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This book surveys the latest advances in radial basis function (RBF) meshless collocation methods which emphasis on recent novel kernel RBFs and new numerical schemes for solving partial differential equations. The RBF collocation methods are inherently free of integration and mesh, and avoid tedious mesh generation involved in standard finite element and boundary element methods. This book focuses primarily on the numerical algorithms, engineering applications, and highlights a large class of novel boundary-type RBF meshless collocation methods. These methods have shown a clear edge over the traditional numerical techniques especially for problems involving infinite domain, moving boundary, thin-walled structures, and inverse problems. Due to the rapid development in RBF meshless collocation methods, there is a need to summarize all these new materials so that they are available to scientists, engineers, and graduate students who are interest to apply these newly developed methods for solving real world’s problems. This book is intended to meet this need. Prof. Wen Chen and Dr. Zhuo-Jia Fu work at Hohai University. Prof. C.S. Chen works at the University of Southern Mississippi.

Mathematical physics --- Classical mechanics. Field theory --- Applied physical engineering --- Computer science --- theoretische fysica --- toegepaste mechanica --- computers --- informatica --- informaticaonderzoek --- ingenieurswetenschappen --- mechanica --- computerkunde --- Radial basis functions. --- Collocation methods.

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The basin of attraction of an equilibrium of an ordinary differential equation can be determined using a Lyapunov function. A new method to construct such a Lyapunov function using radial basis functions is presented in this volume intended for researchers and advanced students from both dynamical systems and radial basis functions. Besides an introduction to both areas and a detailed description of the method, it contains error estimates and many examples.

Lyapunov functions. --- Radial basis functions. --- Electronic books. -- local. --- Lyapunov functions --- Radial basis functions --- Engineering & Applied Sciences --- Mathematics --- Physical Sciences & Mathematics --- Applied Mathematics --- Calculus --- Mathematical Theory --- Basis functions, Radial --- Functions, Radial basis --- Radial basis function method --- Functions, Liapunov --- Liapunov functions --- Mathematics. --- Approximation theory. --- Dynamics. --- Ergodic theory. --- Differential equations. --- Dynamical Systems and Ergodic Theory. --- Approximations and Expansions. --- Ordinary Differential Equations. --- 517.91 Differential equations --- Differential equations --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Math --- Science --- Approximation theory --- Differentiable dynamical systems. --- Differential Equations. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Global analysis (Mathematics) --- Topological dynamics

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This book presents the first “How To” guide to the use of radial basis functions (RBF). It provides a clear vision of their potential, an overview of ready-for-use computational tools and precise guidelines to implement new engineering applications of RBF. Radial basis functions (RBF) are a mathematical tool mature enough for useful engineering applications. Their mathematical foundation is well established and the tool has proven to be effective in many fields, as the mathematical framework can be adapted in several ways. A candidate application can be faced considering the features of RBF: multidimensional space (including 2D and 3D), numerous radial functions available, global and compact support, interpolation/regression. This great flexibility makes RBF attractive – and their great potential has only been partially discovered. This is because of the difficulty in taking a first step toward RBF as they are not commonly part of engineers’ cultural background, but also due to the numerical complexity of RBF problems that scales up very quickly with the number of RBF centers. Fast RBF algorithms are available to alleviate this and high-performance computing (HPC) can provide further aid. Nevertheless, a consolidated tradition in using RBF in engineering applications is still missing and the beginner can be confused by the literature, which in many cases is presented with language and symbolisms familiar to mathematicians but which can be cryptic for engineers. The book is divided in two main sections. The first covers the foundations of RBF, the tools available for their quick implementation and guidelines for facing new challenges; the second part is a collection of practical RBF applications in engineering, covering several topics, including response surface interpolation in n-dimensional spaces, mapping of magnetic loads, mapping of pressure loads, up-scaling of flow fields, stress/strain analysis by experimental displacement fields, implicit surfaces, mesh to cad deformation, mesh morphing for crack propagation in 3D, ice and snow accretion using computational fluid dynamics (CFD) data, shape optimization for external aerodynamics, and use of adjoint data for surface sculpting. For each application, the complete path is clearly and consistently exposed using the systematic approach defined in the first section.

Radial basis functions. --- Mathematics. --- Algorithms. --- Computer mathematics. --- Structural mechanics. --- Computational Science and Engineering. --- Algorithm Analysis and Problem Complexity. --- Structural Mechanics. --- Basis functions, Radial --- Functions, Radial basis --- Radial basis function method --- Approximation theory --- Computer science. --- Computer software. --- Mechanics. --- Mechanics, Applied. --- Solid Mechanics. --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Software, Computer --- Computer systems --- Informatics --- Science --- Algorism --- Algebra --- Arithmetic --- Computer mathematics --- Electronic data processing --- Mathematics --- Foundations

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This book is the first to be devoted to the theory and applications of spherical (radial) basis functions (SBFs), which is rapidly emerging as one of the most promising techniques for solving problems where approximations are needed on the surface of a sphere. The aim of the book is to provide enough theoretical and practical details for the reader to be able to implement the SBF methods to solve real world problems. The authors stress the close connection between the theory of SBFs and that of the more well-known family of radial basis functions (RBFs), which are well-established tools for solving approximation theory problems on more general domains. The unique solvability of the SBF interpolation method for data fitting problems is established and an in-depth investigation of its accuracy is provided. Two chapters are devoted to partial differential equations (PDEs). One deals with the practical implementation of an SBF-based solution to an elliptic PDE and another which describes an SBF approach for solving a parabolic time-dependent PDE, complete with error analysis. The theory developed is illuminated with numerical experiments throughout. Spherical Radial Basis Functions, Theory and Applications will be of interest to graduate students and researchers in mathematics and related fields such as the geophysical sciences and statistics.

Mathematics. --- Approximations and Expansions. --- Partial Differential Equations. --- Numerical Analysis. --- Global Analysis and Analysis on Manifolds. --- Geophysics/Geodesy. --- Physical geography. --- Global analysis. --- Differential equations, partial. --- Numerical analysis. --- Mathématiques --- Géographie physique --- Analyse numérique --- Radial basis functions. --- Spherical functions. --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Operations Research --- Basis functions, Radial --- Functions, Radial basis --- Radial basis function method --- Functions, Spherical --- Geophysics. --- Approximation theory. --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Partial differential equations. --- Approximation theory --- Spherical harmonics --- Transcendental functions --- Spheroidal functions --- Geography --- Mathematical analysis --- Partial differential equations --- Math --- Science --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Geological physics --- Terrestrial physics --- Earth sciences --- Physics --- Geometry, Differential --- Topology --- Differential equations, Partial. --- Global analysis (Mathematics) --- Manifolds (Mathematics)

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As faster and more efficient numerical algorithms become available, the understanding of the physics and the mathematical foundation behind these new methods will play an increasingly important role. This Special Issue provides a platform for researchers from both academia and industry to present their novel computational methods that have engineering and physics applications.

radial basis functions --- finite difference methods --- traveling waves --- non-uniform grids --- chaotic oscillator --- one-step method --- multi-step method --- computer arithmetic --- FPGA --- high strain rate impact --- modeling and simulation --- smoothed particle hydrodynamics --- finite element analysis --- hybrid nanofluid --- heat transfer --- non-isothermal --- shrinking surface --- MHD --- radiation --- multilayer perceptrons --- quaternion neural networks --- metaheuristic optimization --- genetic algorithms --- micropolar fluid --- constricted channel --- MHD pulsatile flow --- strouhal number --- flow pulsation parameter --- multiple integral finite volume method --- finite difference method --- Rosenau-KdV --- conservation --- solvability --- convergence --- transmission electron microscopy (TEM) --- convolutional neural networks (CNN) --- anomaly detection --- principal component analysis (PCA) --- machine learning --- deep learning --- neural networks --- Gallium-Arsenide (GaAs) --- radiation-based flowmeter --- two-phase flow --- feature extraction --- artificial intelligence --- time domain --- Boltzmann equation --- collision integral --- convolutional neural network --- annular regime --- scale layer-independent --- petroleum pipeline --- volume fraction --- dual energy technique --- prescribed heat flux --- similarity solutions --- dual solutions --- stability analysis --- RBF-FD --- node sampling --- lebesgue constant --- complex regions --- finite-difference methods --- data assimilation --- model order reduction --- finite elements analysis --- high dimensional data --- welding