In recent years the study of numerical methods for solving ordinary differential equations has seen many new developments. This second edition of the author's pioneering text is fully revised and updated to acknowledge many of these developments. It includes a complete treatment of linear multistep methods whilst maintaining its unique and comprehensive emphasis on Runge-Kutta methods and general linear methods.
Although the specialist topics are taken to an advanced level, the entry point to the volume as a whole is not especially demanding.
n mathematics, an ordinary differential equation (abbreviated ODE) is an equation containing a function of one independent variable and its derivatives. There are many general forms an ODE can take, and these are classified in practice (see below). The derivatives are ordinary because partial derivatives only apply to functions of many independent variables (see Partial differential equation).
Document "The Mathematical Theory of Maxwell’s Equations" give you the knowledge: The Variational Expansion into Wave Functions, Scattering From a Perfect Conductor, Approach to the Cavity Problem, Boundary Integral Equation Methods for Lipschitz Domains,...
The goal of this book is to develop robust, accurate and efficient numerical methods to price a
number of derivative products in quantitative finance.We focus on one-factor and multi-factor
models for a wide range of derivative products such as options, fixed income products, interest
rate products and ‘real’ options. Due to the complexity of these products it is very difficult to
find exact or closed solutions for the pricing functions. Even if a closed solution can be found
it may be very difficult to compute. For this and other reasons we need to resort to approximate
Computational fluid dynamics (CFD) is concerned with the efficient numerical solution of the partial differential equations that describe fluid dynamics. CFD techniques are commonly used in the many areas of engineering where fluid behavior is an important factor. Traditional fields of application include aerospace and automotive design, and more recently, bioengineering and consumer and medical electronics.
This book presents and develops major numerical methods currently used for solving
problems arising in quantitative finance. Our presentation splits into two parts.
Part I is methodological, and offers a comprehensive toolkit on numerical methods
and algorithms. This includes Monte Carlo simulation, numerical schemes for
partial differential equations, stochastic optimization in discrete time, copula functions,
transform-based methods and quadrature techniques.
Part II is practical, and features a number of self-contained cases.
This section attempts to answer some of the questions you might formulate when you turn the first page: What does this toolbox do? Can I use it? What problems can I solve?, etc.
What Does this Toolbox Do?
The Partial Differential Equation (PDE) Toolbox provides a powerful and flexible environment for the study and solution of partial differential equations in two space dimensions and time. The equations are discretized by the Finite Element Method (FEM). The objectives of the PDE Toolbox are to provide you with tools that:
• Define a PDE problem, i.e.
Many mechanics and physics problems have variational formulations making them appropriate for numerical treatment by finite element techniques and efficient iterative methods. This book describes the mathematical background and reviews the techniques for solving problems, including those that require large computations such as transonic flows for compressible fluids and the Navier-Stokes equations for incompressible viscous fluids.
The goal of this book is to expose the reader to modern computational tools for
solving differential equation models that arise in chemical engineering, e.g.,
diffusion-reaction, mass-heat transfer, and fluid flow. The emphasis is placed
on the understanding and proper use of software packages. In each chapter we
outline numerical techniques that either illustrate a computational property of
interest or are the underlying methods of a computer package. At the close of
each chapter a survey of computer packages is accompanied by examples of
This work is intended to survey the basic theory that underlies the multitude of
parameter-rich models that dominate the hydrological literature today. It is concerned
with the application of the equation of continuity (which is the fundamental theorem of
hydrology) in its complete form combined with a simplified representation of the
principle of conservation of momentum. Since the equation of continuity can be
expressed in linear form by a suitable choice of state variables and is also parameterfree,
it can be readily formulated at all scales of interest.
This material is taught in the BSc. Mathematics degree programme at the Manchester Metropolitan University, UK. The Finite Volume Method (FVM) is taught after the Finite Difference Method (FDM) where important concepts such as convergence, consistency and stability are presented. The FDM material is contained in the online textbook, ‘Introductory Finite Difference Methods for PDEs’ which is free to download from:
The emergence of a new paradigm in science offers vast perspectives for future
investigations, as well as providing fresh insight into existing areas of knowledge,
discovering hitherto unknown relations between them. We can observe
this kind of process in connection with the appearance of the concept of solitons
This text is intended to provide an introduction to the standard methods that are used for the solution of first-order partial
differential equations. Some of these ideas are likely to be introduced, probably in a course on mathematical methods
during the second year of a degree programme with, perhaps, more detail in a third year. The material has been written
to provide a general – but broad – introduction to the relevant ideas, and not as a text closely linked to a specific module
or course of study. Indeed, the intention is to present the material so that it can be used as an...
The three texts in this one cover, entitled ‘The series solution of second order, ordinary differential equations and special
functions’ (Part I), ‘An introduction to Sturm-Liouville theory’ (Part II) and ‘Integral transforms’ (Part III), are three of
the ‘Notebook’ series available as additional and background reading to students at Newcastle University (UK).
Two decades ago when we wrote Spectral Methods in Fluid Dynamics (1988),
the subject was still fairly novel. Motivated by the many favorable comments
we have received and the continuing interest in that book (which will be
referred to as CHQZ1), and yet desiring to present a more modern perspective,
we embarked on the project which resulted in our recent book (Canuto
et al. (2006), referred to as CHQZ2) and the present new book (referred to
The present volume, compiled in honor of an outstanding historian of science,
physicist and exceptional human being, Sam Schweber, is unique in assembling
a broad spectrum of positions on the history of science by some of its leading
representatives. Readers will find it illuminating to learn how prominent authors
judge the current status and the future perspectives of their field. Students will find
this volume helpful as a guide in a fragmented field that continues to be dominated
by idiosyncratic expertise and still lacks a methodical canon.
What is computational physics? Here, we take it to mean techniques for simulating
continuous physical systems on computers. Since mathematical physics
expresses these systems as partial differential equations, an equivalent statement
is that computational physics involves solving systems of partial differential
equations on a computer.
This book is meant to provide an introduction to computational physics
to students in plasma physics and related disciplines. We present most of the
basic concepts needed for numerical solution of partial differential equations.
Tham khảo sách 'examples of applications of the power series method by solution of differential equations with polynomial coefﬁcients calculus 3c-4', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả