This highly useful text shows the reader how to formulate a partial differential equation from the physical problem (constructing the mathematical model) and how to solve the equation (along with initial and boundary conditions). Written for advanced undergraduate and graduate students, as well as professionals working in the applied sciences, this clearly written book offers realistic, practical coverage of diffusion-type problems, hyperbolic-type problems, elliptic-type problems, and numerical and approximate methods. Each chapter contains a selection of relevant problems (answers are provided) and suggestions for further reading.
The book begins with a short review of calculus and ordinary differential equations, then moves on to explore integral curves and surfaces of vector fields, quasi-linear and linear equations of first order, series solutions and the Cauchy Kovalevsky theorem. It then delves into linear partial differential equations, examines the Laplace, wave and heat equations, and concludes with a brief treatment of hyperbolic systems of equations.
Among the most important features of the text are the challenging problems at the end of each section which require a wide variety of responses from students, from providing details of the derivation of an item presented to solving specific problems associated with partial differential equations. Requiring only a modest mathematical background, the text will be indispensable to those who need to use partial differential equations in solving physical problems. It will provide as well the mathematical fundamentals for those who intend to pursue the study of more advanced topics, including modern theory.
Subsequent chapters explore asymptotic behavior of solutions, semi-linear equations and free boundary problems, and the extension of results concerning fundamental solutions and the Cauchy problem to systems of parabolic equations. The final chapter concerns questions of existence and uniqueness for the first boundary value problem and the differentiability of solutions, in terms of both elliptic and parabolic equations. The text concludes with an appendix on nonlinear equations and bibliographies of related works.
Containing a careful selection of standard and timely topics, the Pocket Book of Integrals and Mathematical Formulas, Fourth Edition presents many numerical and statistical tables, scores of worked examples, and the most useful mathematical formulas for engineering and scientific applications. This fourth edition of a bestseller provides even more comprehensive coverage with the inclusion of several additional topics, all while maintaining its accessible, clear style and handy size.
New to the Fourth Edition
• An expanded chapter on series that covers many fascinating properties of the natural numbers that follow from number theory
• New applications such as geostationary satellite orbits and drug kinetics
• An expanded statistics section that discusses nonlinear regression as well as the normal approximation of the binomial distribution
• Revised format of the table of integrals for easier use of the forms and functions
Easy to Use on the Go
The book addresses a range of areas, from elementary algebra, geometry, matrices, and trigonometry to calculus, vector analysis, differential equations, and statistics. Featuring a convenient, portable size, it is sure to remain in the pockets or on the desks of all who use mathematical formulas and tables of integrals and derivatives.
Table of Contents:
Chapter 1. Fourier series
1.1 Basic concepts
1.2 Fourier series and Fourier coefficients
1.3 A minimizing property of the Fourier coefficients. The Riemann-Lebesgue theorem
1.4 Convergence of Fourier series
1.5 The Parseval formula
1.6 Determination of the sum of certain trigonometric series
Chapter 2. Orthogonal systems
2.1 Integration of complex-valued functions of a real variable
2.2 Orthogonal systems
2.3 Complete orthogonal systems
2.4 Integration of Fourier series
2.5 The Gram-Schmidt orthogonalization process
2.6 Sturm-Liouville problems
Chapter 3. Orthogonal polynomials
3.1 The Legendre polynomials
3.2 Legendre series
3.3 The Legendre differential equation. The generating function of the Legendre polynomials
3.4 The Tchebycheff polynomials
3.5 Tchebycheff series
3.6 The Hermite polynomials. The Laguerre polynomials
Chapter 4. Fourier transforms
4.1 Infinite interval of integration
4.2 The Fourier integral formula: a heuristic introduction
4.3 Auxiliary theorems
4.4 Proof of the Fourier integral formula. Fourier transforms
4.5 The convention theorem. The Parseval formula
Chapter 5. Laplace transforms
5.1 Definition of the Laplace transform. Domain. Analyticity
5.2 Inversion formula
5.3 Further properties of Laplace transforms. The convolution theorem
5.4 Applications to ordinary differential equations
Chapter 6. Bessel functions
6.1 The gamma function
6.2 The Bessel differential equation. Bessel functions
6.3 Some particular Bessel functions
6.4 Recursion formulas for the Bessel functions
6.5 Estimation of Bessel functions for large values of x. The zeros of the Bessel functions
6.6 Bessel series
6.7 The generating function of the Bessel functions of integral order
6.8 Neumann functions
Chapter 7. Partial differential equations of first order
7.2 The differential equation of a family of surfaces
7.3 Homogeneous differential equations
7.4 Linear and quasilinear differential equations
Chapter 8. Partial differential equations of second order
8.1 Problems in physics leading to partial differential equations
8.3 The wave equation
8.4 The heat equation
8.5 The Laplace equation
Answers to exercises; Bibliography; Conventions; Symbols; Index
Written on an advanced level, the book is aimed at advanced undergraduates and graduate students with a background in calculus, linear algebra, ordinary differential equations, and complex analysis. Over 260 carefully chosen exercises, with answers, encompass both routing and more challenging problems to help students test their grasp of the material.
In addition to expanded explanations, the 11th edition includes new problems, updated figures and examples to help motivate students. The program is primarily intended for undergraduate students of mathematics, science, or engineering, who typically take a course on differential equations during their first or second year of study. The main prerequisite for engaging with the program is a working knowledge of calculus, gained from a normal two?] or three?] semester course sequence or its equivalent. Some familiarity with matrices will also be helpful in the chapters on systems of differential equations.
Topics include ordinary differential equations in more than two variables, partial differential equations of the first and second orders, Laplace's equation, the wave equation, and the diffusion equation. A helpful Appendix offers information on systems of surfaces, and solutions to the odd-numbered problems appear at the end of the book. Readers pursuing independent study will particularly appreciate the worked examples that appear throughout the text.
This book surveys some of these new developments in analytical and numerical methods, and relates the two through a series of PDE examples. The PDEs that have been selected are largely "named'' since they carry the names of their original contributors. These names usually signify that the PDEs are widely recognized and used in many application areas. The authors’ intention is to provide a set of numerical and analytical methods based on the concept of a traveling wave, with a central feature of conversion of the PDEs to ODEs.
The Matlab and Maple software will be available for download from this website shortly.
www.pdecomp.netIncludes a spectrum of applications in science, engineering, applied mathematicsPresents a combination of numerical and analytical methodsProvides transportable computer codes in Matlab and Maple
Ingeniously relying on elementary algebra and just a smidgen of calculus, Professor Walker demonstrates how the underlying ideas behind wavelet analysis can be applied to solve significant problems in audio and image processing, as well in biology and medicine.
Nearly twice as long as the original, this new edition provides
· 104 worked examples and 222 exercises, constituting a veritable book of review material
· Two sections on biorthogonal wavelets
· A mini-course on image compression, including a tutorial on arithmetic compression
· Extensive material on image denoising, featuring a rarely covered technique for removing isolated, randomly positioned clutter
· Concise yet complete coverage of the fundamentals of time-frequency analysis, showcasing its application to audio denoising, and musical theory and synthesis
· An introduction to the multiresolution principle, a new mathematical concept in musical theory
· Expanded suggestions for research projects
· An enhanced list of references
· FAWAV: software designed by the author, which allows readers to duplicate described applications and experiment with other ideas.
To keep the book current, Professor Walker has created a supplementary website. This online repository includes ready-to-download software, and sound and image files, as well as access to many of the most important papers in the field.
The first part explores Markov processes and Brownian motion; the stochastic integral and stochastic differential equations; elliptic and parabolic partial differential equations and their relations to stochastic differential equations; the Cameron-Martin-Girsanov theorem; and asymptotic estimates for solutions. The section concludes with a look at recurrent and transient solutions.
Volume 2 begins with an overview of auxiliary results in partial differential equations, followed by chapters on nonattainability, stability and spiraling of solutions; the Dirichlet problem for degenerate elliptic equations; small random perturbations of dynamical systems; and fundamental solutions of degenerate parabolic equations. Final chapters examine stopping time problems and stochastic games and stochastic differential games. Problems appear at the end of each chapter, and a familiarity with elementary probability is the sole prerequisite.
Volume III concentrates on the classical aspects of gauge theory, describing the four fundamental forces by the curvature of appropriate fiber bundles. This must be supplemented by the crucial, but elusive quantization procedure.
The book is arranged in four sections, devoted to realizing the universal principle force equals curvature:
Part I: The Euclidean Manifold as a Paradigm
Part II: Ariadne's Thread in Gauge Theory
Part III: Einstein's Theory of Special Relativity
Part IV: Ariadne's Thread in Cohomology
For students of mathematics the book is designed to demonstrate that detailed knowledge of the physical background helps to reveal interesting interrelationships among diverse mathematical topics. Physics students will be exposed to a fairly advanced mathematics, beyond the level covered in the typical physics curriculum.
Quantum Field Theory builds a bridge between mathematicians and physicists, based on challenging questions about the fundamental forces in the universe (macrocosmos), and in the world of elementary particles (microcosmos).
Featuring a thoroughly revised presentation of topics, Beginning Partial Differential Equations, Third Edition provides a challenging, yet accessible, combination of techniques, applications, and introductory theory on the subjectof partial differential equations. The new edition offers nonstandard coverageon material including Burger’s equation, the telegraph equation, damped wavemotion, and the use of characteristics to solve nonhomogeneous problems.
The Third Edition is organized around four themes: methods of solution for initial-boundary value problems; applications of partial differential equations; existence and properties of solutions; and the use of software to experiment with graphics and carry out computations. With a primary focus on wave and diffusion processes, Beginning Partial Differential Equations, Third Edition also includes:Proofs of theorems incorporated within the topical presentation, such as the existence of a solution for the Dirichlet problem The incorporation of Maple™ to perform computations and experiments Unusual applications, such as Poe’s pendulum Advanced topical coverage of special functions, such as Bessel, Legendre polynomials, and spherical harmonics Fourier and Laplace transform techniques to solve important problems
Beginning of Partial Differential Equations, Third Edition is an ideal textbook for upper-undergraduate and first-year graduate-level courses in analysis and applied mathematics, science, and engineering.
This book is suitable for researchers and graduate students working in complex approximation and its applications, mathematical analysis and numerical analysis.
The University of Toronto Undergraduate Competition was founded to provide additional competition experience for undergraduates preparing for the Putnam competition, and is particularly useful for the freshman or sophomore undergraduate. Lecturers, instructors, and coaches for mathematics competitions will find this presentation useful. Many of the problems are of intermediate difficulty and relate to the first two years of the undergraduate curriculum. The problems presented may be particularly useful for regular class assignments. Moreover, this text contains problems that lie outside the regular syllabus and may interest students who are eager to learn beyond the classroom.
Starting with a consideration of the existence and uniqueness theorem, the text advances to the behavior of the characteristics of a first-order equation, boundary problems for second-order linear equations, asymptotic methods, and differential equations in the complex field. The author discusses only ordinary differential equations, excluding coverage of the methods of integration and stressing the importance of reading the properties of the integrals directly from the equations. An extensive bibliography and helpful indexes conclude the text.
Featuring more than 100 stunning color illustrations and requiring only a modest background in math, Creating Symmetry begins by addressing the enigma of a simple curve, whose curious symmetry seems unexplained by its formula. Farris describes how complex numbers unlock the mystery, and how they lead to the next steps on an engaging path to constructing waveforms. He explains how to devise waveforms for each of the 17 possible wallpaper types, and then guides you through a host of other fascinating topics in symmetry, such as color-reversing patterns, three-color patterns, polyhedral symmetry, and hyperbolic symmetry. Along the way, Farris demonstrates how to marry waveforms with photographic images to construct beautiful symmetry patterns as he gradually familiarizes you with more advanced mathematics, including group theory, functional analysis, and partial differential equations. As you progress through the book, you'll learn how to create breathtaking art images of your own.
Fun, accessible, and challenging, Creating Symmetry features numerous examples and exercises throughout, as well as engaging discussions of the history behind the mathematics presented in the book.
This book is not just about math, not just about computing, and not just about applications, but about all three--in other words, computational science. Whether used as an undergraduate textbook, for self-study, or for reference, it builds the foundation you need to make numerical modeling and simulation integral parts of your investigational toolbox.
Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.
This concise text offers undergraduates in mathematics and science a thorough and systematic first course in elementary differential equations. Presuming a knowledge of basic calculus, the book first reviews the mathematical essentials required to master the materials to be presented.
The next four chapters take up linear equations, those of the first order and those with constant coefficients, variable coefficients, and regular singular points. The last two chapters address the existence and uniqueness of solutions to both first order equations and to systems and n-th order equations.
Throughout the book, the author carries the theory far enough to include the statements and proofs of the simpler existence and uniqueness theorems. Dr. Coddington, who has taught at MIT, Princeton, and UCLA, has included many exercises designed to develop the student's technique in solving equations. He has also included problems (with answers) selected to sharpen understanding of the mathematical structure of the subject, and to introduce a variety of relevant topics not covered in the text, e.g. stability, equations with periodic coefficients, and boundary value problems.
* Focuses on the most often used features of Mathematica for the beginning Mathematica user
* New applications from a variety of fields, including engineering, biology, and physics
* All applications were completed using recent versions of Mathematica
Suitable for courses on differential equations with applications to mathematical biology or as an introduction to mathematical biology, Differential Equations and Mathematical Biology, Second Edition introduces students in the physical, mathematical, and biological sciences to fundamental modeling and analytical techniques used to understand biological phenomena. In this edition, many of the chapters have been expanded to include new and topical material.
New to the Second Edition
A section on spiral waves Recent developments in tumor biology More on the numerical solution of differential equations and numerical bifurcation analysis MATLAB® files available for download online Many additional examples and exercises
This textbook shows how first-order ordinary differential equations (ODEs) are used to model the growth of a population, the administration of drugs, and the mechanism by which living cells divide. The authors present linear ODEs with constant coefficients, extend the theory to systems of equations, model biological phenomena, and offer solutions to first-order autonomous systems of nonlinear differential equations using the Poincaré phase plane. They also analyze the heartbeat, nerve impulse transmission, chemical reactions, and predator–prey problems. After covering partial differential equations and evolutionary equations, the book discusses diffusion processes, the theory of bifurcation, and chaotic behavior. It concludes with problems of tumor growth and the spread of infectious diseases.
Mathematica’s diversity makes it particularly well suited to performing calculations encountered when solving many ordinary and partial differential equations. In some cases, Mathematica’s built-in functions can immediately solve a differential equation by providing an explicit, implicit, or numerical solution. In other cases, mathematica can be used to perform the calculations encountered when solving a differential equation.
Because one goal of elementary differential equations courses is to introduce students to basic methods and algorithms so that they gain proficiency in them, nearly every topic covered this book introduces basic commands, also including typical examples of their application. A study of differential equations relies on concepts from calculus and linear algebra, so this text also includes discussions of relevant commands useful in those areas. In many cases, seeing a solution graphically is most meaningful, so the book relies heavily on Mathematica’s outstanding graphics capabilities.Demonstrates how to take advantage of the advanced features of Mathematica 10Introduces the fundamental theory of ordinary and partial differential equations using Mathematica to solve typical problems of interest to students, instructors, scientists, and practitioners in many fieldsShowcases practical applications and case studies drawn from biology, physics, and engineering
Many of the fundamental laws of physics, chemistry, biology, and economics can be formulated as differential equations. This plain-English guide explores the many applications of this mathematical tool and shows how differential equations can help us understand the world around us. Differential Equations For Dummies is the perfect companion for a college differential equations course and is an ideal supplemental resource for other calculus classes as well as science and engineering courses. It offers step-by-step techniques, practical tips, numerous exercises, and clear, concise examples to help readers improve their differential equation-solving skills and boost their test scores.
Petrovsky was a leading figure in Russian mathematics responsible for many advances in the field of partial differential equations. In these masterly lectures, his commentary and discussion of various aspects of the problems under consideration will prove valuable in deepening students’ understanding and appreciation of these problems.
Aiming to fill in the knowledge gaps left by contributed volumes on the topic and increase the accessibility of the extensive journal literature covering BEM applied to plates, author John T. Katsikadelis draws heavily on his pioneering work in the field to provide a complete introduction to theory and application.
Beginning with a chapter of preliminary mathematical background to make the book a self-contained resource, Katsikadelis moves on to cover the application of BEM to basic thin plate problems and more advanced problems. Each chapter contains several examples described in detail and closes with problems to solve. Presenting the BEM as an efficient computational method for practical plate analysis and design, Boundary Element Method for Plate Analysis is a valuable reference for researchers, students and engineers working with BEM and plate challenges within mechanical, civil, aerospace and marine engineering.One of the first resources dedicated to boundary element analysis of plates, offering a systematic and accessible introductory to theory and applicationAuthored by a leading figure in the field whose pioneering work has led to the development of BEM as an efficient computational method for practical plate analysis and designIncludes mathematical background, examples and problems in one self-contained resource
This second edition covers several new topics: new section on capacity theory and elements of potential theory now includes the concepts of quasi-open sets and quasi-continuity; increased number of examples in the areas of linearized elasticity system, obstacles problems, convection-diffusion, and semilinear equations; new section on mass transportation problems and the Kantorovich relaxed formulation of the Monge problem; new subsection on stochastic homogenization establishes the mathematical tools coming from ergodic theory; and an entirely new and comprehensive chapter (17) devoted to gradient flows and the dynamical approach to equilibria.
The book is intended for Ph.D. students, researchers, and practitioners who want to approach the field of variational analysis in a systematic way.