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Fast solvers for elliptic PDEs form a pillar of scientific computing. They enable detailed and accurate simulations of electromagnetic fields, fluid flows, biochemical processes, and much more. This textbook provides an introduction to fast solvers from the point of view of integral equation formulations, which lead to unparalleled accuracy and speed in many applications. The focus is on fast algorithms for handling dense matrices that arise in the discretization of integral operators, such as the fast multipole method and fast direct solvers. While the emphasis is on techniques for dense matrices, the text also describes how similar techniques give rise to linear complexity algorithms for computing the inverse or the LU factorization of a sparse matrix resulting from the direct discretization of an elliptic PDE. This is the first textbook to detail the active field of fast direct solvers, introducing readers to modern linear algebraic techniques for accelerating computations, such as randomized algorithms, interpolative decompositions, and data-sparse hierarchical matrix representations. Written with an emphasis on mathematical intuition rather than theoretical details, it is richly illustrated and provides pseudocode for all key techniques. Fast Direct Solvers for Elliptic PDEs is appropriate for graduate students in applied mathematics and scientific computing, engineers and scientists looking for an accessible introduction to integral equation methods and fast solvers, and researchers in computational mathematics who want to quickly catch up on recent advances in randomized algorithms and techniques for working with data-sparse matrices.
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• Preliminaries
• Function spaces
• Decomposition of the nonlinearity
• Statement of the main estimates
• Proof of the main theorem
• Interlude : Bilinear null form estimates
• Proof of the bilinear estimates
• Proof of the trilinear estimates
• Solvability of paradifferential covariant half-wave equations

In this paper, the authors prove global well-posedness of the massless Maxwell-Dirac equation in the Coulomb gauge on $\mathbb{R}^{1+d} (d\geq 4)$ for data with small scale-critical Sobolev norm, as well as modified scattering of the solutions. Main components of the authors' proof are A) uncovering null structure of Maxwell-Dirac in the Coulomb gauge, and B) proving solvability of the underlying covariant Dirac equation. A key step for achieving both is to exploit (and justify) a deep analogy between Maxwell-Dirac and Maxwell-Klein-Gordon (for which an analogous result was proved earlier by Krieger-Sterbenz-Tataru, which says that the most difficult part of Maxwell-Dirac takes essentially the same form as Maxwell-Klein-Gordon.
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• Main results
• Phase plane analysis
• Proofs of propositions 2.8, 2.12
• Preliminaries on the limit sets and zero number
• Proofs of the main theorems

• Introduction and main result Functional setting Transversality properties of degenerate KAM theory Nash-Moser theorem and measure estimates Approximate inverse The linearized operator in the normal directions Almost diagonalization and invertibility of $\mathcal{L}_{\omega}$ The Nash-Moser iteration Appendix A. Tame estimates for the flow of pseudo-PDEs Bibliography.
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• Introduction and main result
• Functional setting
• Transversality properties of degenerate KAM theory
• Nash-Moser theorem and measure estimates
• Approximate inverse
• The linearized operator in the normal directions
• Almost diagonalization and invertibility of Lω
• The Nash-Moser iteration.

The authors prove the existence and the linear stability of small amplitude time quasi-periodic standing wave solutions (i.e. periodic and even in the space variable $x$) of a 2-dimensional ocean with infinite depth under the action of gravity and surface tension. Such an existence result is obtained for all the values of the surface tension belonging to a Borel set of asymptotically full Lebesgue measure.
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• Remarks on the Mizohata-Takeuchi conjecture and related problems / Neal Bez and Mitsuru Sugimoto
• Asymptotics for the Schrödinger equation with a repulsive delta potential and a long-range dissipative nonlinearity / Masahiro Ikeda
• The sharp lower bound of the lifespan of solutions to semilinear wave equations with low powers in two space dimensions / Takuto Imai, Masakazu Kato, Hiroyuki Takamura and Kyouhei Wakasa
• Remarks on the asymptotic behavior of global solutions to systems of semilinear wave equations / Soichiro Katayama
• Instability of standing waves for a system of nonlinear Schrödinger equations in a degenerate case / Shotaro Kawahara and Masahito Ohta
• Energy structure and asymptotic profile of the linearized system of thermo-elastic equation in lower space dimensions / Yuki Kimura and Takayoshi Ogawa
• Decay estimate and asymptotic behavior of small solutions to Schrödinger equations with subcritical dissipative nonlinearity / Naoyasu Kita and Yoshihisa Nakamura
• Modification of the vector-field method related to quadratically perturbed wave equations in two space dimensions / Hideo Kubo
• Remarks on derivative nonlinear Schrödinger systems with multiple masses / Chunhua Li and Hideaki Sunagawa
• Properties of solutions to the Camassa-Holm equation on the line in a class containing the peakons / Felipe Linares, Gustavo Ponce and Thomas C Sideris
• Remarks on the Hardy type inequalities with remainder terms in the framework of equalities / Shuji Machihara, Tohru Ozawa and Hidemitsu Wadade
• On the scattering problem of mass-subcritical Hartree equation / Satoshi Masaki
• Global solutions for nonlinear Schrödinger equations in De Sitter spacetime / Makoto Nakamura
• On the factorization technique for the dispersive nonlinear equations / Pavel I Naumkin
• Remark on the scattering operator for the quintic nonlinear Dirac equation in one space dimension / Hironobu Sasaki
• Blowup for a complex Ginzburg-Landau equation focusing on the parabolicity / Takuya Tomidokoro and Tomomi Yokota
• Kato Type smoothing estimates for magnetic Schrödinger equations with rough potentials / Takeshi Wada
• Second order asymptotic expansion for wave equations with time-dependent dissipation in one-space dimension / Yuta Wakasugi

This volume is edited as the proceedings of the international conference 'Asymptotic Analysis for Nonlinear Dispersive and Wave Equations' held in September, 2014 at Department of Mathematics, Osaka University, Osaka, Japan. The conference was devoted to the honor of Professor Nakao Hayashi (Osaka University) on the occasion of his 60th birth year, and includes the newest results up to 2017 related to the fields of nonlinear partial differential equations of hyperbolic and dispersive type. In particular, the asymptotic expansion of solutions for those equations has been the main contribution of Professor Hayashi and his collaborators. The contents is 18 papers related to the asymptotic analysis and qualitative research paper concerning the problems of nonlinear wave equations and nonlinear dispersive equations such as nonlinear Schroedinger equations, the Hartree equation, the Camassa-Holm equation, the Ginzburg-Landau equations. Among others, the outstanding method developed by Professor Hayashi and his collaborators is introduced by one of his main collaborator, Professor P I Naumkin. This volume is suitable for any students and young researchers who are starting the research on the asymptotic analysis of nonlinear wave and dispersive equations for knowing the out-lined theory of these fields. Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America.
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• Introduction Preliminaries Meromorphic continuation of the resolvent Boundary layer operators Dynamical resonance free regions Existence of resonances for the Delta potential Appendix A. Model cases Appendix B. Semiclassical intersecting Lagrangian distributions Appendix C. The semiclassical Melrose-Taylor parametrix Bibliography.
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The author studies high energy resonances for the operators $-\Delta_{\partial\Omega, \delta}:=-\Delta \delta_{\partial\Omega}\otimes V\quad \textrm{and}\quad -\Delta_{\partial\Omega, \delta'}:=-\Delta \delta_{\partial\Omega}'\otimes V\partial_\nu$ where $\Omega\subset{\mathbb{R}}^{d}$ is strictly convex with smooth boundary, $V:L^{2}(\partial\Omega)\to L^{2}(\partial\Omega)$ may depend on frequency, and $\delta_{\partial\Omega}$ is the surface measure on $\partial\Omega$.
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• Introduction.- Analysis on Homogeneous Groups.- Hardy Inequalities on Homogeneous Groups.- Rellich, Caarelli-Kohn-Nirenberg, and Sobolev Type Inequalities.- Fractional Hardy Inequalities.- Integral Hardy Inequalities on Homogeneous Groups.- Horizontal Inequalities on Stratied Groups.- Hardy-Rellich Inequalities and Fundamental Solutions.- Geometric Hardy Inequalities on Stratied Groups.- Uncertainty Relations on Homogeneous Groups.- Function Spaces on Homogeneous Groups.- Elements of Potential Theory on Stratified Groups.- Hardy and Rellich Inequalities for Sums of Squares.- Bibliography.- Index.
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This open access book provides an extensive treatment of Hardy inequalities and closely related topics from the point of view of Folland and Stein's homogeneous (Lie) groups. The place where Hardy inequalities and homogeneous groups meet is a beautiful area of mathematics with links to many other subjects. While describing the general theory of Hardy, Rellich, Caffarelli-Kohn-Nirenberg, Sobolev, and other inequalities in the setting of general homogeneous groups, the authors pay particular attention to the special class of stratified groups. In this environment, the theory of Hardy inequalities becomes intricately intertwined with the properties of sub-Laplacians and subelliptic partial differential equations. These topics constitute the core of this book and they are complemented by additional, closely related topics such as uncertainty principles, function spaces on homogeneous groups, the potential theory for stratified groups, and the potential theory for general Hoermander's sums of squares and their fundamental solutions. This monograph is the winner of the 2018 Ferran Sunyer i Balaguer Prize, a prestigious award for books of expository nature presenting the latest developments in an active area of research in mathematics. As can be attested as the winner of such an award, it is a vital contribution to literature of analysis not only because it presents a detailed account of the recent developments in the field, but also because the book is accessible to anyone with a basic level of understanding of analysis. Undergraduate and graduate students as well as researchers from any field of mathematical and physical sciences related to analysis involving functional inequalities or analysis of homogeneous groups will find the text beneficial to deepen their understanding.
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• Introduction Distributions and function spaces Spectral theory of nonselfadjoint operators Linear stability of NLS solitary waves Solitary waves of nonlinear Schrodinger equation Limiting absorption principle Carleman-Berthier-Georgescu estimates The Dirac matrices The Soler model Bi-frequency solitary waves Bifurcations of eigenvalues from the essential spectrum Nonrelativistic asymptotics of solitary waves Spectral stability in the nonrelativistic limit Bibliography Index List of symbols.
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This monograph gives a comprehensive treatment of spectral (linear) stability of weakly relativistic solitary waves in the nonlinear Dirac equation. It turns out that the instability is not an intrinsic property of the Dirac equation that is only resolved in the framework of the second quantization with the Dirac sea hypothesis. Whereas general results about the Dirac-Maxwell and similar equations are not yet available, we can consider the Dirac equation with scalar self-interaction, the model first introduced in 1938. In this book we show that in particular cases solitary waves in this model may be spectrally stable (no linear instability). This result is the first step towards proving asymptotic stability of solitary waves. The book presents the necessary overview of the functional analysis, spectral theory, and the existence and linear stability of solitary waves of the nonlinear Schrodinger equation. It also presents the necessary tools such as the limiting absorption principle and the Carleman estimates in the form applicable to the Dirac operator, and proves the general form of the Dirac-Pauli theorem. All of these results are used to prove the spectral stability of weakly relativistic solitary wave solutions of the nonlinear Dirac equation.
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• Review of Fractional Differentiation
• Special Functions
• The Gamma Function
• The Beta Function
• The Complementary Error Function
• The Mittag-Leffler Function
• Laplace Transformation and Convolution
• Riemann-Liouville Fractional Differentiation
• Caputo Fractional Derivative
• Classical Fractional Derivatives
• Partial Riemann-Liouville Fractional Derivative
• Fractional Operators with Variable Order
• Tempered Fractional Differentiation
• Properties of Tempered Fractional Derivative and Integral
• Laplace Transforms of the Tempered Fractional Calculus
• Caputo-Fabrizio Fractional Differentiation
• Caputo-Fabrizio Fractional Derivative in Caputo Sense
• The Laplace Transform of the Caputo-Fabrizio Fractional Derivative
• Fourier Transform of Fractional Gradient, Divergence and Laplacian
• Caputo-Fabrizio Fractional Derivative in Riemann-Liouville Sense
• Stretch Fractional Differentiation
• The Atangana-Baleanu Fractional Derivative and Integral
• The Riesz Potential and Riesz Fractional Derivatives
• The Atangana-Gómez Fractional Derivative
• References
• Finite Difference Approximations
• Finite Difference Approximation Schemes
• Taylor Series and Finite Difference Approximation
• Higher Order Finite Difference Approximation
• Error Analysis
• Illustrative Example
• Order of Accuracy and Consistency
• Matrix Notation
• Numerical Stability and Convergence Analysis
• Von Neumann (Fourier Series) Stability Analysis
• Matrix Stability Analysis
• Convergence
• Symmetry
• Numerical Results
• Finite Difference Approximations Schemes for Fractional Equations
• Matrix Representation of the Finite Difference Schemes
• Convergence Analysis of Fractional Finite Difference Schemes
• Stability Analysis of Fractional Finite Difference Schemes
• Numerical Approximation of Time-Fractional Sub-diffusion Process with the Second-Order Implicit Difference Method
• Second-Order Implicit Difference Approximation
• References
• Numerical Approximation of Riemann-Liouville Differentiation
• Numerical Approximation for Time Derivative
• Numerical Approximation for Space First-Order Derivative
• Numerical Approximation for Space Second-Order Derivative
• Crank-Nicholson Scheme for Time-Fractional Differential Equations in Riemann-Liouville Sense
• A New Definition of Fractional Time Derivative in Riemann-Liouville Sense
• References
• Numerical Approximation of Caputo Differentiation
• Numerical Approximation for Time Derivative
• Numerical Approximation for Space First-Order Derivative
• Numerical Methods for Fractional Evolution Equations
• Fractional Euler and Adams Methods
• The New Fractional Adams-Bashforth Scheme with Caputo Derivative
• Existence and Uniqueness of Solutions
• References
• Numerical Approximation of Caputo-Fabrizio Differentiation
• Numerical Approximation for the Caputo-Fabrizio Fractional Derivative in Caputo Sense
• Numerical Approximation for Time Derivative
• Numerical Approximation for Space First-Order Derivative
• Numerical Approximation for Space Second-Order Derivative
• Three-Step Adams-Bashforth Scheme with Caputo-Fabrizio Fractional Derivative
• Stability Analysis
• Applications
• References
• Numerical Approximation of Atangana-Baleanu Differentiation
• Atangana-Baleanu Fractional Derivative in Caputo Sense

• Uniqueness and Existence of Solution via Chaotic Process
• Numerical Experiments
• Example 1
• Reference
• Application to Ordinary Fractional Differential Equations
• Numerical Approximation of Fractional Ordinary Differential Equation with the Caputo Derivative
• Numerical Schemes and Stability Analysis
• Modelling the Spread of Viruses in Computer via the Caputo Fractional Derivative
• Modelling the Spread of River Blindness Disease with the Caputo Fractional Derivative
• Modelling of Nonlinear Interpersonal Relationship with Time-Fractional Derivative
• Modelling of El Niño Chaotic Dynamical System with the Caputo, Caputo-Fabrizio and Atangana-Baleanu Fractional Derivatives
• A Novel Fractional Model for the Lassa Hemorrhagic Fever
• Modelling of Ebola Hemorrhagic Fever : Fractional Derivative Approach
• References
• Application to Partial Fractional Differential Equation
• Space-Fractional Diffusion Equation with New (Atangana-Gomez) Fractional Derivative in Riemann-Liouville Sense
• Space-Fractional Diffusion Equation with the Riemann-Liouville Derivative
• Fourier Transform Methods
• Finite Difference Methods
• Predictor-Corrector Method of Approximation
• Fourier Spectral Method for Space-Fractional Reaction-Diffusion
• Application of Caputo-Fabrizio Derivative to Nonlinear Reaction-Diffusion
• Derivation of the Solution via Iterative Method
• Stability Analysis via Fixed Point Theorem
• Application of Caputo-Fabrizio Derivative to Transmission Line Model with Losses
• Stability Analysis of the Numerical Scheme with the Caputo-Fabrizio Derivative
• Application of the Caputo-Fabrizio Derivative in Caputo Sense to Time-Fractional Advection-Diffusion Equation
• Stability Anlalysis of the Numerical Method with the Caputo-Fabrizio Derivative for Caputo Type
• Convergence Analysis of the Numerical Scheme with the Caputo-Fabrizio Derivative for Caputo Type
• Applications of Fractional Derivatives to Diffusion-Advection Equation
• Riemann-Liouville Approach
• Caputo-Fabrizio-Riemann Approach
• Atangana-Baleanu-Riemann Approach
• Application to Partial Fractional Differential Equation
• The Fisher Equation
• The Gray-Scott Model
• Application of Riesz Fractional Derivative to Schrödinger Equation
• One-Dimensional Fractional-in-Space Schrödinger Equation
• Two-Dimensional Fractional-in-Space Schrödinger Equation
• Numerical Methods of Discretization
• Numerical Experiments
• One-Dimensional Results for Fractional Schrödinger Equation
• Two-Dimensional Results for Fractional Schrödinger Equation
• Three-Dimensional Results for Fractional Schrödinger Equation
• References
• Bibliography

This book discusses numerical methods for solving partial differential and integral equations, as well as ordinary differential and integral equations, involving fractional differential and integral operators. Differential and integral operators presented in the book include those with exponential decay law, known as Caputo-Fabrizio differential and integral operators, those with power law, known as Riemann-Liouville fractional operators, and those for the generalized Mittag-Leffler function, known as the Atangana-Baleanu fractional operators. The book reviews existing numerical schemes associated with fractional operators including those with power law, while also highlighting new trends in numerical schemes for recently introduced differential and integral operators. In addition, the initial chapters address useful properties of each differential and integral fractional operator. Methods discussed in the book are subsequently used to solved problems arising in many fields of science, technology, and engineering, including epidemiology, chaos, solitons, fractals, diffusion, groundwater, and fluid mechanics. Given its scope, the book offers a valuable resource for graduate students of mathematics and engineering, and researchers in virtually all fields of science, technology, and engineering, as well as an excellent addition to libraries.
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• Verification by Finite-Dimensional Projection
• Basic Principle of the Verification
• Fixed-Point Formulation of the Problem
• Finite-Dimensional Projection and Constructive Error Estimates
• Aubin-Nitsche Trick
• FS-Int : A Simple Enclosure Method of Solutions
• Criterion for Infinite-Dimensional Part
• Criterion for Finite-Dimensional Part
• Notes on FS-Int Programming
• An Application to the Second-Order Elliptic Boundary Value Problems
• Dirichlet Boundary Value Problems
• Constructive Error Estimations
• One-Dimensional Case
• Two-Dimensional Case
• Linear Triangular Element
• Fourier Basis
• Legendre Basis
• A Simple Example of FS-Int
• Other Examples of FS-Int
• Case of Non-convex Domain
• An Example for ... -Error Estimations on Non-convex Polygon
• Finite-Element Approximation
• Computation of the Constant c₂(h)
• Computation of the Constants c₃ and c₄
• Computation of the Constant K₁
• Computation of the Constant K₂
• Computation of the Constant K₃
• An Example
• Embedding Constants
• Newton-Type Approaches in Finite Dimension
• FN-Int : An Enclosure Method with Newton-Type Operator
• Deriving Newton-Type Operator Nh
• Fixed-Point Formulation by Newton-Type Operator
• Notes on FN-Int Programming
• A Simple Example of FN-Int
• Other Examples of FN-Int
• Simplifying Grad-Shafranov Equation
• Verification for Bifurcated Solutions
• Some Refinements
• Residual Formulation
• Direct Residual Formulation
• An Example of Direct Residual Formulation
• X*-Type Residual Formulation
• X*-Type a Posteriori Residual Formulation
• Examples
• Norm Estimation for the Finite-Dimensional Part
• Some Difficulties in FN-Int
• Alternative Computation for Finite-Dimensional Part
• Notes on FN-Norm Programming
• Examples
• Verification of the Local Uniqueness by FN-Int
• Residual Vector
• Candidate Set and Verification Condition
• Notes on FN-IntU Programming
• Example
• Verification of the Local Uniqueness for FN-Norm
• Candidate Set and Verification Condition
• Notes on FN-NormU Programming
• Examples
• Infinite-Dimensional Newton-Type Method
• A Verification Algorithm Based on Sequential Iterations
• Fixed-Point Form
• Verification Condition
• Local Uniqueness of the Solution
• Verification Algorithm IS-Res
• Notes on IS-Res Programming
• Extension of the Region for Local Uniqueness
• Examples
• Two-Point Boundary Value Problem
• Second-Order Elliptic Boundary Value Problem
• Fourth-Order Elliptic Problem
• A Verification Algorithm Based on Newton-Like Iterations
• Fixed-Point Formulation
• Verification Condition
• Local Uniqueness of the Solution
• Verification Algorithm
• Extension of the Region for Local Uniqueness
• The Invertibility of L and Computation of M
• A Method Based on Fixed-Point Formulation
• A Method of Direct Computation of M
• In the Case of Second-Order Elliptic Boundary Value Problems
• Examples
• Second-Order Elliptic Boundary Value Problems
• A Reaction Diffusion System
• Two-Dimensional Problem
• Fourth-Order Elliptic Problem
• Applications to the Computer-Assisted Proofs in Analysis
• Nonlinear Elliptic Boundary Value Problems
• Emden Equation
• Elliptic Equations with Neumann Boundary Conditions and Systems
• Stationary Navier-Stokes Problem
• Kolmogorov Problem
• Driven Cavity Problem
• Fixed-Point Formulation Based on IN-Linz
• Invertibility Condition of L and Computation of M
• Examples
• Heat Convection Problems
• Verification of Two-Dimensional Non Trivial Solutions
• Existence Proof of a Bifurcation Point
• Three-Dimensional Problems
• Enclosing, Excluding Eigenvalue Problems
• Second-Order Elliptic Eigenvalue Problem
• Orr-Sommerfeld Problem
• Eigenvalue Exclosure
• Other Applications
• Evolutional Equations
• Full Discretization of the Heat Equation
• Notations and Finite-Dimensional Projections
• Full-Discretization Scheme
• Error Estimates for Full-Discretization
• Preliminary Results for Semidiscretization

• Constructive Estimates for Full-Discretization
• Norm Estimates for the Inverse of Linear Parabolic Operator
• Discretized Linear Problem
• A Posteriori Estimates for ...
• Numerical Examples
• A Posteriori Estimates of the Inverse Parabolic Operator
• Verification Results for Solutions of Nonlinear Parabolic Equations
• A Related Fully Discrete Galerkin Scheme
• A Full Discretization Scheme for Heat Equation
• Application to the Verified Computation of Nonlinear Problems
• Numerical Examples
• Some Remarks on Other Approaches
• Computer-Assisted Proofs for Nonlinear Elliptic Boundary Value Problems via Eigenvalue Bounds
• Semilinear Elliptic Boundary Value Problems : Abstract Approach and Strong Solutions
• Abstract Formulation
• Strong Solutions
• Examples
• Emden's Equation on an Unbounded L-Shaped Domain
• A Nonlinear Schrödinger Equation on R²
• Gelfand's Equation on a Non-convex Polygonal Domain
• Embedding Constants
• Proof of Lemma 7.1
• Fourth-Order Problems
• Other Problem Types
• Solution Branches
• A Computer-Assisted Uniqueness Proof
• Turning and Bifurcation Points
• Turning Points
• Gelfand's Equation on a Square
• Symmetry-Breaking Bifurcations
• Symmetric Solution Branch
• Symmetry-Breaking Solution Branch
• An Alternative Approach
• A Duffing-Type Equation
• Non-self-Adjoint Eigenvalue Problems
• Eigenpair Enclosures
• Eigenvalue Exclosures
• The Orr-Sommerfeld Equation with Blasius Profile
• Goerisch Setting and Homotopy for Problem (9.131)
• Numerical Results and Instability Proof
• Systems and Non-symmetric Problems of Second Order
• Bound K for L⁻¹, Bijectivity of L
• Strong Formulation
• Weak Formulation
• Local Lipschitz Bound for Fʹ
• Eigenvalue Bounds for Self-Adjoint Eigenvalue Problems
• Eigenvalue Bounds for Self-Adjoint Operators
• Basic Properties
• Bounds for Eigenvalues in General Location
• Bounds to Eigenvalues Below ... (A)
• Poincaré's Min-Max Principle, Comparison Problems
• Eigenvalue Problems with Bilinear Forms
• The Associated Self-Adjoint Operator
• Eigenvalue Bounds for the Bilinear Form Problem
• Goerisch's Extension
• Comparison Problems, Homotopy Method
• Examples of Homotopies
• Coefficient Homotopy
• Domain Deformation Homotopy
• Domain Decomposition Homotopy
• Fourth-Order Problems, Boundary Homotopy
• Related Work and Tools
• Computer-Assisted Proofs for Dynamical Systems
• Topological Methods
• Fixed-Point Approaches
• Stationary Solutions of Problem (11.1)
• Time-Periodic Orbits
• Stable and Unstable Manifold
• Connecting Orbits
• Basic Tools
• Fixed-Point Formulation
• Some Pitfalls in Numerical Computation
• "Spurious" Solution in a Discretized Problem
• Rounding Errors in Matlab Linear Solver
• Rump's Example
• Interval Arithmetic
• Interval Representation
• The Four Operations of Interval Arithmetic
• Some Properties of Interval Arithmetic
• Introduction to Intlab
• Verifications for Finite-Dimensional Problems
• Nonlinear Systems in Rn
• Linear Systems
• Matrix Eigenvalue Problems
• Validation of Positive Definiteness of Matrices
• Spectral Norms
• For General Matrices
• For Hermitian Matrices
• For General Matrix G
• For Hermitian Matrix G
• An Upper Bound of Absolute Maximum Eigenvalues for Generalized Matrix Eigenvalue Problem -- g References
• Index

In the last decades, various mathematical problems have been solved by computer-assisted proofs, among them the Kepler conjecture, the existence of chaos, the existence of the Lorenz attractor, the famous four-color problem, and more. In many cases, computer-assisted proofs have the remarkable advantage (compared with a "theoretical" proof) of additionally providing accurate quantitative information. The authors have been working more than a quarter century to establish methods for the verified computation of solutions for partial differential equations, mainly for nonlinear elliptic problems of the form - u=f(x, u, u) with Dirichlet boundary conditions. Here, by "verified computation" is meant a computer-assisted numerical approach for proving the existence of a solution in a close and explicit neighborhood of an approximate solution. The quantitative information provided by these techniques is also significant from the viewpoint of a posteriori error estimates for approximate solutions of the concerned partial differential equations in a mathematically rigorous sense. In this monograph, the authors give a detailed description of the verified computations and computer-assisted proofs for partial differential equations that they developed. In Part I, the methods mainly studied by the authors Nakao and Watanabe are presented. These methods are based on a finite dimensional projection and constructive a priori error estimates for finite element approximations of the Poisson equation. In Part II, the computer-assisted approaches via eigenvalue bounds developed by the author Plum are explained in detail. The main task of this method consists of establishing eigenvalue bounds for the linearization of the corresponding nonlinear problem at the computed approximate solution. Some brief remarks on other approaches are also given in Part III. Each method in Parts I and II is accompanied by appropriate numerical examples that confirm the actual usefulness of the authors' methods. Also in some examples practical computer algorithms are supplied so that readers can easily implement the verification programs by themselves.
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• Preface Mohamed Ben Ayed, Mohamed Ali Jendoubi, Yomna Rebai, Hassna Riahi and Hatem Zaag-- Abbas Bahri: a dedicated life Mohamed Ben Ayed--
• 1. Blow-up rate for a semilinear wave equation with exponential nonlinearity in one space dimension Asma Azaiez, Nader Masmoudi and Hatem Zaag--
• 2. On the role of anisotropy in the weak stability of the Navier-Stokes system Hajer Bahouri, Jean-Yves Chemin and Isabelle Gallagher--
• 3. The motion law of fronts for scalar reaction-diffusion equations with multiple wells: the degenerate case Fabrice Bethuel and Didier Smets--
• 4. Finite-time blowup for some nonlinear complex Ginzburg-Landau equations Thierry Cazenave and Seifeddine Snoussi--
• 5. Asymptotic analysis for the Lane-Emden problem in dimension two Francesca de Marchis, Isabella Ianni and Filomena Pacella--
• 6. A data assimilation algorithm: the paradigm of the 3D Leray- model of turbulence Aseel Farhat, Evelyn Lunasin and Edriss S. Titi--
• 7. Critical points at infinity methods in CR geometry Najoua Gamara--
• 8. Some simple problems for the next generations Alain Haraux--
• 9. Clustering phenomena for linear perturbation of the Yamabe equation Angela Pistoia and Giusi Vaira--
• 10. Towards better mathematical models for physics Luc Tartar.
• (source: Nielsen Book Data)

In this edited volume leaders in the field of partial differential equations present recent work on topics in PDEs arising from geometry and physics. The papers originate from a 2015 research school organized by CIMPA and MIMS in Hammamet, Tunisia to celebrate the 60th birthday of the late Professor Abbas Bahri. The opening chapter commemorates his life and work. While the research presented in this book is cutting-edge, the treatment throughout is at a level accessible to graduate students. It includes short courses offering readers a unique opportunity to learn the state of the art in evolution equations and mathematical models in physics, which will serve as an introduction for students and a useful reference for established researchers. Finally, the volume includes many open problems to inspire the next generation.
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• Singular perturbations of elliptic boundary problems
• Second-order elliptic boundary value problems with a small leading part
• Introduction
• Application to stochastic differential equations
• The survival probability and the eigenvalue problem
• Discussion
• A primer of asymptotics for ODEs
• The laplace expansion of integrals
• The asymptotics of a first-order initial value problem
• Matched asymptotic expansions
• An Application to stochastic differential equations : the exit problem in R1
• Asymptotics of a second-order boundary value problem
• Asymptotics of a homogeneous second-order boundary value problem
• Asymptotics of the inhomogeneous boundary value problem
• Examples and applications to stochastic equations
• Small diffusion with the flow : the homogeneous boundary value problem
• Small diffusion against the flow
• Small diffusion against the flow : the inhomogeneous boundary value problem
• The boundary value problem with a sharp potential barrier
• The problem for a smooth potential barrier at the boundary
• The second eigenvalue of the fokker-planck operator
• A diffusion model of random signals
• Loss of lock in a first-order phase-locked loop in phase-modulated radio signals
• Annotations
• Singular perturbations in higher dimensions
• Introduction
• The WKB method
• The eikonal equation
• The transport equation
• The characteristic equations
• Boundary layers at non-characteristic boundaries
• Boundary layers at characteristic boundaries in the plane
• The boundary value problem with non-characteristic boundaries
• The boundary value problem in planar domains with characteristic boundaries
• Loss of lock in a second-order phase-locked loop
• The phase plane of the reduced problem
• The mean time to lose lock
• The boundary layer structure of ...
• Asymptotic solution of the stationary fokker-planck equation
• The eikonal equation for (3.136)
• The eikonal on the separatrix
• The transport equation
• Derivation of (3.114)
• Green's function for the boundary value problem is the exit density
• Annotations
• An attractor inside an unstable limit cycle
• The reduced equation : an underdamped forced pendulum
• Asymptotics of the fokker-planck equation near the limit cycle
• The boundary value problem for the fokker-planck equation in ...
• Annotations
• Eigenvalues of a non-self-adjoint elliptic operator
• Introduction
• Eigenvalues and the survival probability
• The principal eigenvalue and the structure of the field a(x)
• The precise WKB structure of the principal eigenfunction
• The eikonal equation
• The transport equation
• The boundary layer equation
• The first eigenfunction of the adjoint problem
• Higher-order eigenvalues
• Oscillatory escape time
• Spontaneous activity in the cerebral cortex
• Numerical study of oscillatory decay
• A model of up-state dynamics in a neuronal network
• The phase space of the model
• Brownian simulations of oscillation phenomena in (4.80)
• The exit density from a focus near a limit cycle
• The mean first passage time ...(x)
• Numerical study of the eikonal equation
• Normal flux on ... : the exit time density
• Computation of the second eigenvalue
• Brownian dynamics simulations
• A two-term approximation of the exit-time density
• Exit time densities in three ranges of noise amplitude
• Appendices
• The density of exit points
• Expansion of the field near the boundary and no cycling
• The lacobian of b... at ...
• The real part ...
• Annotations
• Short-time asymptotics of the heat kernel
• The one-dimensional case
• The ray method for short time asymptotics of green's function
• The trace
• Simply connected domains
• Multiply connected domains
• Recovering ... from P(t)
• Discussion
• Construction of the short-time asymptotic of the fokker-planck equation with a periodic potential
• Annotations
• Mixed boundary conditions for elliptic and parabolic equations
• The mixed boundary value problem
• Introduction
• Formulation of the mixed boundary value problem
• The narrow escape time problem
• A pathological example
• The matched asymptotics approach
• Higher-order asymptotics in the unit ball
• The narrow escape time through multiple absorbing windows
• Annotations
• The mixed boundary value problem in R2
• A neumann-dirichlet boundary value problem
• The neumann function
• The mixed boundary value problem on a riemannian manifold in R2
• Exit though several windows
• The helmholtz equation for two windows
• Asymptotic solution of the helmholtz equation
• The mixed boundary value problem for poisson's equation in dire straits
• The case of a bottleneck
• The case of several bottlenecks
• The mixed boundary value problem on a surface of revolution
• A composite domain with a bottleneck
• The narrow escape time from domains in r2 and r3 with bottlenecks
• The principal eigenvalue and bottlenecks
• Connecting head and neck
• The principal eigenvalue in dumbbell-shaped domains
• Diffusion of a needle in dire straits
• The diffusion law of a needle in a planar strip
• The turnaround time ...
• Applications of the narrow escape time
• Annotation to the narrow escape time problem
• Narrow escape in R3
• The neumann function in regular domains in R3
• Elliptic absorbing window
• Second-order asymptotics for a circular window
• The first eigenvalue for two small dirichlet windows
• Multiple absorbing windows
• Higher-order expansion of the net through many small windows on a sphere
• Application to leakage in a conductor of brownian particles
• Activation through a narrow opening
• The neumann function
• Solution of the mixed boundary value problem
• Deep well : a markov chain model
• The mixed boundary value problem in a solid funnel-shaped domain
• The mixed boundary value problem with a dirichlet ribbon
• Selected applications in molecular biophysics
• Leakage from a ccylinder
• Applications of the mixed boundary value problem
• Annotations
• Short-time asymptotics of the heat kernel and extreme statistics of the NET
• Introduction
• The pdf of the first escape time
• The pdf of the first arrival time in an interval
• Asymptotics of the expected shortest time
• Escape from a ray
• Escape from an interval ...
• The FAT in a bounded domain in R...
• Asymptotics in R3
• Asymptotics in R2
• Statistics of the arrival time of the second particle
• Poissonian-like approximation
• Pr... of N Brownian i.i.d. Trajectories in a segment
• Applications of the FAT in biophysics
• Annotations
• The poisson-nernst-planck equations in a ball
• Introduction
• Synopsis of results
• Poisson-nernst-planck equations in a ball
• The steady-state solution
• Existence of solutions
• The distribution of voltage and charge in a dielectric ball
• Scaling laws for the maximal number of charges
• Ionic flux in a small window at high charge
• Flow through a narrow window at high charge
• Current in a voltage-clamped dendritic spine
• Appendix 1 : reverse liouville-gelfand-bratu equation
• Small a expansion of ...(x)
• Numerical scheme for the solution of (10.13)
• Steady solution in a ball with a cusp-shaped funnel
• Reduced equations in an uncharged cusp-shaped funnel
• Asymptotics of voltage between funnel and center
• Poisson-nernst-planck solutions in a 3d cusp-shaped funnel
• Asymptotic analysis of the pnp equations in a cusp-shaped funnel
• The potential drop in ...
• Annotations
• Reconstruction of surface diffusion from projected data
• Projection of diffusion from a curve to a line
• Driftless diffusion on a curve
• The case of diffusion with drift
• Reconstruction of a parabola from projected diffusion data
• Appendix 2
• Reconstruction of projected stochastic dynamics
• Reconstruction of a surface from planar projections of diffusion trajectories
• The drift field
• The reconstruction procedure
• Annotations
• Asymptotic formulas in molecular and cellular biology
• Introduction
• From molecular to cellular description
• Flux through narrow passages identifies cellular compartments
• Examples of asymptotic formulas : fluxes into small targets
• Formulas in two dimensions
• Narrow escape formulas in three-dimensions
• Cusp-shaped funnel : hidden targets control rates in R...
• DNA repair in a confined chromatin structure in R...
• Asymmetric dumbbell-shaped cell division
• Annotations
• Bibliography
• Index.

This is a monograph on the emerging branch of mathematical biophysics combining asymptotic analysis with numerical and stochastic methods to analyze partial differential equations arising in biological and physical sciences. In more detail, the book presents the analytic methods and tools for approximating solutions of mixed boundary value problems, with particular emphasis on the narrow escape problem. Informed throughout by real-world applications, the book includes topics such as the Fokker-Planck equation, boundary layer analysis, WKB approximation, applications of spectral theory, as well as recent results in narrow escape theory. Numerical and stochastic aspects, including mean first passage time and extreme statistics, are discussed in detail and relevant applications are presented in parallel with the theory. Including background on the classical asymptotic theory of differential equations, this book is written for scientists of various backgrounds interested in deriving solutions to real-world problems from first principles.
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• 1. Notes on Weyl algebras and D-modules / Markus Brodmann
• 2. Inverse systems of local rings / Juan Elias
• 3. Lectures on the representation type of a projective variety / Rosa M. Miró-Roig
• 4. Simplicial toric varieties which are set-theoretic complete intersections / Marcel Morales.

This book presents four lectures on recent research in commutative algebra and its applications to algebraic geometry. Aimed at researchers and graduate students with an advanced background in algebra, these lectures were given during the Commutative Algebra program held at the Vietnam Institute of Advanced Study in Mathematics in the winter semester 2013 -2014. The first lecture is on Weyl algebras (certain rings of differential operators) and their D-modules, relating non-commutative and commutative algebra to algebraic geometry and analysis in a very appealing way. The second lecture concerns local systems, their homological origin, and applications to the classification of Artinian Gorenstein rings and the computation of their invariants. The third lecture is on the representation type of projective varieties and the classification of arithmetically Cohen -Macaulay bundles and Ulrich bundles. Related topics such as moduli spaces of sheaves, liaison theory, minimal resolutions, and Hilbert schemes of points are also covered. The last lecture addresses a classical problem: how many equations are needed to define an algebraic variety set-theoretically? It systematically covers (and improves) recent results for the case of toric varieties.
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• Introduction Wave equations The heat equation Laplace's equation Properties of the Fourier transform Wave equations on $\mathbb{R}^n$ Dispersion Conservation laws and shocks Bibliography Index.
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Does entropy really increase no matter what we do? Can light pass through a Big Bang? What is certain about the Heisenberg uncertainty principle? Many laws of physics are formulated in terms of differential equations, and the questions above are about the nature of their solutions. This book puts together the three main aspects of the topic of partial differential equations, namely theory, phenomenology, and applications, from a contemporary point of view. In addition to the three principal examples of the wave equation, the heat equation, and Laplace's equation, the book has chapters on dispersion and the Schrodinger equation, nonlinear hyperbolic conservation laws, and shock waves. The book covers material for an introductory course that is aimed at beginning graduate or advanced undergraduate level students. Readers should be conversant with multivariate calculus and linear algebra. They are also expected to have taken an introductory level course in analysis. Each chapter includes a comprehensive set of exercises, and most chapters have additional projects, which are intended to give students opportunities for more in-depth and open-ended study of solutions of partial differential equations and their properties.
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• Preface--
• 1. Mathematical preliminaries--
• 2. A survey of the fractional calculus--
• 3. From normal to anomalous diffusion--
• 4. Fractional diffusion equations: elementary applications--
• 5. Fractional diffusion equations: surface effects--
• 6. Fractional nonlinear diffusion equation--
• 7. Anomalous diffusion: anisotropic case--
• 8. Fractional Schroedinger equations--
• 9. Anomalous diffusion and impedance spectroscopy--
• 10. The Poisson-Nernst-Planck anomalous (PNPA) models-- References-- Index.
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Anomalous diffusion has been detected in a wide variety of scenarios, from fractal media, systems with memory, transport processes in porous media, to fluctuations of financial markets, tumour growth, and complex fluids. Providing a contemporary treatment of this process, this book examines the recent literature on anomalous diffusion and covers a rich class of problems in which surface effects are important, offering detailed mathematical tools of usual and fractional calculus for a wide audience of scientists and graduate students in physics, mathematics, chemistry and engineering. Including the basic mathematical tools needed to understand the rules for operating with the fractional derivatives and fractional differential equations, this self-contained text presents the possibility of using fractional diffusion equations with anomalous diffusion phenomena to propose powerful mathematical models for a large variety of fundamental and practical problems in a fast-growing field of research.
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• Gaussian Sobolev p-space.- Gaussian Campanato (p, k)-class.- Gaussian p-capacity.- Restriction of Gaussian Sobolev p-space.- Gaussian 1-capacity to Gaussian -capacity.- Gaussian BV-capacity.
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This monograph develops the Gaussian functional capacity theory with applications to restricting the Gaussian Campanato/Sobolev/BV space. Included in the text is a new geometric characterization of the Gaussian 1-capacity and the Gaussian Poincare 1-inequality. Applications to function spaces and geometric measures are also presented. This book will be of use to researchers who specialize in potential theory, elliptic differential equations, functional analysis, probability, and geometric measure theory.
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• Preface / Chiara Bianchini, Antoine Henrot, Rolando Magnanini
• Stable solutions to some elliptic problems : minimal cones, the Allen-Cahn equation, and blow-up solutions / Xavier Cabré and Giorgio Poggesi
• Isoperimetric inequalities for eigenvalues of the Laplacian / Antoine Henrot
• Topological aspects of critical points and level sets in elliptic PDEs / Alberto Encisco and Daniel Peralta-Salas
• Symmetry properties for solutions of higher-order elliptic boundary value problems / Wolfgang Reichel
• Recent trends in free boundary regularity / Henrik Shahgholian.

The aim of this book is to present different aspects of the deep interplay between Partial Differential Equations and Geometry. It gives an overview of some of the themes of recent research in the field and their mutual links, describing the main underlying ideas, and providing up-to-date references.Collecting together the lecture notes of the five mini-courses given at the CIME Summer School held in Cetraro (Cosenza, Italy) in the week of June 19-23, 2017, the volume presents a friendly introduction to a broad spectrum of up-to-date and hot topics in the study of PDEs, describing the state-of-the-art in the subject. It also gives further details on the main ideas of the proofs, their technical difficulties, and their possible extension to other contexts. Aiming to be a primary source for researchers in the field, the book will attract potential readers from several areas of mathematics.
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• Introduction: Some motivating questions-- The Cauchy-Kovalevskaya theorem with estimates-- Remarks on the Cauchy-Kovalevskaya theorem-- Zerner's theorem-- The method of globalizing families-- Holmgren's uniqueness theorem-- The continuity method of F. John-- The Bony-Schapira theorem-- Applications of the Bony-Schapira theorem: Part I - Vekua hulls-- Applications of the Bony-Schapira theorem: Part II - Szego's theorem revisited-- The reflection principle-- The reflection principle (continued)-- Cauchy problems and the Schwarz potential conjecture-- The Schwarz potential conjecture for spheres-- Potential theory on ellipsoids: Part I - The mean value property-- Potential theory on ellipsoids: Part II - There is no gravity in the cavity-- Potential theory on ellipsoids: Part III - The Dirichlet problem-- Singularities encountered by the analytic continuation of solutions to the Dirichlet problem-- An introduction to J. Leray's principle on propagation of singularities through $\mathbb{C}^n$-- Global propagation of singularities in $\mathbb{C}^n$-- Quadrature domains and Laplacian growth-- Other varieties of quadrature domains-- Bibliography-- Index.
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Why do solutions of linear analytic PDE suddenly break down? What is the source of these mysterious singularities, and how do they propagate? Is there a mean value property for harmonic functions in ellipsoids similar to that for balls? Is there a reflection principle for harmonic functions in higher dimensions similar to the Schwarz reflection principle in the plane? How far outside of their natural domains can solutions of the Dirichlet problem be extended? Where do the continued solutions become singular and why? This book invites graduate students and young analysts to explore these and many other intriguing questions that lead to beautiful results illustrating a nice interplay between parts of modern analysis and themes in ``physical'' mathematics of the nineteenth century. To make the book accessible to a wide audience including students, the authors do not assume expertise in the theory of holomorphic PDE, and most of the book is accessible to anyone familiar with multivariable calculus and some basics in complex analysis and differential equations.
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• An introduction to a fluid-structure model.- Linear parabolic-hyperbolic fluid-structure interaction models.- Flow-plate interactions: well-posedness and long-time behavior.- Some aspects in nonlinear acoustics coupling and shape optimization.
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This book is devoted to the study of coupled partial differential equation models, which describe complex dynamical systems occurring in modern scientific applications such as fluid/flow-structure interactions. The first chapter provides a general description of a fluid-structure interaction, which is formulated within a realistic framework, where the structure subject to a frictional damping moves within the fluid. The second chapter then offers a multifaceted description, with often surprising results, of the case of the static interface; a case that is argued in the literature to be a good model for small, rapid oscillations of the structure. The third chapter describes flow-structure interaction where the compressible Navier-Stokes equations are replaced by the linearized Euler equation, while the solid is taken as a nonlinear plate, which oscillates in the surrounding gas flow. The final chapter focuses on a the equations of nonlinear acoustics coupled with linear acoustics or elasticity, as they arise in the context of high intensity ultrasound applications.
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