Topic

# Differential equation

About: Differential equation is a research topic. Over the lifetime, 88062 publications have been published within this topic receiving 2005802 citations. The topic is also known as: differential equation (elementary mathematics).

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01 Jan 1941

TL;DR: In this paper, the authors present a theory for linear PDEs: Sobolev spaces Second-order elliptic equations Linear evolution equations, Hamilton-Jacobi equations and systems of conservation laws.

Abstract: Introduction Part I: Representation formulas for solutions: Four important linear partial differential equations Nonlinear first-order PDE Other ways to represent solutions Part II: Theory for linear partial differential equations: Sobolev spaces Second-order elliptic equations Linear evolution equations Part III: Theory for nonlinear partial differential equations: The calculus of variations Nonvariational techniques Hamilton-Jacobi equations Systems of conservation laws Appendices Bibliography Index.

25,691 citations

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TL;DR: The PSC algorithm as mentioned in this paper approximates the Hamilton-Jacobi equations with parabolic right-hand-sides by using techniques from the hyperbolic conservation laws, which can be used also for more general surface motion problems.

Abstract: New numerical algorithms are devised (PSC algorithms) for following fronts propagating with curvature-dependent speed. The speed may be an arbitrary function of curvature, and the front can also be passively advected by an underlying flow. These algorithms approximate the equations of motion, which resemble Hamilton-Jacobi equations with parabolic right-hand-sides, by using techniques from the hyperbolic conservation laws. Non-oscillatory schemes of various orders of accuracy are used to solve the equations, providing methods that accurately capture the formation of sharp gradients and cusps in the moving fronts. The algorithms handle topological merging and breaking naturally, work in any number of space dimensions, and do not require that the moving surface be written as a function. The methods can be used also for more general Hamilton-Jacobi-type problems. The algorithms are demonstrated by computing the solution to a variety of surface motion problems.

12,372 citations

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31 May 1995

TL;DR: This paper presents background history of space-grid time-domain techniques for Maxwell's equations scaling to very large problem sizes defense applications dual-use electromagnetics technology, and the proposed three-dimensional Yee algorithm for solving these equations.

Abstract: Part 1 Reinventing electromagnetics: background history of space-grid time-domain techniques for Maxwell's equations scaling to very large problem sizes defense applications dual-use electromagnetics technology. Part 2 The one-dimensional scalar wave equation: propagating wave solutions finite-difference approximation of the scalar wave equation dispersion relations for the one-dimensional wave equation numerical group velocity numerical stability. Part 3 Introduction to Maxwell's equations and the Yee algorithm: Maxwell's equations in three dimensions reduction to two dimensions equivalence to the wave equation in one dimension. Part 4 Numerical stability: TM mode time eigenvalue problem space eigenvalue problem extension to the full three-dimensional Yee algorithm. Part 5 Numerical dispersion: comparison with the ideal dispersion case reduction to the ideal dispersion case for special grid conditions dispersion-optimized basic Yee algorithm dispersion-optimized Yee algorithm with fourth-order accurate spatial differences. Part 6 Incident wave source conditions for free space and waveguides: requirements for the plane wave source condition the hard source total-field/scattered field formulation pure scattered field formulation choice of incident plane wave formulation. Part 7 Absorbing boundary conditions for free space and waveguides: Bayliss-Turkel scattered-wave annihilating operators Engquist-Majda one-way wave equations Higdon operator Liao extrapolation Mei-Fang superabsorption Berenger perfectly-matched layer (PML) absorbing boundary conditions for waveguides. Part 8 Near-to-far field transformation: obtaining phasor quantities via discrete fourier transformation surface equivalence theorem extension to three dimensions phasor domain. Part 9 Dispersive, nonlinear, and gain materials: linear isotropic case recursive convolution method linear gyrontropic case linear isotropic case auxiliary differential equation method, Lorentz gain media. Part 10 Local subcell models of the fine geometrical features: basis of contour-path FD-TD modelling the simplest contour-path subcell models the thin wire conformal modelling of curved surfaces the thin material sheet relativistic motion of PEC boundaries. Part 11 Explicit time-domain solution of Maxwell's equations using non-orthogonal and unstructured grids, Stephen Gedney and Faiza Lansing: nonuniform, orthogonal grids globally orthogonal global curvilinear co-ordinates irregular non-orthogonal unstructured grids analysis of printed circuit devices using the planar generalized Yee algorithm. Part 12 The body of revolution FD-TD algorithm, Thomas Jurgens and Gregory Saewert: field expansion difference equations for on-axis cells numerical stability PML absorbing boundary condition. Part 13 Modelling of electromagnetic fields in high-speed electronic circuits, Piket-May and Taflove. (part contents).

10,961 citations

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01 Jan 1964

TL;DR: In this article, the Poincare-Bendixson theory is used to explain the existence of linear differential equations and the use of Implicity Function and fixed point Theorems.

Abstract: Foreword to the Classics Edition Preface to the First Edition Preface to the Second Edition Errata I: Preliminaries II: Existence III: Differential In qualities and Uniqueness IV: Linear Differential Equations V: Dependence on Initial Conditions and Parameters VI: Total and Partial Differential Equations VII: The Poincare-Bendixson Theory VIII: Plane Stationary Points IX: Invariant Manifolds and Linearizations X: Perturbed Linear Systems XI: Linear Second Order Equations XII: Use of Implicity Function and Fixed Point Theorems XIII: Dichotomies for Solutions of Linear Equations XIV: Miscellany on Monotomy Hints for Exercises References Index.

8,982 citations

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01 Jan 2004

TL;DR: The fourth volume in a series of volumes devoted to self-contained and up-to-date surveys in the theory of ODEs was published by as discussed by the authors, with an additional effort to achieve readability for mathematicians and scientists from other related fields so that the chapters have been made accessible to a wider audience.

Abstract: This handbook is the fourth volume in a series of volumes devoted to self contained and up-to-date surveys in the theory of ordinary differential equations, with an additional effort to achieve readability for mathematicians and scientists from other related fields so that the chapters have been made accessible to a wider audience. It covers a variety of problems in ordinary differential equations. It provides pure mathematical and real world applications. It is written for mathematicians and scientists of many related fields.

7,736 citations