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1.

図書

図書
Christian Constanda
出版情報: Boca Raton : Chapman & Hall/CRC, c2010  xviii, 325 p. ; 24 cm
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Foreword
Preface to the Second Edition
Preface to the First Edition
Ordinary Differential Equations: Brief Review / Chapter 1:
First-Order Equations / 1.1:
Homogeneous Linear Equations with Constant Coefficients / 1.2:
Nonhomogeneous Linear Equations with Constant Coefficients / 1.3:
Cauchy-Euler Equations / 1.4:
Functions and Operators / 1.5:
Exercises
Fourier Series / Chapter 2:
The Full Fourier Series / 2.1:
Fourier Sine Series / 2.2:
Fourier Cosine Series / 2.3:
Convergence and Differentiation / 2.4:
Sturm-Liouville Problems / Chapter 3:
Regular Sturm-Liouville Problems / 3.1:
Other Problems / 3.2:
Bessel Functions / 3.3:
Legendre Polynomials / 3.4:
Spherical Harmonics / 3.5:
Some Fundamental Equations of Mathematical Physics / Chapter 4:
The Heat Equation / 4.1:
The Laplace Equation / 4.2:
The Wave Equation / 4.3:
Other Equations / 4.4:
The Method of Separation of Variables / Chapter 5:
Equations with More than Two Variables / 5.1:
Linear Nonhomogeneous Problems / Chapter 6:
Equilibrium Solutions / 6.1:
Nonhomogeneous Problems / 6.2:
The Method of Eigenfunction Expansion / Chapter 7:
The Fourier Transformations / 7.1:
The Full Fourier Transformation / 8.1:
The Fourier Sine and Cosine Transformations / 8.2:
Other Applications / 8.3:
The Laplace Transformation / Chapter 9:
Definition and Properties / 9.1:
Applications / 9.2:
The Method of Green's Functions / Chapter 10:
General Second-Order Linear Partial Differential Equations with Two Independent Variables / 10.1:
The Canonical Form / 11.1:
Hyperbolic Equations / 11.2:
Parabolic Equations / 11.3:
Elliptic Equations / 11.4:
The Method of Characteristics / Chapter 12:
First-Order Linear Equations / 12.1:
First-Order Quasilinear Equations / 12.2:
The One-Dimensional Wave Equation / 12.3:
Other Hyperbolic Equations / 12.4:
Perturbation and Asymptotic Methods / Chapter 13:
Asymptotic Series / 13.1:
Regular Perturbation Problems / 13.2:
Singular Perturbation Problems / 13.3:
Complex Variable Methods / Chapter 14:
Systems of Equations / 14.1:
Answers to Odd-Numbered Exercises
Appendix
Bibliography
Index
Foreword
Preface to the Second Edition
Preface to the First Edition
2.

図書

図書
J. Bourgain
出版情報: Providence, RI : American Mathematical Society, c1999  viii, 182 p. ; 26 cm
シリーズ名: Colloquium publications / American Mathematical Society ; v. 46
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Introduction and summary
An overview of results on the Cauchy problem for NLS
Further comments 3D $H^1$-critical defocusing NLS Global wellposedness below energy norm
Nonlinear Schrodinger equation with periodic boundary conditions
Growth of Sobolev norms in linear
Schrodinger equations with smooth time dependent potential
Zakharov systems
References
Index
Introduction and summary
An overview of results on the Cauchy problem for NLS
Further comments 3D $H^1$-critical defocusing NLS Global wellposedness below energy norm
3.

図書

図書
Ronghua Li, Zhongying Chen, Wei Wu
出版情報: New York : Marcel Dekker, c2000  xv, 442 p. ; 24 cm
シリーズ名: Monographs and textbooks in pure and applied mathematics ; 226
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Preface
Preliminaries / 1:
Sobolev Spaces / 1.1:
Smooth approximations. Fundamental lemma of variational methods / 1.1.1:
Generalized derivatives and Sobolev spaces / 1.1.2:
Imbedding and trace theorems / 1.1.3:
Finite element spaces / 1.1.4:
Interpolation error estimates in Sobolev spaces / 1.1.5:
Variational Problems and Their Approximations / 1.2:
Abstract variational form / 1.2.1:
Green's formulas and variational problems / 1.2.2:
Well-posedness of variational problems / 1.2.3:
Approximation methods. A necessary and sufficient condition for approximate-solvability / 1.2.4:
Galerkin methods / 1.2.5:
Generalized Galerkin methods / 1.2.6:
Bibliography and Comments
Two Point Boundary Value Problems / 2:
Basic Ideas of the Generalized Difference Method / 2.1:
A variational form / 2.1.1:
Generalized Galerkin variational principles / 2.1.2:
Generalized difference methods / 2.1.4:
Linear Element Difference Schemes / 2.2:
Trial and test function spaces / 2.2.1:
Difference equations / 2.2.2:
Convergence estimates / 2.2.3:
Quadratic Element Difference Schemes / 2.3:
Trial and test spaces / 2.3.1:
Convergence order estimates / 2.3.2:
Cubic Element Difference Schemes / 2.4:
Some lemmas / 2.4.1:
Existence, uniqueness and stability / 2.4.4:
Numerical examples / 2.4.5:
Estimates in L[superscript 2] and Maximum Norms / 2.5:
L[superscript 2]-estimates / 2.5.1:
Maximum norm estimates / 2.5.2:
Superconvergence / 2.6:
Optimal stress points / 2.6.1:
Superconvergence for linear element difference schemes / 2.6.2:
Superconvergence for cubic element difference schemes / 2.6.3:
Generalized Difference Methods for a Fourth Order Equation / 2.7:
Generalized difference equations / 2.7.1:
Positive definiteness of a(u[subscript h], II*[subscript h] u[subscript h]) / 2.7.2:
Second Order Elliptic Equations / 2.7.3:
Introduction / 3.1:
Generalized Difference Methods on Triangular Meshes / 3.2:
Generalized difference equation / 3.2.1:
a priori estimates / 3.2.3:
Error estimates / 3.2.4:
Generalized Difference Methods on Quadrilateral Meshes / 3.3:
Numerical example / 3.3.1:
L[superscript 2] and Maximum Norm Estimates / 3.5:
L[superscript 2] estimates / 3.6.1:
A maximum estimate and some remarks / 3.6.2:
Superconvergences / 3.7:
Weak estimate of interpolations / 3.7.1:
Superconvergence estimates / 3.7.2:
Fourth Order and Nonlinear Elliptic Equations / 4:
Mixed Generalized Difference Methods Based on Ciarlet-Raviart Variational Principle / 4.1:
Mixed generalized difference equations / 4.1.1:
Mixed Generalized Difference Methods Based on Hermann-Miyoshi Variational Principle / 4.1.2:
Numerical experiments / 4.2.1:
Nonconforming Generalized Difference Method Based on Zienkiewicz Elements / 4.3:
Variational principle / 4.3.1:
Generalized difference schemes based on Zienkiewicz elements / 4.3.2:
Error analyses / 4.3.3:
Numerical experiment / 4.3.4:
Nonconforming Generalized Difference Methods Based on Adini Elements / 4.4:
Generalized difference scheme / 4.4.1:
Error estimate / 4.4.2:
Second Order Nonlinear Elliptic Equations / 4.4.3:
Parabolic Equations / 4.5.1:
Semi-discrete Generalized Difference Schemes / 5.1:
Problem and schemes / 5.1.1:
L[superscript 2]-error estimate / 5.1.2:
H[superscript 1]-error estimate / 5.1.4:
Fully-discrete Generalized Difference Schemes / 5.2:
Fully-discrete schemes / 5.2.1:
Error estimates for backward Euler generalized difference schemes / 5.2.2:
Error estimates for Crank-Nicolson generalized difference schemes / 5.2.3:
Mass Concentration Methods / 5.3:
Construction of schemes / 5.3.1:
Error estimates for semi-discrete schemes / 5.3.2:
Error estimates for fully-discrete schemes / 5.3.3:
High Order Element Difference Schemes / 5.4:
Cubic element difference schemes for one-dimensional parabolic equations / 5.4.1:
Quadratic element difference schemes for two-dimensional parabolic equations / 5.4.2:
Generalized Difference Methods for Nonlinear Parabolic Equations / 5.5:
Hyperbolic Equations / 5.5.1:
Generalized Difference Methods for Second Order Hyperbolic Equations / 6.1:
Semi-discrete generalized difference scheme / 6.1.1:
Fully-discrete generalized difference scheme / 6.1.2:
Generalized Upwind Schemes for First Order Hyperbolic Equations / 6.2:
Generalized upwind schemes / 6.2.1:
Semi-discrete error estimates / 6.2.2:
Fully-discrete error estimates / 6.2.3:
Generalized Upwind Schemes for First Order Hyperbolic Systems / 6.3:
Integral forms / 6.3.1:
Generalized upwind difference schemes / 6.3.2:
Estimation of a bilinear form / 6.3.3:
Some practical difference schemes / 6.3.4:
A numerical example / 6.3.5:
Finite Volume Methods for Nonlinear Conservative Hyperbolic Equations / 6.4:
Convection-Dominated Diffusion Problems / 7:
One-Dimensional Characteristic Difference Schemes / 7.1:
Difference methods based on algebraic interpolations / 7.1.1:
Upwind difference schemes / 7.1.2:
Generalized Upwind Difference Schemes for Steady-state Problems / 7.2:
Construction of the difference schemes / 7.2.1:
Convergence and error estimate / 7.2.2:
Extreme value theorem and uniform convergence / 7.2.3:
Mass conservation / 7.2.4:
Generalized Upwind Difference Schemes for Nonsteady-state Problems / 7.3:
Construction of difference schemes / 7.3.1:
Highly Accurate Generalized Upwind Schemes / 7.3.2:
Upwind Schemes for Nonlinear Convection Problems / 7.4.1:
Applications / 8:
Planar Elastic Problems / 8.1:
Displacement methods / 8.1.1:
Mixed methods / 8.1.2:
Computation of Electromagnetic Fields / 8.2:
Numerical Simulation of Underground Water Pollution / 8.3:
Upwind weighted multi-element balancing method / 8.3.1:
Stokes Equation / 8.4:
Nonconforming generalized difference method / 8.4.1:
Coupled Sound-Heat Problems / 8.4.2:
Regularized Long Wave Equations / 8.6:
Semi-discrete generalized difference schemes / 8.6.1:
Fully-discrete generalized difference schemes / 8.6.2:
Hierarchical Basis Methods / 8.6.3:
Hierarchical Basis / 8.7.1:
Application to difference equations / 8.7.2:
Iteration methods / 8.7.3:
Bibliography / 8.7.4:
Index
Preface
Preliminaries / 1:
Sobolev Spaces / 1.1:
4.

図書

図書
Daniel J. Duffy
出版情報: Chichester : Wiley, c2006  xv, 423 p. ; 26 cm.
シリーズ名: Wiley finance series
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Goals of this Book and Global Overview / 0:
What is this book? / 0.1:
Why has this book been written? / 0.2:
For whom is this book intended? / 0.3:
Why should I read this book? / 0.4:
The structure of this book / 0.5:
What this book does not cover / 0.6:
Contact, feedback and more information / 0.7:
The Continuous Theory of Partial Differential Equations / Part I:
An Introduction to Ordinary Differential Equations / 1:
Introduction and objectives / 1.1:
Two-point boundary value problem / 1.2:
Linear boundary value problems / 1.3:
Initial value problems / 1.4:
Some special cases / 1.5:
Summary and conclusions / 1.6:
An Introduction to Partial Differential Equations / 2:
Partial differential equations / 2.1:
Specialisations / 2.3:
Parabolic partial differential equations / 2.4:
Hyperbolic equations / 2.5:
Systems of equations / 2.6:
Equations containing integrals / 2.7:
Second-Order Parabolic Differential Equations / 2.8:
Linear parabolic equations / 3.1:
The continuous problem / 3.3:
The maximum principle for parabolic equations / 3.4:
A special case: one-factor generalised Black-Scholes models / 3.5:
Fundamental solution and the Green's function / 3.6:
Integral representation of the solution of parabolic PDEs / 3.7:
Parabolic equations in one space dimension / 3.8:
An Introduction to the Heat Equation in One Dimension / 3.9:
Motivation and background / 4.1:
The heat equation and financial engineering / 4.3:
The separation of variables technique / 4.4:
Transformation techniques for the heat equation / 4.5:
An Introduction to the Method of Characteristics / 4.6:
First-order hyperbolic equations / 5.1:
Second-order hyperbolic equations / 5.3:
Applications to financial engineering / 5.4:
Propagation of discontinuities / 5.5:
Finite Difference Methods: The Fundamentals / 5.7:
An Introduction to the Finite Difference Method / 6:
Fundamentals of numerical differentiation / 6.1:
Caveat: accuracy and round-off errors / 6.3:
Where are divided differences used in instrument pricing? / 6.4:
Nonlinear initial value problems / 6.5:
Scalar initial value problems / 6.7:
An Introduction to the Method of Lines / 6.8:
Classifying semi-discretisation methods / 7.1:
Semi-discretisation in space using FDM / 7.3:
Numerical approximation of first-order systems / 7.4:
General Theory of the Finite Difference Method / 7.5:
Some fundamental concepts / 8.1:
Stability and the Fourier transform / 8.3:
The discrete Fourier transform / 8.4:
Stability for initial boundary value problems / 8.5:
Finite Difference Schemes for First-Order Partial Differential Equations / 8.6:
Scoping the problem / 9.1:
Why first-order equations are different: Essential difficulties / 9.3:
A simple explicit scheme / 9.4:
Some common schemes for initial value problems / 9.5:
Some common schemes for initial boundary value problems / 9.6:
Monotone and positive-type schemes / 9.7:
Extensions, generalisations and other applications / 9.8:
FDM for the One-Dimensional Convection-Diffusion Equation / 9.9:
Approximation of derivatives on the boundaries / 10.1:
Time-dependent convection-diffusion equations / 10.3:
Fully discrete schemes / 10.4:
Specifying initial and boundary conditions / 10.5:
Semi-discretisation in space / 10.6:
Semi-discretisation in time / 10.7:
Exponentially Fitted Finite Difference Schemes / 10.8:
Motivating exponential fitting / 11.1:
Exponential fitting and time-dependent convection-diffusion / 11.3:
Stability and convergence analysis / 11.4:
Approximating the derivative of the solution / 11.5:
Special limiting cases / 11.6:
Applying FDM to One-Factor Instrument Pricing / 11.7:
Exact Solutions and Explicit Finite Difference Method for One-Factor Models / 12:
Exact solutions and benchmark cases / 12.1:
Perturbation analysis and risk engines / 12.3:
The trinomial method: Preview / 12.4:
Using exponential fitting with explicit time marching / 12.5:
Approximating the Greeks / 12.6:
Appendix: the formula for Vega / 12.7:
An Introduction to the Trinomial Method / 13:
Motivating the trinomial method / 13.1:
Trinomial method: Comparisons with other methods / 13.3:
The trinomial method for barrier options / 13.4:
Exponentially Fitted Difference Schemes for Barrier Options / 13.5:
What are barrier options? / 14.1:
Initial boundary value problems for barrier options / 14.3:
Using exponential fitting for barrier options / 14.4:
Time-dependent volatility / 14.5:
Some other kinds of exotic options / 14.6:
Comparisons with exact solutions / 14.7:
Other schemes and approximations / 14.8:
Extensions to the model / 14.9:
Advanced Issues in Barrier and Lookback Option Modelling / 14.10:
Kinds of boundaries and boundary conditions / 15.1:
Discrete and continuous monitoring / 15.3:
Continuity corrections for discrete barrier options / 15.4:
Complex barrier options / 15.5:
The Meshless (Meshfree) Method in Financial Engineering / 15.6:
Motivating the meshless method / 16.1:
An introduction to radial basis functions / 16.3:
Semi-discretisations and convection-diffusion equations / 16.4:
Applications of the one-factor Black-Scholes equation / 16.5:
Advantages and disadvantages of meshless / 16.6:
Extending the Black-Scholes Model: Jump Processes / 16.7:
Jump-diffusion processes / 17.1:
Partial integro-differential equations and financial applications / 17.3:
Numerical solution of PIDE: Preliminaries / 17.4:
Techniques for the numerical solution of PIDEs / 17.5:
Implicit and explicit methods / 17.6:
Implicit-explicit Runge-Kutta methods / 17.7:
Using operator splitting / 17.8:
Splitting and predictor-corrector methods / 17.9:
FDM for Multidimensional Problems / 17.10:
Finite Difference Schemes for Multidimensional Problems / 18:
Elliptic equations / 18.1:
Diffusion and heat equations / 18.3:
Advection equation in two dimensions / 18.4:
Convection-diffusion equation / 18.5:
An Introduction to Alternating Direction Implicit and Splitting Methods / 18.6:
What is ADI, really? / 19.1:
Improvements on the basic ADI scheme / 19.3:
ADI for first-order hyperbolic equations / 19.4:
ADI classico and three-dimensional problems / 19.5:
The Hopscotch method / 19.6:
Boundary conditions / 19.7:
Advanced Operator Splitting Methods: Fractional Steps / 19.8:
Initial examples / 20.1:
Problems with mixed derivatives / 20.3:
Predictor-corrector methods (approximation correctors) / 20.4:
Partial integro-differential equations / 20.5:
More general results / 20.6:
Modern Splitting Methods / 20.7:
A different kind of splitting: The IMEX schemes / 21.1:
Applicability of IMEX schemes to Asian option pricing / 21.4:
Applying FDM to Multi-Factor Instrument Pricing / 21.5:
Options with Stochastic Volatility: The Heston Model / 22:
An introduction to Ornstein-Uhlenbeck processes / 22.1:
Stochastic differential equations and the Heston model / 22.3:
Using finite difference schemes: Prologue / 22.4:
A detailed example / 22.6:
Finite Difference Methods for Asian Options and Other 'Mixed' Problems / 22.7:
An introduction to Asian options / 23.1:
My first PDE formulation / 23.3:
Using operator splitting methods / 23.4:
Cheyette interest models / 23.5:
New developments / 23.6:
Multi-Asset Options / 23.7:
A taxonomy of multi-asset options / 24.1:
Common framework for multi-asset options / 24.3:
An overview of finite difference schemes for multi-asset problems / 24.4:
Numerical solution of elliptic equations / 24.5:
Solving multi-asset Black-Scholes equations / 24.6:
Special guidelines and caveats / 24.7:
Finite Difference Methods for Fixed-Income Problems / 24.8:
An introduction to interest rate modelling / 25.1:
Single-factor models / 25.3:
Some specific stochastic models / 25.4:
An introduction to multidimensional models / 25.5:
The thorny issue of boundary conditions / 25.6:
Introduction to approximate methods for interest rate models / 25.7:
Free and Moving Boundary Value Problems / 25.8:
Background to Free and Moving Boundary Value Problems / 26:
Notation and definitions / 26.1:
Some preliminary examples / 26.3:
Solutions in financial engineering: A preview / 26.4:
Numerical Methods for Free Boundary Value Problems: Front-Fixing Methods / 26.5:
An introduction to front-fixing methods / 27.1:
A crash course on partial derivatives / 27.3:
Functions and implicit forms / 27.4:
Front fixing for the heat equation / 27.5:
Front fixing for general problems / 27.6:
Multidimensional problems / 27.7:
Front fixing and American options / 27.8:
Other finite difference schemes / 27.9:
Viscosity Solutions and Penalty Methods for American Option Problems / 27.10:
Definitions and main results for parabolic problems / 28.1:
An introduction to semi-linear equations and penalty method / 28.3:
Implicit, explicit and semi-implicit schemes / 28.4:
Multi-asset American options / 28.5:
Variational Formulation of American Option Problems / 28.6:
A short history of variational inequalities / 29.1:
A first parabolic variational inequality / 29.3:
Functional analysis background / 29.4:
Kinds of variational inequalities / 29.5:
Variational inequalities using Rothe's methods / 29.6:
American options and variational inequalities / 29.7:
Design and Implementation In C++ / 29.8:
Finding the Appropriate Finite Difference Schemes for your Financial Engineering Problem / 30:
The financial model / 30.1:
The viewpoints in the continuous model / 30.3:
The viewpoints in the discrete model / 30.4:
Auxiliary numerical methods / 30.5:
New Developments / 30.6:
Design and Implementation of First-Order Problems / 30.7:
Software requirements / 31.1:
Modular decomposition / 31.3:
Useful C++ data structures / 31.4:
One-factor models / 31.5:
Multi-factor models / 31.6:
Generalisations and applications to quantitative finance / 31.7:
Appendix: Useful data structures in C++ / 31.8:
Moving to Black-Scholes / 32:
The PDE model / 32.1:
The FDM model / 32.3:
Algorithms and data structures / 32.4:
The C++ model / 32.5:
Test case: The two-dimensional heat equation / 32.6:
Finite difference solution / 32.7:
Moving to software and method implementation / 32.8:
Generalisations / 32.9:
C++ Class Hierarchies for One-Factor and Two-Factor Payoffs / 32.10:
Abstract and concrete payoff classes / 33.1:
Using payoff classes / 33.3:
Lightweight payoff classes / 33.4:
Super-lightweight payoff functions / 33.5:
Payoff functions for multi-asset option problems / 33.6:
Caveat: non-smooth payoff and convergence degradation / 33.7:
Appendices / 33.8:
An introduction to integral and partial integro-differential equations / A1:
An introduction to the finite element method / A2:
Bibliography
Index
Goals of this Book and Global Overview / 0:
What is this book? / 0.1:
Why has this book been written? / 0.2:
5.

図書

図書
Daniele Funaro
出版情報: Berlin ; New York : Springer-Verlag, c1997  x, 211 p. ; 24 cm
シリーズ名: Lecture notes in computational science and engineering ; 1
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6.

図書

図書
Olle Stormark
出版情報: Cambridge ; New York : Cambridge University Press, 2000  xv, 572 p.
シリーズ名: Encyclopedia of mathematics and its applications / edited by G.-C. Rota ; v. 80
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7.

図書

図書
Bhimsen K. Shivamoggi
出版情報: Boston : Birkhäuser, c2003  xiv, 354 p. ; 24 cm
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8.

図書

図書
Arthur G. Hansen
出版情報: Englewood Cliffs, N.J. : Prentice-Hall, c1964  xiv, 114 p. ; 24 cm
シリーズ名: Prentice-Hall series in engineering of the physical sciences
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9.

図書

図書
Victor P. Pikulin, Stanislav I. Pohozaev ; translated from the Russian by Andrei Iacob
出版情報: Basel : Birkhäuser Verlag, c2001  viii, 206 p. ; 24 cm
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10.

図書

図書
Jens M. Melenk
出版情報: Berlin ; Tokyo : Springer, c2002  xiv, 318 p. ; 24 cm
シリーズ名: Lecture notes in mathematics ; 1796
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Introduction / 1:
Finite Element Approximation / Part I:
hp-FEM for Reaction Diffusion Problems: Principal Results / 2:
hp Approximation / 3:
Regularity in Countably Normed Spaces / Part II:
The Countably Normed Spaces blb,e / 4:
Regularity Theory in Countably Normed Spaces / 5:
Regularity in Terms of Asymptotic Expansions / Part III:
Exponentially Weighted Countably Normed Spaces / 6:
Appendix
References
Index.
Introduction / 1:
Finite Element Approximation / Part I:
hp-FEM for Reaction Diffusion Problems: Principal Results / 2:
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