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Books on the topic 'Integral Equation Approach'

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Published: 6 September 2023

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1

Słobodzian, Piotr M. Electromagnetic analysis of shielded microwave structures: The surface integral equation approach. Wrocław: Oficyna Wydawnicza Politechniki Wrocławskiej, 2007.

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2

Sin-Chung, Chang, and United States. National Aeronautics and Space Administration., eds. The Space-time solution element method-a new numerical approach for the Navier-Stokes equations. [Washington, DC]: National Aeronautics and Space Administration, 1995.

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3

Sin-Chung, Chang, and United States. National Aeronautics and Space Administration., eds. The Space-time solution element method-a new numerical approach for the Navier-Stokes equations. [Washington, DC]: National Aeronautics and Space Administration, 1995.

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4

Bogdanov, L. V. Analytic-Bilinear Approach to Integrable Hierarchies. Dordrecht: Springer Netherlands, 1999.

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5

Assing, Sigurd. Continuous strong Markov processes in dimension one: A stochastic calculus approach. Berlin: Springer, 1998.

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6

Stochastic integration and differential equations: A new approach. 2nd ed. Berlin: Springer-Verlag, 1992.

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7

The computational complexity of differential and integral equations: An information-based approach. Oxford [England]: Oxford University Press, 1991.

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8

Protter, Philip E. Stochastic integration and differential equations: A new approach. Berlin: Springer-Verlag, 1990.

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9

1950-, Panagiotopoulos P. D., ed. The boundary integral approach to static and dynamic contact problems: Equality and inequality methods. Basel: Birkhäuser, 1992.

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10

Spain) UIMP-RSME Lluis Santaló Summer (2012 Santander. Recent advances in real complexity and computation: UIMP-RSME Lluis A. Santaló Summer School, Recent advances in real complexity and computation, July 16-20, 2012, Universidad Internacional Menéndez Pelayo, Santander, Spain. Edited by Montaña, Jose Luis, 1961- editor of compilation and Pardo, L. M. (Luis M.), editor of compilation. Providence, Rhode Island: American Mathematical Society, 2013.

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11

Banjai, Lehel, and Francisco-Javier Sayas. Integral Equation Methods for Evolutionary PDE: A Convolution Quadrature Approach. Springer International Publishing AG, 2022.

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12

Messali, N. An integral equation approach to continuous system identification and model reduction. 1988.

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13

The Space-time solution element method-a new numerical approach for the Navier-Stokes equations. [Washington, DC]: National Aeronautics and Space Administration, 1995.

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14

Rajeev, S. G. Fluid Mechanics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198805021.001.0001.

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Starting with a review of vector fields and their integral curves, the book presents the basic equations of the subject: Euler and Navier–Stokes. Some solutions are studied next: ideal flows using conformal transformations, viscous flows such as Couette and Stokes flow around a sphere, shocks in the Burgers equation. Prandtl’s boundary layer theory and the Blasius solution are presented. Rayleigh–Taylor instability is studied in analogy with the inverted pendulum, with a digression on Kapitza’s stabilization. The possibility of transients in a linearly stable system with a non-normal operator is studied using an example by Trefethen et al. The integrable models (KdV, Hasimoto’s vortex soliton) and their hamiltonian formalism are studied. Delving into deeper mathematics, geodesics on Lie groups are studied: first using the Lie algebra and then using Milnor’s approach to the curvature of the Lie group. Arnold’s deep idea that Euler’s equations are the geodesic equations on the diffeomorphism group is then explained and its curvature calculated. The next three chapters are an introduction to numerical methods: spectral methods based on Chebychev functions for ODEs, their application by Orszag to solve the Orr–Sommerfeld equation, finite difference methods for elementary PDEs, the Magnus formula and its application to geometric integrators for ODEs. Two appendices give an introduction to dynamical systems: Arnold’s cat map, homoclinic points, Smale’s horse shoe, Hausdorff dimension of the invariant set, Aref ’s example of chaotic advection. The last appendix introduces renormalization: Ising model on a Cayley tree and Feigenbaum’s theory of period doubling.
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15

Benson, Trevor, and Alexander Nerukh. Non-Stationary Electromagnetics: An Integral Equations Approach. Jenny Stanford Publishing, 2018.

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16

Benson, Trevor, and Alexander Nerukh. Non-Stationary Electromagnetics: An Integral Equations Approach. Taylor & Francis Group, 2018.

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17

Non-Stationary Electromagnetics: An Integral Equations Approach. Jenny Stanford Publishing, 2018.

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18

Morawetz, Klaus. Nonlocal Collision Integral. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797241.003.0013.

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The kinetic equation with the nonlocal shifts is presented as the final result on the way to derive the kinetic equation with nonlocal corrections. The exclusive dependence of the nonlocal and non-instant corrections on the scattering phase shift confirms the results from the theory of gases. With the approximation on the level of the Brueckner reaction matrix, the corresponding non-instant and nonlocal scattering integral in parallel with the classical Enskog’s equation, can be treated with Monte-Carlo simulation techniques. Neglecting the shifts, the Landau theory of quasiparticle transport appears. In this sense the presented kinetic theory unifies both approaches. An intrinsic symmetry is found from the optical theorem which allows for representing the collision integral equivalently either in particle-hole symmetric or space-time symmetric form.
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19

Algebro-geometric approach to nonlinear integrable equations. Berlin: Springer-Verlag, 1994.

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20

Approximation of Additive Convolution-Like Operators: Real C*-Algebra Approach (Frontiers in Mathematics). Birkhäuser, 2008.

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21

Zeitlin, Vladimir. Wave Turbulence. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198804338.003.0013.

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Main notions and ideas of wave (weak) turbulence theory are explained with the help of Hamiltonian approach to wave dynamics, and are applied to waves in RSW model. Derivation of kinetic equations under random-phase approximation is explained. Short inertia–gravity waves on the f plane, short equatorial inertia–gravity waves, and Rossby waves on the beta plane are then considered along these lines. In all of these cases, approximate solutions of kinetic equation, annihilating the collision integral, can be obtained by scaling arguments, giving power-law energy spectra. The predictions of turbulence of inertia–gravity waves on the f plane are compared with numerical simulations initialised by ensembles of random waves. Energy spectra much steeper than theoretical are observed. Finite-size effects, which prevent energy transfer from large to short scales, provide a plausible explanation. Long waves thus evolve towards breaking and shock formation, yet the number of shocks is insufficient to produce shock turbulence.
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22

Suris, Yuri B. Problem of Integrable Discretization: Hamiltonian Approach. Springer Basel AG, 2012.

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23

Approximation Of Additive Convolutionlike Operators Real Calgebra Approach. Birkhauser Boston, 2008.

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24

Susnjara, Anna. Deterministic and Stochastic Modeling in Computational Electromagnetics: Integral and Differential Equation Approaches. Wiley & Sons, Incorporated, John, 2023.

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25

Susnjara, Anna. Deterministic and Stochastic Modeling in Computational Electromagnetics: Integral and Differential Equation Approaches. Wiley & Sons, Incorporated, John, 2023.

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26

Susnjara, Anna. Deterministic and Stochastic Modeling in Computational Electromagnetics: Integral and Differential Equation Approaches. Wiley & Sons, Incorporated, John, 2023.

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27

Antes, H. The Boundary Integral Approach to Static and Dynamic Contact Problems. Springer, 2012.

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28

The Problem of Integrable Discretization: Hamiltonian Approach (Progress in Mathematics). Birkhäuser Basel, 2003.

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29

Mann, Peter. Differential Equations. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0035.

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This chapter presents the general formulation of the calculus of variations as applied to mechanics, relativity and field theories. The calculus of variations is a common mathematical technique used throughout classical mechanics. First developed by Euler to determine the shortest paths between fixed points along a surface, it was applied by Lagrange to mechanical problems in analytical mechanics. The variational problems in the chapter have been simplified for ease of understanding upon first introduction, in order to give a general mathematical framework. This chapter takes a relaxed approach to explain how the Euler–Lagrange equation is derived using this method. It also discusses first integrals. The chapter closes by defining the functional derivative, which is used in classical field theory.
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30

Antes, H., and P. D. Panagiotopoulos. Boundary Integral Approach to Static and Dynamic Contact Problems: Equality and Inequality Methods. Birkhauser Verlag, 2013.

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31

The Problem of Integrable Discretization: Hamiltonian Approach (Progress in Mathematics (Boston, Mass.), V. 219.). Birkhauser, 2003.

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32

Boudou, Alain, and Yves Romain. On Product Measures Associated with Stationary Processes. Edited by Frédéric Ferraty and Yves Romain. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780199568444.013.15.

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This article considers the connections between product measures and stationary processes. It first provides an overview of historical facts and relevant terminology, basic concepts and the mathematical approach. In particular, it discusses random measures, the projection-valued spectral measure (PVSM), convolution products, and the association between shift operators and PVSMs. It then presents the main results and their first potential applications, focusing on stochastic integrals, the image of a random measure under measurable mapping, the existence of a transport-type theorem, and the transpose of a continuous homomorphism between groups. It also describes the PVSM associated with a unitary operator, the convolution product of two PVSMs, the unitary operators generated by a PVSM, extension of the convolution product of two PVSMs, an equation where the unknown quantity is a PVSM, and the convolution product of two random measures. The article concludes with an analysis of mathematical developments related to the previous results.
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33

Boudreau, Joseph F., and Eric S. Swanson. Numerical quadrature. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198708636.003.0005.

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This chapter discusses the numerous applications of numerical quadrature (integration) in classical mechanics, in semiclassical approaches to quantum mechanics, and in statistical mechanics; and then describes several ways of implementing integration in C++, for both proper and improper integrals. Various algorithms are described and analyzed, including simple classical quadrature algorithms as well as those enhanced with speedups and convergence tests. Classical orthogonal polynomials, whose properties are reviewed, are the basis of a sophisticated technique known as Gaussian integration. Practical implementations require the roots of these polynomials, so an algorithm for finding them from three-term recurrence relations is presented. On the computational side, the concept of polymorphism is introduced and exploited (prior to the detailed treatment later in the text). The nondimensionalization of physical problems, which is a common and important means of simplifying a problem, is discussed using Compton scattering and the Schrödinger equation as an example.
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34

Mashhoon, Bahram. Nonlocal Gravity. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198803805.001.0001.

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A postulate of locality permeates through the special and general theories of relativity. First, Lorentz invariance is extended in a pointwise manner to actual, namely, accelerated observers in Minkowski spacetime. This hypothesis of locality is then employed crucially in Einstein’s local principle of equivalence to render observers pointwise inertial in a gravitational field. Field measurements are intrinsically nonlocal, however. To go beyond the locality postulate in Minkowski spacetime, the past history of the accelerated observer must be taken into account in accordance with the Bohr-Rosenfeld principle. The observer in general carries the memory of its past acceleration. The deep connection between inertia and gravitation suggests that gravity could be nonlocal as well and in nonlocal gravity the fading gravitational memory of past events must then be taken into account. Along this line of thought, a classical nonlocal generalization of Einstein’s theory of gravitation has recently been developed. In this nonlocal gravity (NLG) theory, the gravitational field is local, but satisfies a partial integro-differential field equation. A significant observational consequence of this theory is that the nonlocal aspect of gravity appears to simulate dark matter. The implications of NLG are explored in this book for gravitational lensing, gravitational radiation, the gravitational physics of the Solar System and the internal dynamics of nearby galaxies as well as clusters of galaxies. This approach is extended to nonlocal Newtonian cosmology, where the attraction of gravity fades with the expansion of the universe. Thus far only some of the consequences of NLG have been compared with observation.
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