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Published: 25 May 2024

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1

Schiesser, W. E. A compendium of partial differential equation models: Method of lines analysis with MATLAB. Cambridge: Cambridge University Press, 2009.

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2

C, Sorensen D., and Institute for Computer Applications in Science and Engineering., eds. An asymptotic induced numerical method for the convection-diffusion-reaction equation. Hampton, VA: Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 1988.

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3

N, Bellomo, and Gatignol Renée, eds. Lecture notes on the discretization of the Boltzmann equation. River Edge, NJ: World Scientific, 2003.

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4

United States. National Aeronautics and Space Administration., ed. Compact finite volume methods for the diffusion equation. Greensboro, NC: Dept. of Mechanical Engineering, N.C. A&T State University, 1989.

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5

T, Patera Anthony, Peraire Jaume, and Langley Research Center, eds. A posteriori finite element bounds for sensitivity derivatives of partial-differential-equation outputs. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1998.

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6

Parallel-vector equation solvers for finite element engineering applications. New York: Kluwer Academic / Plenum Publishers, 2002.

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7

Wang, Baoxiang. Harmonic analysis method for nonlinear evolution equations, I. Singapore: World Scientific Pub. Co., 2011.

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8

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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9

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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10

Yeffet, Amir. A non-dissipative staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell's equations. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1999.

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11

Yeffet, Amir. A non-dissipative staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell's equations. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1999.

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12

Yeffet, Amir. A non-dissipative staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell's equations. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1999.

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13

Yeffet, Amir. A non-dissipative staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell's equations. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1999.

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14

Yeffet, Amir. A non-dissipative staggered fourth-order accurate explicit finite difference scheme for the time-domain Maxwell's equations. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1999.

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15

R, Radespiel, Turkel E, and Institute for Computer Applications in Science and Engineering., eds. Comparison of several dissipation algorithms for central difference schemes. Hampton, VA: Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 1997.

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16

N, Tiwari S., and Langley Research Center, eds. Radiative interactions in chemically reacting compressible nozzle flows using Monte Carlo simulations. Norfolk, Va: Institute for Computational and Applied Mechanics, Old Dominion University, 1994.

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17

Center, Langley Research, ed. Proper orthogonal decomposition in optimal control of fluids. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1999.

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18

Differential equation based method for accurate approximations in optimization. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1990.

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19

Schiesser, W. E., and Graham W. Griffiths. Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab. Cambridge University Press, 2009.

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20

Wu, Sean F. Helmholtz Equation Least Squares Method: For Reconstructing and Predicting Acoustic Radiation. Springer, 2015.

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21

Donninger, Roland, and Joachim Krieger. Vector Field Method on the Distorted Fourier Side and Decay for Wave Equations with Potentials. American Mathematical Society, 2016.

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22

Griffiths, Graham W., and William E. Schiesser. Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab. Cambridge University Press, 2009.

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23

Griffiths, Graham W., and William E. Schiesser. Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab. Cambridge University Press, 2009.

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24

Griffiths, Graham W., and William E. Schiesser. Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab. Cambridge University Press, 2009.

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25

Griffiths, Graham W., and William E. Schiesser. Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab. Cambridge University Press, 2009.

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26

Wu, Sean F. The Helmholtz Equation Least Squares Method: For Reconstructing and Predicting Acoustic Radiation. Springer, 2016.

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27

Nguyen, Duc Thai. Parallel-Vector Equation Solvers for Finite Element Engineering Applications. Springer, 2012.

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28

Ohira, Toru. A master equation approach to stochastic neurodynamics. 1993.

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29

(Editor), N. Bellomo, and Renee Gatignol (Editor), eds. Lecture Notes on the Discretization of the Boltzmann Equation (Series on Advances in Mathematics for Applied Sciences). World Scientific Publishing Company, 2003.

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30

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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31

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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32

Escudier, Marcel. Laminar boundary layers. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198719878.003.0017.

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This chapter starts by introducing the concept of a boundary layer and the associated boundary-layer approximations. The laminar boundary-layer equations are then derived from the Navier-Stokes equations. The assumption of velocity-profile similarity is shown to reduce the partial differential boundary-layer equations to ordinary differential equations. The results of numerical solutions to these equations are discussed: Blasius’ equation, for zero-pressure gradient, and the Falkner-Skan equation for wedge flows. Von Kármán’s momentum-integral equation is derived and used to obtain useful results for the zero-pressure-gradient boundary layer. Pohlhausen’s quartic-profile method is then discussed, followed by the approximate method of Thwaites. The chapter concludes with a qualitative account of the way in which aerodynamic lift is generated.
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33

Rajeev, S. G. Finite Difference Methods. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198805021.003.0014.

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This chapter offers a peek at the vast literature on numerical methods for partial differential equations. The focus is on finite difference methods (FDM): approximating differential operators by functions of difference operators. Padé approximants (Fornberg) give a unifying principle for deriving the various stencils used by numericists. Boundary value problems for the Poisson equation and initial value problems for the diffusion equation are solved using FDM. Numerical instability of explicit schemes are explained physically and implicit schemes introduced. A discrete version of theClebsch formulation of incompressible Euler equations is proposed. The chapter concludes with the radial basis function method and its application to a discrete version of the Lagrangian formulation of Navier–Stokes.
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34

Eriksson, Olle, Anders Bergman, Lars Bergqvist, and Johan Hellsvik. Atomistic Spin Dynamics. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198788669.001.0001.

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The purpose of this book is to provide a theoretical foundation and an understanding of atomistic spin-dynamics, and to give examples of where the atomistic Landau-Lifshitz-Gilbert equation can and should be used. The contents involve a description of density functional theory both from a fundamental viewpoint as well as a practical one, with several examples of how this theory can be used for the evaluation of ground state properties like spin and orbital moments, magnetic form-factors, magnetic anisotropy, Heisenberg exchange parameters, and the Gilbert damping parameter. This book also outlines how interatomic exchange interactions are relevant for the effective field used in the temporal evolution of atomistic spins. The equation of motion for atomistic spin-dynamics is derived starting from the quantum mechanical equation of motion of the spin-operator. It is shown that this lead to the atomistic Landau-Lifshitz-Gilbert equation, provided a Born-Oppenheimer-like approximation is made, where the motion of atomic spins is considered slower than that of the electrons. It is also described how finite temperature effects may enter the theory of atomistic spin-dynamics, via Langevin dynamics. Details of the practical implementation of the resulting stochastic differential equation are provided, and several examples illustrating the accuracy and importance of this method are given. Examples are given of how atomistic spin-dynamics reproduce experimental data of magnon dispersion of bulk and thin-film systems, the damping parameter, the formation of skyrmionic states, all-thermal switching motion, and ultrafast magnetization measurements.
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35

Rajeev, S. G. Spectral Methods. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198805021.003.0013.

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Thenumerical solution of ordinary differential equations (ODEs)with boundary conditions is studied here. Functions are approximated by polynomials in a Chebychev basis. Sections then cover spectral discretization, sampling, interpolation, differentiation, integration, and the basic ODE. Following Trefethen et al., differential operators are approximated as rectangular matrices. Boundary conditions add additional rows that turn them into square matrices. These can then be diagonalized using standard linear algebra methods. After studying various simple model problems, this method is applied to the Orr–Sommerfeld equation, deriving results originally due to Orszag. The difficulties of pushing spectral methods to higher dimensions are outlined.
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36

Mann, Peter. Vector Calculus. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0034.

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This chapter gives a non-technical overview of differential equations from across mathematical physics, with particular attention to those used in the book. It is a common trend in physics and nature, or perhaps just the way numbers and calculus come together, that, to describe the evolution of things, most theories use a differential equation of low order. This chapter is useful for those with no prior knowledge of the differential equations and explains the concepts required for a basic exposition of classical mechanics. It discusses separable differential equations, boundary conditions and initial value problems, as well as particular solutions, complete solutions, series solutions and general solutions. It also discusses the Cauchy–Lipschitz theorem, flow and the Fourier method, as well as first integrals, complete integrals and integral curves.
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37

Boudreau, Joseph F., and Eric S. Swanson. Continuum dynamics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198708636.003.0019.

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The theory and application of a variety of methods to solve partial differential equations are introduced in this chapter. These methods rely on representing continuous quantities with discrete approximations. The resulting finite difference equations are solved using algorithms that stress different traits, such as stability or accuracy. The Crank-Nicolson method is described and extended to multidimensional partial differential equations via the technique of operator splitting. An application to the time-dependent Schrödinger equation, via scattering from a barrier, follows. Methods for solving boundary value problems are explored next. One of these is the ubiquitous fast Fourier transform which permits the accurate solution of problems with simple boundary conditions. Lastly, the finite element method that is central to modern engineering is developed. Methods for generating finite element meshes and estimating errors are also discussed.
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38

Optimization of Objective Functions: Analytics. Numerical Methods. Design of Experiments. Moscow, Russia: Fizmatlit Publisher, 2009.

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