Topic
Numerical partial differential equations
About: Numerical partial differential equations is a research topic. Over the lifetime, 20199 publications have been published within this topic receiving 703797 citations.
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Abstract: Maxwell's equations are replaced by a set of finite difference equations. It is shown that if one chooses the field points appropriately, the set of finite difference equations is applicable for a boundary condition involving perfectly conducting surfaces. An example is given of the scattering of an electromagnetic pulse by a perfectly conducting cylinder.
14,070 citations
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02 Mar 2006
TL;DR: In this article, the authors present a method for solving Fractional Differential Equations (DFE) using Integral Transform Methods for Explicit Solutions to FractionAL Differentially Equations.
Abstract: 1. Preliminaries. 2. Fractional Integrals and Fractional Derivatives. 3. Ordinary Fractional Differential Equations. Existence and Uniqueness Theorems. 4. Methods for Explicitly solving Fractional Differential Equations. 5. Integral Transform Methods for Explicit Solutions to Fractional Differential Equations. 6. Partial Fractional Differential Equations. 7. Sequential Linear Differential Equations of Fractional Order. 8. Further Applications of Fractional Models. Bibliography Subject Index
11,492 citations
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TL;DR: The Duality Principle relating stochastic estimation and deterministic control problems plays an important role in the proof of theoretical results and properties of the variance equation are of great interest in the theory of adaptive systems.
Abstract: A nonlinear differential equation of the Riccati type is derived for the covariance matrix of the optimal filtering error. The solution of this \"variance equation\" completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or nonstationary statistics. The variance equation is closely related to the Hamiltonian (canonical) differential equations of the calculus of variations. Analytic solutions are available in some cases. The significance of the variance equation is illustrated by examples which duplicate, simplify, or extend earlier results in this field. The Duality Principle relating stochastic estimation and deterministic control problems plays an important role in the proof of theoretical results. In several examples, the estimation problem and its dual are discussed side-by-side. Properties of the variance equation are of great interest in the theory of adaptive systems. Some aspects of this are considered briefly.
6,152 citations
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TL;DR: In this paper, the authors present a statistical approach to the study of integro-differential equations that occur in various branches of the natural sciences, such as biology and chemistry.
Abstract: We shall present here the motivation and a general description of a method dealing with a class of problems in mathematical physics. The method is, essentially, a statistical approach to the study of differential equations, or more generally, of integro-differential equations that occur in various branches of the natural sciences.
5,432 citations
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01 Jan 1978
TL;DR: A self-contained presentation of the methods of asymptotics and perturbation theory, methods useful for obtaining approximate analytical solutions to differential and difference equations is given in this paper.
Abstract: This book gives a self-contained presentation of the methods of asymptotics and perturbation theory, methods useful for obtaining approximate analytical solutions to differential and difference equations. Parts and chapter titles are as follows: fundamentals - ordinary differential equations, difference equations; local analysis - approximate solution of linear differential equations, approximate solution of nonlinear differential equations, approximate solution of difference equations, asymptotic expansion of integrals; perturbation methods - perturbation series, summation series; and global analysis - boundary layer theory, WKB theory, multiple-scale analysis. An appendix of useful formulas is included. 147 figures, 43 tables. (RWR)
4,776 citations