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Separation of variables methods for solving partial differential equations are of immense theoretical and practical importance in mathematical physics. They are the most powerful tool known for obtaining explicit solutions of the partial differential equations of mathematical physics. The purpose of this book is to give an up-to-date presentation of the theory of separation of variables and its relation to superintegrability. Collating and presenting it in a unified, updated and a more accessible manner, the results scattered in the literature that the authors have prepared is an invaluable resource for mathematicians and mathematical physicists in particular, as well as science, engineering, geological and biological researchers interested in explicit solutions.

Separation of variables. --- Differential equations, Partial --- Mathematical physics. --- SCIENCE / Physics / Mathematical & Computational. --- Numerical solutions.

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Although the solution of Partial Differential Equations by numerical methods is the standard practice in industries, analytical methods are still important for the critical assessment of results derived from advanced computer simulations and the improvement of the underlying numerical techniques. Literature devoted to analytical methods, however, often focuses on theoretical and mathematical aspects and is therefore useless to most engineers. Analytical Methods for Heat Transfer and Fluid Flow Problems addresses engineers and engineering students. It describes useful analytical methods by applying them to real-world problems rather than solving the usual over-simplified classroom problems. The book demonstrates the applicability of analytical methods even for complex problems and guides the reader to a more intuitive understanding of approaches and solutions.

Fluid mechanics. --- Heat --- Mechanics, Analytic. --- Separation of variables. --- Transmission. --- warmteoverdracht --- matrijzen --- 66.021.4 --- 66.021.4 Heat transfer --- Heat transfer --- Fluid mechanics --- Mechanics, Analytic --- Separation of variables --- Variables, Separation of --- Differential equations, Partial --- Analytical mechanics --- Kinetics --- Thermal transfer --- Transmission of heat --- Energy transfer --- Hydromechanics --- Continuum mechanics --- Transmission --- Numerical solutions --- Thermodynamics. --- Heat engineering. --- Heat transfer. --- Mass transfer. --- Applied mathematics. --- Engineering mathematics. --- Automotive engineering. --- Fluids. --- Engineering Fluid Dynamics. --- Engineering Thermodynamics, Heat and Mass Transfer. --- Mathematical and Computational Engineering. --- Automotive Engineering. --- Fluid- and Aerodynamics. --- Hydraulics --- Mechanics --- Physics --- Hydrostatics --- Permeability --- Engineering --- Engineering analysis --- Mathematical analysis --- Mass transport (Physics) --- Thermodynamics --- Transport theory --- Mechanical engineering --- Chemistry, Physical and theoretical --- Dynamics --- Heat-engines --- Quantum theory --- Mathematics

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Konrad Schöbel aims to lay the foundations for a consequent algebraic geometric treatment of variable separation, which is one of the oldest and most powerful methods to construct exact solutions for the fundamental equations in classical and quantum physics. The present work reveals a surprising algebraic geometric structure behind the famous list of separation coordinates, bringing together a great range of mathematics and mathematical physics, from the late 19th century theory of separation of variables to modern moduli space theory, Stasheff polytopes and operads. "I am particularly impressed by his mastery of a variety of techniques and his ability to show clearly how they interact to produce his results.”   (Jim Stasheff)   Contents The Foundation: The Algebraic Integrability Conditions The Proof of Concept: A Complete Solution for the 3-Sphere The Generalisation: A Solution for Spheres of Arbitrary Dimension The Perspectives: Applications and Generalisations   Target Groups Scientists in the fields of Mathematical Physics and Algebraic Geometry   The Author Konrad Schöbel studied physics and mathematics at Friedrich-Schiller University Jena (Germany) and Universidad de Granada (Spain) and obtained his PhD at the Université de Provence Aix-Marseille I (France). He now holds a postdoc position at Friedrich-Schiller University Jena and works as a research and development engineer for applications in clinical ultrasound diagnostics.

Applied Physics --- Operations Research --- Engineering & Applied Sciences --- Civil & Environmental Engineering --- Separation of variables. --- Differential equations. --- 517.91 Differential equations --- Differential equations --- Variables, Separation of --- Mathematics. --- Algebra. --- Geometry. --- Mathematical physics. --- Mathematical Physics. --- Physical mathematics --- Physics --- Mathematics --- Euclid's Elements --- Mathematical analysis --- Math --- Science --- Differential equations, Partial --- Numerical solutions

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The authors explore a unifying model which couples phase separation and damage processes in a system of partial differential equations. The model has technological applications to solder materials where interactions of both phenomena have been observed and cannot be neglected for a realistic description. The equations are derived in a thermodynamically consistent framework and suitable weak formulations for various types of this coupled system are presented. In the main part, existence of weak solutions is proven and degenerate limits are investigated. Contents Modeling of Phase Separation and Damage Processes Notion of Weak Solutions Existence of Weak Solutions Degenerate Limit Target Groups Researchers, academics and scholars in the field of (applied) mathematics Material scientists in the field of modeling damaging processes The Authors Christian Heinemann earned his doctoral degree at the Humboldt-Universität zu Berlin under the supervision of Prof. Dr. Jürgen Sprekels and Dr. Christiane Kraus. He is a member of the research staff at the Young Scientists' Group at the Weierstrass Institute for Applied Analysis and Stochastics in Berlin. Christiane Kraus is a leader of the Young Scientists' Group at the Weierstrass Institute for Applied Analysis and Stochastics in Berlin.

Separation of variables. --- Differential equations, Partial. --- Partial differential equations --- Variables, Separation of --- Differential equations, Partial --- Numerical solutions --- Global analysis (Mathematics). --- Materials. --- Analysis. --- Mathematical Physics. --- Metallic Materials. --- Engineering --- Engineering materials --- Industrial materials --- Engineering design --- Manufacturing processes --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Materials --- Mathematical analysis. --- Analysis (Mathematics). --- Mathematical physics. --- Metals. --- Metallic elements --- Chemical elements --- Ores --- Metallurgy --- Physical mathematics --- Physics --- 517.1 Mathematical analysis --- Mathematical analysis --- Mathematics

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This book focuses on applications of the theory of fractional calculus in numerical analysis and various fields of physics and engineering. Inequalities involving fractional calculus operators containing the Mittag–Leffler function in their kernels are of particular interest. Special attention is given to dynamical models, magnetization, hypergeometric series, initial and boundary value problems, and fractional differential equations, among others.

Research & information: general --- Mathematics & science --- fractional derivative --- generalized Mittag-Leffler kernel (GMLK) --- Legendre polynomials --- Legendre spectral collocation method --- dynamical systems --- random time change --- inverse subordinator --- asymptotic behavior --- Mittag–Leffler function --- data fitting --- magnetization --- magnetic fluids --- Gamma function --- Psi function --- Pochhammer symbol --- hypergeometric function 2F1 --- generalized hypergeometric functions tFu --- Gauss’s summation theorem for 2F1(1) --- Kummer’s summation theorem for 2F1(−1) --- generalized Kummer’s summation theorem for 2F1(−1) --- Stirling numbers of the first kind --- Hilfer–Hadamard fractional derivative --- Riemann–Liouville fractional derivative --- Caputo fractional derivative --- fractional differential equations --- inclusions --- nonlocal boundary conditions --- existence and uniqueness --- fixed point --- gamma function --- Beta function --- Mittag-Leffler function --- Generalized Mittag-Leffler functions --- generalized hypergeometric function --- Fox–Wright function --- recurrence relations --- Riemann–Liouville fractional calculus operators --- (α, h-m)-p-convex function --- Fejér–Hadamard inequality --- extended generalized fractional integrals --- Mittag–Leffler functions --- initial value problems --- Laplace transform --- exact solution --- Chebyshev inequality --- Pólya-Szegö inequality --- fractional integral operators --- Wright function --- Srivastava’s polynomials --- fractional calculus operators --- Lavoie–Trottier integral formula --- Oberhettinger integral formula --- fractional partial differential equation --- boundary value problem --- separation of variables --- Mittag-Leffler --- Abel-Gontscharoff Green’s function --- Hermite-Hadamard inequalities --- convex function --- κ-Riemann-Liouville fractional integral --- Dirichlet averages --- B-splines --- dirichlet splines --- Riemann–Liouville fractional integrals --- hypergeometric functions of one and several variables --- generalized Mittag-Leffler type function --- Srivastava–Daoust generalized Lauricella hypergeometric function --- fractional calculus --- Hermite–Hadamard inequality --- Fox H function --- subordinator and inverse stable subordinator --- Lamperti law --- order statistic --- n/a --- Gauss's summation theorem for 2F1(1) --- Kummer's summation theorem for 2F1(−1) --- generalized Kummer's summation theorem for 2F1(−1) --- Hilfer-Hadamard fractional derivative --- Riemann-Liouville fractional derivative --- Fox-Wright function --- Riemann-Liouville fractional calculus operators --- Fejér-Hadamard inequality --- Mittag-Leffler functions --- Pólya-Szegö inequality --- Srivastava's polynomials --- Lavoie-Trottier integral formula --- Abel-Gontscharoff Green's function --- Riemann-Liouville fractional integrals --- Srivastava-Daoust generalized Lauricella hypergeometric function --- Hermite-Hadamard inequality