This basic result is applied to some inverse problems of reconstruction of time- and space-dependent sources.
Seminar video report
University of Bolonga, Italy
simone.ciani3@unibo.it
Anisotropic De Giorgi Classes and the Regularity of Solutions to Some Anisotropic Elliptic Equations
Thursday, 20 April 2023 at 15:00
On line: https://meet.google.com/mna-fdrh-tcm
Abstract
We introduce anisotropic De Giorgi classes through general energy inequalities, to describe an old problem that remained open. We motivate the introduction of such classes by recent results on solutions to anisotropic singular equations of the kind
$$ \sum^{s}_{i=1} \partial_{ii} u + \sum^{N}_{i=s+1} \partial_{i} \left( A_{i} (x, u, \nabla u)\right) = 0, \quad x\in\Omega\subset\subset \mathbb{R}^{N} \text{ for } 1\ge s \ge (N-1)$$
where each operator \(A_{i}\) behaves as the \(p\)-Laplacian with \(1 \lt p \lt 2\). The resolution of this special problem borrows parabolic potential techniques of expansion of positivity to obtain the interior Hölder continuity and some integral and pointwise Harnack inequalities.
Seminar video report
Institute of Hydromechanics of the NAS of Ukraine, Ukraine
yuriy.a.semenov@outlook.com
Nonlinear 2D Problems of Free Surface/Interface Flows
Monday, 10 April 2023 at 13:00
On line: https://meet.google.com/nif-zfwd-mrg
Abstract
The mechanics of flexure-gravity waves are both challenging and exciting. In the last decade, this topic has received much attention due to ice melting in the Arctic Regions and opening new routs for ships. Most of the studies are based on linear theories, while the nonlinear effects play an important role in the comprehensive physical understanding of the phenomena. The development of the nonlinear theory of the flexural-gravity waves requires advanced analytical tools for solving fluid/structure interaction problems. In this seminar talk, we focus our attention on the methodology for solving fluid/structure interaction problems involving free-surface and interface flows with an elastic sheet on the liquid surface. We consider methodology development from the classical hodograph method to the advanced integral hodograph method which was successfully applied for solving water-entry problems, and free surface problems for high-speed planning hulls and hydrofoil crafts. We show how the boundary-value problem can be reduced to a system of integral equations, which are then solved numerically. Special attention will be given to analysing the classical problem of flexural-gravity waves generated by an obstruction on the bottom of the channel. The existence of the solution and the interface shape for both sub and super-critical flow regions will be discussed.
Seminar video report
Supplementary materials: Presentation
University of Limerick, Ireland
eugene.benilov@ul.ie
Solitary and Periodic Waves in a Fifth-Order Korteweg-de Vries Equation
Thursday, 23 March 2023 at 15:00
On line: http://meet.google.com/nif-zfwd-mrg
Abstract
I consider a fifth-order KdV equation, where the fifth-order derivative term is multiplied by a small parameter ε. It has been conjectured that this equation admits a non-local solitary wave solution which has a central core and an oscillatory tail either behind or in front of the core. I shall prove that this solution cannot be exactly steady; instead, the amplitude of the central core decays due to the energy flux generated in the oscillatory tail. The decay rate is calculated in the limit ε → 0. In order to verify the analytical results, I have developed a high-precision spectral method for numerical integration of this equation. The asymptotic and numerical result show good agreement.
Seminar video report
Claremont Graduate University
marina.chugunova@cgu.edu
Mathematical Modeling of Pressure Regimes in Fontan Blood Flow Circulation
Friday, 24 February 2023 at 18:00
On line: https://meet.google.com/nif-zfwd-mrg
Abstract
Babies born with a single functioning heart ventricle instead of the usual two require a series of surgeries during the first few years of life to redirect their blood flow. The resulting circulation, in which systemic venous blood flows directly into the pulmonary arteries, bypassing the heart, is referred to as the Fontan circulation. We develop two mathematical lumped parameter models for blood pressure distribution in the Fontan blood flow circulation: an ODE based spatially homogeneous model and a PDE based spatially inhomogeneous model. Numerical simulations of the ODE model with physiologically consistent input parameters and cardiac cycle pressure-volume outputs reveal the existence of a critical value for pulmonary resistance above which the cardiac output dramatically decreases. Joint work with: M.G. Doyle, J.P. Keener, S.L. Roche and R.M. Taranets.
Seminar video report