Analysis
Department of Mathematics and Statistics
Texas Tech University
We study systems of nonlinear ordinary differential equations where the dominant term, with respect to large spatial variables, causes blow-ups and is positively homogeneous of a degree $1+\alpha$ for some $\alpha>0$. We prove that the asymptotic behavior of a solution $y(t)$ near a finite blow-up time $T_*$ is $(T_*-t)^{-1/\alpha}\xi_*$ for some nonzero vector $\xi_*$. Specific error estimates for $|(T_*-t)^{1/\alpha}y(t)-\xi_*|$ are provided. In some typical cases, they can be a positive power of $(T_*-t)$ or $1/|\ln(T_*-t)|$. This depends on whether the decaying rate of the lower order term, relative to the size of the dominant term, is of a power or logarithmic form. Similar results are obtained for a class of nonlinear differential inequalities with finite time blow-up solutions. Our results cover larger classes of nonlinear equations, differential inequalities and error estimates than those in the previous work.
We study systems of nonlinear ordinary differential equations where the dominant term, with respect to large spatial variables, causes blow-ups and is positively homogeneous of a degree $1+\alpha$ for some $\alpha>0$. We prove that the asymptotic behavior of a solution $y(t)$ near a finite blow-up time $T_*$ is $(T_*-t)^{-1/\alpha}\xi_*$ for some nonzero vector $\xi_*$. Specific error estimates for $|(T_*-t)^{1/\alpha}y(t)-\xi_*|$ are provided. In some typical cases, they can be a positive power of $(T_*-t)$ or $1/|\ln(T_*-t)|$. This depends on whether the decaying rate of the lower order term, relative to the size of the dominant term, is of a power or logarithmic form. Similar results are obtained for a class of nonlinear differential inequalities with finite time blow-up solutions. An application to a model of inhomogeneous populations will be given.
When are two geometric shapes identical or similar? This question has intrigued mathematicians from ancient times with its roots going back to Euclid's Elements. In particular, in Elements Euclid provided several answers to the question: When are two triangles congruent and when they are similar? The answer to this innocent-looking question depends on what tools for measurement are available and which characteristics of a triangle we measure. In a classical Euclidean setting, we measure lengths of sides and angles of triangles. But often these measurements are not available. For example, if we hear a triangular drum playing on a neighboring street can we recognize the shape from its tones? Or, staying away from a triangular oven, we feel temperature when one of its sides is heated. If we know the temperature for each of the sides, can we recognize the shape of a triangle? These and some other questions, as well as some open problems, will be discussed in this talk. This topic belongs to the intersection of areas of Elementary Geometry, Complex Analysis, Mathematical Physics, and PDE's. Furthermore, Transcendental Special Functions appear in this study rather naturally. So, all people who are interested in these matters are invited to attend. Because triangles as a generic geometric configuration are simplest possible, most problems are understandable for graduate and undergraduate students (assuming they are familiar with elements of Euclidean geometry) and can be good topics for student research projects.
The present work aims to propose a computational technique for reconstructing the spatially complex-valued potential in the time-dependent Schrödinger equation from two time measurements and Cauchy boundary data. This reconstruction process also aids in determining the intensity function. Our inverse solver is derived based on some special transformations and the so-called Fourier--Klibanov basis, leading to a block system of coupled elliptic PDEs with Cauchy data. The block-coupled elliptic system is approximated by minimizing a weighted Tikhonov-like cost functional in a partially discrete setting. In this scenario, we rely on a general 1D Carleman estimate to prove the Fréchet differentiability, strong convexity of the functional, which further leads to the $L^2$-type error estimate of the gradient descent method. Some numerical results are provided to verify the performance of the proposed inversion. This is a joint work with Dr. Vo Anh Khoa, Texas Tech University.
This Analysis seminar may be attended Monday the 27th at 4:00 PM CDT (UTC-5) via this Zoom link.
Meeting ID: 947 7432 8481
Passcode: 745829