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Stefan Problem
In mathematics and its applications, particularly to phase transitions in matter, a Stefan problem is a particular kind of boundary value problem for a system of partial differential equations (PDE), in which the boundary between the phases can move with time. The classical Stefan problem aims to describe the evolution of the boundary between two phases of a material undergoing a phase change, for example the melting of a solid, such as ice to water. This is accomplished by solving heat equations in both regions, subject to given boundary and initial conditions. At the interface between the phases (in the classical problem) the temperature is set to the phase change temperature. To close the mathematical system a further equation, the Stefan condition, is required. This is an energy balance which defines the position of the moving interface. Note that this evolving boundary is an unknown (hyper-)surface; hence, Stefan problems are examples of free boundary problems. Analogou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transcendental Equation
In applied mathematics, a transcendental equation is an equation over the real (or complex) numbers that is not algebraic, that is, if at least one of its sides describes a transcendental function. Examples include: :\begin x &= e^ \\ x &= \cos x \\ 2^x &= x^2 \end A transcendental equation need not be an equation between elementary functions, although most published examples are. In some cases, a transcendental equation can be solved by transforming it into an equivalent algebraic equation. Some such transformations are sketched below; computer algebra systems may provide more elaborated transformations. In general, however, only approximate solutions can be found. Transformation into an algebraic equation Ad hoc methods exist for some classes of transcendental equations in one variable to transform them into algebraic equations which then might be solved. Exponential equations If the unknown, say ''x'', occurs only in exponents: * applying the natural logarithm t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stefan Number
The Stefan number (St or Ste) is defined as the ratio of sensible heat to latent heat. It is given by the formula \mathrm = \frac, where * ''cp'' is the specific heat, ** cp is the specific heat of solid phase in the freezing process while cp is the specific heat of liquid phase in the melting process. * ∆''T'' is the temperature difference between phases, * ''L'' is the latent heat of melting. It is a dimensionless parameter that is useful in analyzing a Stefan problem. The parameter was developed from Josef Stefan's calculations of the rate of phase change of water into ice Ice is water that is frozen into a solid state, typically forming at or below temperatures of 0 ° C, 32 ° F, or 273.15 K. It occurs naturally on Earth, on other planets, in Oort cloud objects, and as interstellar ice. As a naturally oc ... on the polar ice caps and coined by G.S.H. Lock in 1969. The problems origination is fully described by Vuik and further commentary on its ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sensible Heat
Sensible heat is heat exchanged by a body or thermodynamic system in which the exchange of heat changes the temperature of the body or system, and some macroscopic variables of the body or system, but leaves unchanged certain other macroscopic variables of the body or system, such as volume or pressure. Usage The term is used in contrast to a latent heat, which is the amount of heat exchanged that is hidden, meaning it occurs without change of temperature. For example, during a phase change such as the melting of ice, the temperature of the system containing the ice and the liquid is constant until all ice has melted. Latent and sensible heat are complementary terms. The sensible heat of a thermodynamic process may be calculated as the product of the body's mass (''m'') with its specific heat capacity (''c'') and the change in temperature (\Delta T): : Q_ = m c \Delta T \, . ''Sensible heat'' and ''latent heat'' are not special forms of energy. Rather, they describe exchanges ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nondimensionalization
Nondimensionalization is the partial or full removal of physical dimensions from an equation involving physical quantities by a suitable substitution of variables. This technique can simplify and parameterize problems where measured units are involved. It is closely related to dimensional analysis. In some physical systems, the term ''scaling'' is used interchangeably with ''nondimensionalization'', in order to suggest that certain quantities are better measured relative to some appropriate unit. These units refer to quantities intrinsic to the system, rather than units such as SI units. Nondimensionalization is not the same as converting extensive quantities in an equation to intensive quantities, since the latter procedure results in variables that still carry units. Nondimensionalization can also recover characteristic properties of a system. For example, if a system has an intrinsic resonance frequency, length, or time constant, nondimensionalization can recover these ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The American Mathematical Society
The ''Journal of the American Mathematical Society'' (''JAMS''), is a quarterly peer-reviewed mathematical journal published by the American Mathematical Society. It was established in January 1988. Abstracting and indexing This journal is abstracted and indexed in: 2011. American Mathematical Society. * Mathematical Reviews * Zentralblatt MATH * Science Citation Index * ISI Alerting Services * CompuMath Citation Index * [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Xavier Ros-Oton
Xavier Ros Oton (Barcelona, 1988) is a Catalan mathematician who works on partial differential equations (PDEs). He is an ICREA Research Professor and a Full Professor at the University of Barcelona. Research His research is mainly focused on topics related to the regularity of solutions to nonlinear elliptic and parabolic PDE. Some of his main contributions have been in the context of free boundary problems, integro-differential equations, and the calculus of variations. Career He earned his Bachelor's and master's degree at the Universitat Politècnica de Catalunya in 2010 and 2011, and completed his PhD in 2014 under the supervision of Xavier Cabré. He then moved to the University of Texas at Austin, where he was an R. H. Bing Instructor, and worked with Alessio Figalli and Luis Caffarelli. After that, he was an assistant professor at the University of Zurich. Since 2020, Ros-Oton is an ICREA Research Professor at the University of Barcelona. He is a member of the editori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alessio Figalli
Alessio Figalli (; born 2 April 1984) is an Italian mathematician working primarily on the calculus of variations and partial differential equations. He was awarded the Peccot-Vimont Prize and the Peccot Lectures in 2012, the EMS Prize in 2012, the Stampacchia Medal in 2015, the Feltrinelli Prize in 2017, and the Fields Medal in 2018. He was an invited speaker at the International Congress of Mathematicians 2014. In 2016 he was awarded a European Research Council (ERC) grant, and in 2018 he received the Doctorate ''Honoris Causa'' from the Université Côte d'Azur. In 2019, he received the Doctorate ''Honoris Causa'' from the Polytechnic University of Catalonia. Biography Figalli received his master's degree from the University of Pisa in 2006 (as a student of the Scuola Normale Superiore di Pisa), and earned his doctorate in 2007 under the supervision of Luigi Ambrosio at the Scuola Normale Superiore di Pisa and Cédric Villani at the École Normale Supérieure de Lyon. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Luis Caffarelli
Luis Ángel Caffarelli (; born December 8, 1948) is an Argentine-American mathematician. He studies partial differential equations and their applications. Caffarelli is a professor of mathematics at the University of Texas at Austin, and the winner of the 2023 Abel Prize. Career Caffarelli was born and grew up in Buenos Aires. He obtained his Masters of Science (1968) and Ph.D. (1972) at the University of Buenos Aires. His Ph.D. advisor was Calixto Calderón. He currently holds the Sid Richardson Chair at the University of Texas at Austin and is core faculty at the Oden Institute for Computational Engineering and Sciences. He also has been a professor at the University of Minnesota, the University of Chicago, and the Courant Institute of Mathematical Sciences at New York University. From 1986 to 1996 he was a professor at the Institute for Advanced Study in Princeton. Research Caffarelli published "The regularity of free boundaries in higher dimensions" in 1977 in ''Acta Math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Latent Heat
Latent heat (also known as latent energy or heat of transformation) is energy released or absorbed, by a body or a thermodynamic system, during a constant-temperature process—usually a first-order phase transition, like melting or condensation. Latent heat can be understood as hidden energy which is supplied or extracted to change the state of a substance without changing its temperature or pressure. This includes the latent heat of fusion (solid to liquid), the latent heat of vaporization (liquid to gas) and the latent heat of sublimation (solid to gas). The term was introduced around 1762 by Scottish chemist Joseph Black. Black used the term in the context of calorimetry where a heat transfer caused a volume change in a body while its temperature was constant. In contrast to latent heat, sensible heat is energy transferred as heat, with a resultant temperature change in a body. Usage The terms ''sensible heat'' and ''latent heat'' refer to energy transferred between a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heat Flux
In physics and engineering, heat flux or thermal flux, sometimes also referred to as heat flux density, heat-flow density or heat-flow rate intensity, is a flow of energy per unit area per unit time (physics), time. Its SI units are watts per square metre (W/m2). It has both a direction and a magnitude, and so it is a Vector (geometric), vector quantity. To define the heat flux at a certain point in space, one takes the Limiting case (mathematics), limiting case where the size of the surface becomes infinitesimally small. Heat flux is often denoted \vec_\mathrm, the subscript specifying ''heat'' flux, as opposed to ''Mass flux, mass'' or Transport phenomena, ''momentum'' flux. Heat conduction#Fourier's law, Fourier's law is an important application of these concepts. Fourier's law For most solids in usual conditions, heat is transported mainly by thermal conduction, conduction and the heat flux is adequately described by Fourier's law. Fourier's law in one dimension \phi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
