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Initial value problem

About: Initial value problem is a(n) research topic. Over the lifetime, 27818 publication(s) have been published within this topic receiving 519226 citation(s). The topic is also known as: cauchy problem.
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Book
11 Feb 1992-
Abstract: 1 Generation and Representation.- 1.1 Uniformly Continuous Semigroups of Bounded Linear Operators.- 1.2 Strongly Continuous Semigroups of Bounded Linear Operators.- 1.3 The Hille-Yosida Theorem.- 1.4 The Lumer Phillips Theorem.- 1.5 The Characterization of the Infinitesimal Generators of C0 Semigroups.- 1.6 Groups of Bounded Operators.- 1.7 The Inversion of the Laplace Transform.- 1.8 Two Exponential Formulas.- 1.9 Pseudo Resolvents.- 1.10 The Dual Semigroup.- 2 Spectral Properties and Regularity.- 2.1 Weak Equals Strong.- 2.2 Spectral Mapping Theorems.- 2.3 Semigroups of Compact Operators.- 2.4 Differentiability.- 2.5 Analytic Semigroups.- 2.6 Fractional Powers of Closed Operators.- 3 Perturbations and Approximations.- 3.1 Perturbations by Bounded Linear Operators.- 3.2 Perturbations of Infinitesimal Generators of Analytic Semigroups.- 3.3 Perturbations of Infinitesimal Generators of Contraction Semigroups.- 3.4 The Trotter Approximation Theorem.- 3.5 A General Representation Theorem.- 3.6 Approximation by Discrete Semigroups.- 4 The Abstract Cauchy Problem.- 4.1 The Homogeneous Initial Value Problem.- 4.2 The Inhomogeneous Initial Value Problem.- 4.3 Regularity of Mild Solutions for Analytic Semigroups.- 4.4 Asymptotic Behavior of Solutions.- 4.5 Invariant and Admissible Subspaces.- 5 Evolution Equations.- 5.1 Evolution Systems.- 5.2 Stable Families of Generators.- 5.3 An Evolution System in the Hyperbolic Case.- 5.4 Regular Solutions in the Hyperbolic Case.- 5.5 The Inhomogeneous Equation in the Hyperbolic Case.- 5.6 An Evolution System for the Parabolic Initial Value Problem.- 5.7 The Inhomogeneous Equation in the Parabolic Case.- 5.8 Asymptotic Behavior of Solutions in the Parabolic Case.- 6 Some Nonlinear Evolution Equations.- 6.1 Lipschitz Perturbations of Linear Evolution Equations.- 6.2 Semilinear Equations with Compact Semigroups.- 6.3 Semilinear Equations with Analytic Semigroups.- 6.4 A Quasilinear Equation of Evolution.- 7 Applications to Partial Differential Equations-Linear Equations.- 7.1 Introduction.- 7.2 Parabolic Equations-L2 Theory.- 7.3 Parabolic Equations-Lp Theory.- 7.4 The Wave Equation.- 7.5 A Schrodinger Equation.- 7.6 A Parabolic Evolution Equation.- 8 Applications to Partial Differential Equations-Nonlinear Equations.- 8.1 A Nonlinear Schroinger Equation.- 8.2 A Nonlinear Heat Equation in R1.- 8.3 A Semilinear Evolution Equation in R3.- 8.4 A General Class of Semilinear Initial Value Problems.- 8.5 The Korteweg-de Vries Equation.- Bibliographical Notes and Remarks.

10,921 citations




Journal ArticleDOI
Abstract: A method for solving the initial-value problem of the Korteweg-deVries equation is presented which is applicable to initial data that approach a constant sufficiently rapidly as $|x|\ensuremath{\rightarrow}\ensuremath{\infty}$. The method can be used to predict exactly the "solitons," or solitary waves, which emerge from arbitrary initial conditions. Solutions that describe any finite number of solitons in interaction can be expressed in closed form.

3,602 citations


01 Jan 1967-
Abstract: Keywords: equations : differentielles ; stabilite ; transport ; transfert de chaleur ; mecanique des : fluides ; ondes Reference Record created on 2005-11-18, modified on 2016-08-08

3,446 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202238
20211,245
20201,348
20191,269
20181,125
20171,164

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Topic's top 5 most impactful authors

Ravi P. Agarwal

51 papers, 1.5K citations

Donal O'Regan

42 papers, 809 citations

T. E. Simos

39 papers, 1.8K citations

Philippe G. LeFloch

30 papers, 316 citations

Mouffak Benchohra

25 papers, 653 citations