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Nonlinear diffusion equations, an important class of parabolic equations, come from a variety of diffusion phenomena which appear widely in nature. They are suggested as mathematical models of physical problems in many fields, such as filtration, phase transition, biochemistry and dynamics of biological groups. In many cases, the equations possess degeneracy or singularity. The appearance of degeneracy or singularity makes the study more involved and challenging. Many new ideas and methods have been developed to overcome the special difficulties caused by the degeneracy and singularity, which enrich the theory of partial differential equations. This book provides a comprehensive presentation of the basic problems, main results and typical methods for nonlinear diffusion equations with degeneracy. Some results for equations with singularity are touched upon.

Burgers equation. --- Heat equation.

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Heat provides the energy that drives almost all geological phenomena and sets the temperature at which these phenomena operate. This book explains the key physical principles of heat transport with simple physical arguments and scaling laws that allow quantitative evaluation of heat flux and cooling conditions in a variety of geological settings and systems. The thermal structure and evolution of magma reservoirs, the crust, the lithosphere and the mantle of the Earth are reviewed within the context of plate tectonics and mantle convection - illustrating how theoretical arguments can be combined with field and laboratory data to arrive at accurate interpretations of geological observations. Appendices contain data on the thermal properties of rocks, surface heat flux measurements and rates of radiogenic heat production. This book can be used for advanced courses in geophysics, geodynamics and magmatic processes, and is a reference for researchers in geoscience, environmental science, physics, engineering and fluid dynamics.

Terrestrial heat flow. --- Terrestrial heat transfer --- Burgers equation --- Earth temperature --- Geophysics --- Heat --- Heat budget (Geophysics) --- Heat equation --- Transmission --- Earth --- Internal structure. --- Earth (Planet)

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Asymptotic expansions. --- Burgers equation. --- Turbulence --- Mathematical models. --- Burgers equation --- Asymptotic expansions --- Burgers, Equation de --- Développements asymptotiques --- 532.517 --- -Flow, Turbulent --- Turbulent flow --- Diffusion equation, Nonlinear --- Heat flow equation, Nonlinear --- Nonlinear diffusion equation --- Nonlinear heat flow equation --- Heat equation --- Asymptotes --- Convergence --- Divergent series --- Liquid motion according to type of flow --- -Liquid motion according to type of flow --- 532.517 Liquid motion according to type of flow --- -Diffusion equation, Nonlinear --- Flow, Turbulent --- Développements asymptotiques --- Navier-Stokes equations --- Asymptotic developments --- Difference equations --- Functions --- Numerical analysis --- Mathematical models --- Mathematical physics --- Fluid mechanics --- Modèles mathématiques --- Turbulence - Mathematical models

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The aim of these notes is to include in a uniform presentation style several topics related to the theory of degenerate nonlinear diffusion equations, treated in the mathematical framework of evolution equations with multivalued m-accretive operators in Hilbert spaces. The problems concern nonlinear parabolic equations involving two cases of degeneracy. More precisely, one case is due to the vanishing of the time derivative coefficient and the other is provided by the vanishing of the diffusion coefficient on subsets of positive measure of the domain. From the mathematical point of view the results presented in these notes can be considered as general results in the theory of degenerate nonlinear diffusion equations. However, this work does not seek to present an exhaustive study of degenerate diffusion equations, but rather to emphasize some rigorous and efficient techniques for approaching various problems involving degenerate nonlinear diffusion equations, such as well-posedness, periodic solutions, asymptotic behaviour, discretization schemes, coefficient identification, and to introduce relevant solving methods for each of them.

Burgers equation --- Degenerate differential equations --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Burgers equation. --- Degenerate differential equations. --- Equations of degenerate type --- Diffusion equation, Nonlinear --- Heat flow equation, Nonlinear --- Nonlinear diffusion equation --- Nonlinear heat flow equation --- Mathematics. --- Partial differential equations. --- Applied mathematics. --- Engineering mathematics. --- Calculus of variations. --- Partial Differential Equations. --- Calculus of Variations and Optimal Control; Optimization. --- Applications of Mathematics. --- Isoperimetrical problems --- Variations, Calculus of --- Maxima and minima --- Engineering --- Engineering analysis --- Mathematical analysis --- Partial differential equations --- Math --- Science --- Differential equations, Partial --- Heat equation --- Navier-Stokes equations --- Turbulence --- Differential equations, partial. --- Mathematical optimization. --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Operations research --- Simulation methods --- System analysis

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Numerical methods are a specific form of mathematics that involve creating and use of algorithms to map out the mathematical core of a practical problem. Numerical methods naturally find application in all fields of engineering, physical sciences, life sciences, social sciences, medicine, business, and even arts. The common uses of numerical methods include approximation, simulation, and estimation, and there is almost no scientific field in which numerical methods do not find a use. Results communicated here include topics ranging from statistics (Detecting Extreme Values with Order Statistics in Samples from Continuous Distributions) and Statistical software packages (dCATCH—A Numerical Package for d-Variate near G-Optimal Tchakaloff Regression via Fast NNLS) to new approaches for numerical solutions (Exact Solutions to the Maxmin Problem max‖Ax‖ Subject to ‖Bx‖≤1; On q-Quasi-Newton’s Method for Unconstrained Multiobjective Optimization Problems; Convergence Analysis and Complex Geometry of an Efficient Derivative-Free Iterative Method; On Derivative Free Multiple-Root Finders with Optimal Fourth Order Convergence; Finite Integration Method with Shifted Chebyshev Polynomials for Solving Time-Fractional Burgers’ Equations) to the use of wavelets (Orhonormal Wavelet Bases on The 3D Ball Via Volume Preserving Map from the Regular Octahedron) and methods for visualization (A Simple Method for Network Visualization).

Research & information: general --- Mathematics & science --- Clenshaw–Curtis–Filon --- high oscillation --- singular integral equations --- boundary singularities --- local convergence --- nonlinear equations --- Banach space --- Fréchet-derivative --- finite integration method --- shifted Chebyshev polynomial --- Caputo fractional derivative --- Burgers’ equation --- coupled Burgers’ equation --- maxmin --- supporting vector --- matrix norm --- TMS coil --- optimal geolocation --- probability computing --- Monte Carlo simulation --- order statistics --- extreme values --- outliers --- multiobjective programming --- methods of quasi-Newton type --- Pareto optimality --- q-calculus --- rate of convergence --- wavelets on 3D ball --- uniform 3D grid --- volume preserving map --- Network --- graph drawing --- planar visualizations --- multiple root solvers --- composite method --- weight-function --- derivative-free method --- optimal convergence --- multivariate polynomial regression designs --- G-optimality --- D-optimality --- multiplicative algorithms --- G-efficiency --- Caratheodory-Tchakaloff discrete measure compression --- Non-Negative Least Squares --- accelerated Lawson-Hanson solver