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Scaling of Differential Equations

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Book Series: Simula SpringerBriefs on Computing ISBN: 9783319327259 9783319327266 Year: Pages: 138 DOI: 10.1007/978-3-319-32726-6 Language: English
Publisher: Springer
Subject: Mathematics
Added to DOAB on : 2017-01-24 17:59:55
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The book serves both as a reference for various scaled models with corresponding dimensionless numbers, and as a resource for learning the art of scaling. A special feature of the book is the emphasis on how to create software for scaled models, based on existing software for unscaled models.Scaling (or non-dimensionalization) is a mathematical technique that greatly simplifies the setting of input parameters in numerical simulations. Moreover, scaling enhances the understanding of how different physical processes interact in a differential equation model. Compared to the existing literature, where the topic of scaling is frequently encountered, but very often in only a brief and shallow setting, the present book gives much more thorough explanations of how to reason about finding the right scales. This process is highly problem dependent, and therefore the book features a lot of worked examples, from very simple ODEs to systems of PDEs, especially from fluid mechanics.The text is easily accessible and example-driven. The first part on ODEs fits even a lower undergraduate level, while the most advanced multiphysics fluid mechanics examples target the graduate level. The scientific literature is full of scaled models, but in most of the cases, the scales are just stated without thorough mathematical reasoning. This book explains how the scales are found mathematically.This book will be a valuable read for anyone doing numerical simulations based on ordinary or partial differential equations.

Random Differential Equations in Scientific Computing

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ISBN: 9788376560267 Year: Pages: 650 DOI: 10.2478/9788376560267 Language: English
Publisher: De Gruyter
Subject: Mathematics
Added to DOAB on : 2014-03-05 13:15:31
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This book is a holistic and self-contained treatment of the analysis and numerics of random differential equations from a problem-centred point of view. An interdisciplinary approach is applied by considering state-of-the-art concepts of both dynamical systems and scientific computing. The red line pervading this book is the two-fold reduction of a random partial differential equation disturbed by some external force as present in many important applications in science and engineering. First, the random partial differential equation is reduced to a set of random ordinary differential equations in the spirit of the method of lines. These are then further reduced to a family of (deterministic) ordinary differential equations. The monograph will be of benefit, not only to mathematicians, but can also be used for interdisciplinary courses in informatics and engineering.

Multidimensional Inverse and Ill-Posed Problems for Differential Equations

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Book Series: Inverse and Ill-Posed Problems Series ISSN: 1381-4524 ISBN: 9783110271478 Year: Volume: 4 Pages: 139,00 DOI: 10.1515/9783110271478 Language: English
Publisher: De Gruyter
Subject: Mathematics
Added to DOAB on : 2019-06-27 13:08:38
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This monograph is devoted to statements of multidimensional inverse problems, in particular to methods of their investigation. Questions of the uniqueness of solution, solvability and stability are studied. Methods to construct a solution are given and, in certain cases, inversion formulas are given as well. Concrete applications of the theory developed here are also given. Where possible, the author has stopped to consider the method of investigation of the problems, thereby sometimes losing generality and quantity of the problems, which can be examined by such a method. The book should be of interet to researchers in the field of applied mathematics, geophysics and mathematical biology.

Finite Difference Computing with PDEs: A Modern Software Approach

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Book Series: Texts in Computational Science and Engineering ISSN: 1611-0994 / 2197-179X ISBN: 9783319554556 9783319554563 Year: Pages: 507 DOI: https://doi.org/10.1007/978-3-319-55456-3 Language: English
Publisher: Springer
Subject: Computer Science
Added to DOAB on : 2017-11-24 13:03:18
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This easy-to-read book introduces the basics of solving partial differential equations by means of finite difference methods. Unlike many of the traditional academic works on the topic, this book was written for practitioners. Accordingly, it especially addresses: the construction of finite difference schemes, formulation and implementation of algorithms, verification of implementations, analyses of physical behavior as implied by the numerical solutions, and how to apply the methods and software to solve problems in the fields of physics and biology.

Lições de matemática II

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Book Series: Ensino ISBN: 9789892613178 Year: Pages: 360 DOI: https://doi.org/10.14195/978-989-26-1318-5 Language: Portuguese
Publisher: Coimbra University Press
Subject: Mathematics
Added to DOAB on : 2019-04-23 16:21:06
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A edição deste manual insere-se num projeto, ensaiado no âmbito da unidade curricular de Matemática II da Licenciatura em Gestão da FEUC, com o propósito de incentivar a participação nas aulas e realçar a importância do estudo individual e tutorial. O texto foi organizado com três objetivos: (i) ser elemento de consulta durante as sessões presenciais (aulas); (ii) estimular a componente de trabalho autónomo do aluno, tanto no estudo pré-aula como pós-aula; e ainda, (iii) ser um documento autocontido, pressupondo embora a frequência da unidade curricular de Matemática I, que abordasse três tópicos (séries numéricas e representação de funções em séries de potências, funções reais de duas variáveis reais e complementos de equações diferenciais ordinárias) aparentemente disjuntos. Finalmente, é conveniente referir que, no sentido de incluir alguns conceitos, porventura esquecidos ou pouco amadurecidos, sem sobrecarregar o texto principal foram criados quatro apêndices, designadamente, versando sobre: o conjunto dos números reais e algumas propriedades elementares, sucessões de números reais, algumas noções de topologia em ℝ2 e exponencial complexa.

Neural Masses and Fields: Modelling the Dynamics of Brain Activity

Authors: --- --- ---
Book Series: Frontiers Research Topics ISSN: 16648714 ISBN: 9782889194278 Year: Pages: 237 DOI: 10.3389/978-2-88919-427-8 Language: English
Publisher: Frontiers Media SA
Subject: Neurology --- Science (General)
Added to DOAB on : 2016-01-19 14:05:46
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Biophysical modelling of brain activity has a long and illustrious history and has recently profited from technological advances that furnish neuroimaging data at an unprecedented spatiotemporal resolution. Neuronal modelling is a very active area of research, with applications ranging from the characterization of neurobiological and cognitive processes, to constructing artificial brains in silico and building brain-machine interface and neuroprosthetic devices. Biophysical modelling has always benefited from interdisciplinary interactions between different and seemingly distant fields; ranging from mathematics and engineering to linguistics and psychology. This Research Topic aims to promote such interactions by promoting papers that contribute to a deeper understanding of neural activity as measured by fMRI or electrophysiology.In general, mean field models of neural activity can be divided into two classes: neural mass and neural field models. The main difference between these classes is that field models prescribe how a quantity characterizing neural activity (such as average depolarization of a neural population) evolves over both space and time as opposed to mass models, which characterize activity over time only; by assuming that all neurons in a population are located at (approximately) the same point. This Research Topic focuses on both classes of models and considers several aspects and their relative merits that: span from synapses to the whole brain; comparisons of their predictions with EEG and MEG spectra of spontaneous brain activity; evoked responses, seizures, and fitting data - to infer brain states and map physiological parameters.

Operators of Fractional Calculus and Their Applications

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ISBN: 9783038973409 / 9783038973416 Year: Pages: 136 DOI: 10.3390/books978-3-03897-341-6 Language: English
Publisher: MDPI - Multidisciplinary Digital Publishing Institute
Subject: Mathematics --- Physics (General)
Added to DOAB on : 2019-01-16 12:17:12
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During the past four decades or so, various operators of fractional calculus, such as those named after Riemann–Liouville, Weyl, Hadamard, Grunwald–Letnikov, Riesz, Erdelyi–Kober, Liouville–Caputo, and so on, have been found to be remarkably popular and important due mainly to their demonstrated applications in numerous diverse and widespread fields of the mathematical, physical, chemical, engineering, and statistical sciences. Many of these fractional calculus operators provide several potentially useful tools for solving differential, integral, differintegral, and integro-differential equations, together with the fractional-calculus analogues and extensions of each of these equations, and various other problems involving special functions of mathematical physics, as well as their extensions and generalizations in one and more variables. In this Special Issue, we invite and welcome review, expository, and original research articles dealing with the recent advances in the theory of fractional calculus and its multidisciplinary applications.

Advanced Numerical Methods in Applied Sciences

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ISBN: 9783038976660 / 9783038976677 Year: Pages: 306 DOI: 10.3390/books978-3-03897-667-7 Language: eng
Publisher: MDPI - Multidisciplinary Digital Publishing Institute
Subject: Science (General) --- Mathematics
Added to DOAB on : 2019-06-26 08:44:06
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The use of scientific computing tools is currently customary for solving problems at several complexity levels in Applied Sciences. The great need for reliable software in the scientific community conveys a continuous stimulus to develop new and better performing numerical methods that are able to grasp the particular features of the problem at hand. This has been the case for many different settings of numerical analysis, and this Special Issue aims at covering some important developments in various areas of application.

Keywords

time fractional differential equations --- mixed-index problems --- analytical solution --- asymptotic stability --- conservative problems --- Hamiltonian problems --- energy-conserving methods --- Poisson problems --- Hamiltonian Boundary Value Methods --- HBVMs --- line integral methods --- constrained Hamiltonian problems --- Hamiltonian PDEs --- highly oscillatory problems --- boundary element method --- finite difference method --- floating strike Asian options --- continuous geometric average --- barrier options --- isogeometric analysis --- adaptive methods --- hierarchical splines --- THB-splines --- local refinement --- linear systems --- preconditioners --- Cholesky factorization --- limited memory --- Volterra integral equations --- Volterra integro–differential equations --- collocation methods --- multistep methods --- convergence --- B-spline --- optimal basis --- fractional derivative --- Galerkin method --- collocation method --- spectral (eigenvalue) and singular value distributions --- generalized locally Toeplitz sequences --- discretization of systems of differential equations --- higher-order finite element methods --- discontinuous Galerkin methods --- finite difference methods --- isogeometric analysis --- B-splines --- curl–curl operator --- time harmonic Maxwell’s equations and magnetostatic problems --- low rank completion --- matrix ODEs --- gradient system --- ordinary differential equations --- Runge–Kutta --- tree --- stump --- order --- elementary differential --- edge-histogram --- edge-preserving smoothing --- histogram specification --- initial value problems --- one-step methods --- Hermite–Obreshkov methods --- symplecticity --- B-splines --- BS methods --- hyperbolic partial differential equations --- high order discontinuous Galerkin finite element schemes --- shock waves and discontinuities --- vectorization and parallelization --- high performance computing --- generalized Schur algorithm --- null-space --- displacement rank --- structured matrices --- stochastic differential equations --- stochastic multistep methods --- stochastic Volterra integral equations --- mean-square stability --- asymptotic stability --- numerical analysis --- numerical methods --- scientific computing --- initial value problems --- one-step methods --- Hermite–Obreshkov methods --- symplecticity --- B-splines --- BS methods

Joseph Fourier 250th Birthday. Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst century

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ISBN: 9783038977469 Year: Pages: 260 DOI: 10.3390/books978-3-03897-747-6 Language: eng
Publisher: MDPI - Multidisciplinary Digital Publishing Institute
Subject: Science (General) --- Physics (General)
Added to DOAB on : 2019-04-05 11:17:10
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For the 250th birthday of Joseph Fourier, born in 1768 in Auxerre, France, this MDPI Special Issue will explore modern topics related to Fourier Analysis and Heat Equation. Modern developments of Fourier analysis during the 20th century have explored generalizations of Fourier and Fourier–Plancherel formula for non-commutative harmonic analysis, applied to locally-compact, non-Abelian groups. In parallel, the theory of coherent states and wavelets has been generalized over Lie groups. One should add the developments, over the last 30 years, of the applications of harmonic analysis to the description of the fascinating world of aperiodic structures in condensed matter physics. The notions of model sets, introduced by Y. Meyer, and of almost periodic functions, have revealed themselves to be extremely fruitful in this domain of natural sciences. The name of Joseph Fourier is also inseparable from the study of the mathematics of heat. Modern research on heat equations explores the extension of the classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. In parallel, in geometric mechanics, Jean-Marie Souriau interpreted the temperature vector of Planck as a space-time vector, obtaining, in this way, a phenomenological model of continuous media, which presents some interesting properties. One last comment concerns the fundamental contributions of Fourier analysis to quantum physics: Quantum mechanics and quantum field theory. The content of this Special Issue will highlight papers exploring non-commutative Fourier harmonic analysis, spectral properties of aperiodic order, the hypoelliptic heat equation, and the relativistic heat equation in the context of Information Theory and Geometric Science of Information.

Dynamical Models of Biology and Medicine

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ISBN: 9783039212170 / 9783039212187 Year: Pages: 294 DOI: 10.3390/books978-3-03921-218-7 Language: eng
Publisher: MDPI - Multidisciplinary Digital Publishing Institute
Subject: Science (General) --- Biology
Added to DOAB on : 2019-12-09 11:49:15
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Mathematical and computational modeling approaches in biological and medical research are experiencing rapid growth globally. This Special Issue Book intends to scratch the surface of this exciting phenomenon. The subject areas covered involve general mathematical methods and their applications in biology and medicine, with an emphasis on work related to mathematical and computational modeling of the complex dynamics observed in biological and medical research. Fourteen rigorously reviewed papers were included in this Special Issue. These papers cover several timely topics relating to classical population biology, fundamental biology, and modern medicine. While the authors of these papers dealt with very different modeling questions, they were all motivated by specific applications in biology and medicine and employed innovative mathematical and computational methods to study the complex dynamics of their models. We hope that these papers detail case studies that will inspire many additional mathematical modeling efforts in biology and medicine

Keywords

hemodynamic model --- microcirculation load --- liquid-solid-porous media seepage coupling --- 2-combination --- graphical representation --- cell-based vector --- numerical characterization --- phylogenetic analysis --- intraguild predation --- random perturbations --- persistence --- stationary distribution --- global asymptotic stability --- quorum sensing --- chemostat --- mathematical model --- differential equations --- delay --- bifurcations --- dynamical system --- numerical simulation --- predator-prey model --- switched harvest --- limit cycle --- rich dynamics --- algae growth models --- uncertainty quantification --- asymptotic theory --- bootstrapping --- model comparison tests --- Raphidocelis subcapitata --- Daphnia magna --- spotting --- wildfire --- transport equations --- spotting distribution --- obesity --- mechano-electrochemical model --- articular cartilage --- cartilage degeneration --- cartilage loading --- optimal control --- hepatitis B --- delay differential equations (DDE) --- immune response --- drug therapy --- dynamic model --- flocculation --- global stability --- uniform persistence --- epidermis --- mathematical model --- bacterial inflammation --- bacterial competition --- chronic myeloid leukemia --- tyrosine kinase inhibitors --- immunomodulatory therapies --- combination therapy --- equilibrium points --- mathematical modeling --- prostate cancer --- androgen deprivation therapy --- data fitting --- generalized pseudo amino acid composition --- numerical characterization --- phylogenetic analysis --- identification of DNA-binding proteins --- n/a

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