In this paper, we establish a necessary and sufficient condition for the existence and regularity of the density of the solution to a semilinear stochastic (fractional) heat equation with measure-valued initial conditions. Under a mild cone condition for the diffusion coefficient, we establish the smooth joint density at multiple points. The tool ......
A Dynamical Systems Approach to Theory and Practice
This graduate-level introduction to ordinary differential equations combines both qualitative and numerical analysis of solutions, in line with Poincare's vision for the field over a century ago. Taking into account the remarkable development of dynamical systems since then, the authors present the core topics that every young mathematician of our ......
This book gives an extensive survey of many important topics in the theory of Hamilton-Jacobi equations with particular emphasis on modern approaches and viewpoints. Firstly, the basic well-posedness theory of viscosity solutions for first-order Hamilton-Jacobi equations is covered. Then, the homogenization theory, a very active research topic ......
This text introduces students to the theory and practice of differential equations, which are fundamental to the mathematical formulation of problems in physics, chemistry, biology, economics, and other sciences. The book is ideally suited for undergraduate or beginning graduate students in mathematics, and will also be useful for students in the ......
"A reader with a strong background in mathematics, at least two semesters of calculus, and interest in the social sciences will find the book helpful in learning how this area of mathematics can be used in different applications." -S.L. Sullivan, Catawba College Differential Equations: A Modeling Approach introduces differential equations and differential equation modeling to students and researchers in the social sciences. The text explains the mathematics and theory of differential equations. Graphical methods of analysis are emphasized over formal proofs, making the text even more accessible for newcomers to the subject matter. This volume introduces the subject of ordinary differential equations - as well as systems of such equations - to the social science audience. Social science examples are used extensively, and readers are guided through the most elementary models to much more advanced specifications. Emphasis is placed on easily applied and broadly applicable numerical methods for solving differential equations, thereby avoiding complicated mathematical "tricks" that often do not even work with more interesting nonlinear models. Also, graphical methods of analysis are introduced that allow social scientists to rapidly access the power of sophisticated model specifications. This volume also describes in clear language how to evaluate the stability of a system of differential equations (linear or nonlinear) by using the system's eigenvalues. The mixture of nonlinearity with dynamical systems is a virtual trademark for this author's approach to modeling, and this theme comes through clearly throughout this volume. This volume's clarity of exposition encourages social science students of mathematical modeling to begin working with differential equation models that address complex and sophisticated social theories. Key Features: The text is accessibly written, so that students with minimal mathematical training can understand all of the basic concepts and techniques presented. The author uses social sciences examples to illustrate the relevance of differential equation modeling to readers. Readers can use graphical methods to produce penetrating analysis of differential equation systems. Linear and nonlinear model specifications are explained from a social science perspective. Most interesting differential equation models are nonlinear, and readers need to know how to specify and work with such models in the social sciences.
