Applied Differential Equations with Boundary Value Problems presents a contemporary treatment of ordinary differential equations (ODEs) and an introduction to partial differential equations (PDEs), including their applications in engineering and the sciences. This new edition of the author’s popular textbook adds coverage of boundary value problems. The text covers traditional material, along with novel approaches to mathematical modeling that harness the capabilities of numerical algorithms and popular computer software packages. It contains practical techniques for solving the equations as well as corresponding codes for numerical solvers. Many examples and exercises help students master effective solution techniques, including reliable numerical approximations. This book describes differential equations in the context of applications and presents the main techniques needed for modeling and systems analysis. It teaches students how to formulate a mathematical model, solve differential equations analytically and numerically, analyze them qualitatively, and interpret the results.

Building on the basic techniques of separation of variables and Fourier series, the book presents the solution of boundary-value problems for basic partial differential equations: the heat equation, wave equation, and Laplace equation, considered in various standard coordinate systems--rectangular, cylindrical, and spherical. Each of the equations is derived in the three-dimensional context; the solutions are organized according to the geometry of the coordinate system, which makes the mathematics especially transparent. Bessel and Legendre functions are studied and used whenever appropriate throughout the text. The notions of steady-state solution of closely related stationary solutions are developed for the heat equation; applications to the study of heat flow in the earth are presented. The problem of the vibrating string is studied in detail both in the Fourier transform setting and from the viewpoint of the explicit representation (d'Alembert formula). Additional chapters include the numerical analysis of solutions and the method of Green's functions for solutions of partial differential equations. The exposition also includes asymptotic methods (Laplace transform and stationary phase). With more than 200 working examples and 700 exercises (more than 450 with answers), the book is suitable for an undergraduate course in partial differential equations.

This book is the most comprehensive, up-to-date account of the popular numerical methods for solving boundary value problems in ordinary differential equations. It aims at a thorough understanding of the field by giving an in-depth analysis of the numerical methods by using decoupling principles. Numerous exercises and real-world examples are used throughout to demonstrate the methods and the theory. Although first published in 1988, this republication remains the most comprehensive theoretical coverage of the subject matter, not available elsewhere in one volume. Many problems, arising in a wide variety of application areas, give rise to mathematical models which form boundary value problems for ordinary differential equations. These problems rarely have a closed form solution, and computer simulation is typically used to obtain their approximate solution. This book discusses methods to carry out such computer simulations in a robust, efficient, and reliable manner.

This book highlights an unprecedented number of real-life applications of differential equations together with the underlying theory and techniques. The problems and examples presented here touch on key topics in the discipline, including first order (linear and nonlinear) differential equations, second (and higher) order differential equations, first order differential systems, the Runge–Kutta method, and nonlinear boundary value problems. Applications include growth of bacterial colonies, commodity prices, suspension bridges, spreading rumors, modeling the shape of a tsunami, planetary motion, quantum mechanics, circulation of blood in blood vessels, price-demand-supply relations, predator-prey relations, and many more. Upper undergraduate and graduate students in Mathematics, Physics and Engineering will find this volume particularly useful, both for independent study and as supplementary reading. While many problems can be solved at the undergraduate level, a number of challenging real-life applications have also been included as a way to motivate further research in this vast and fascinating field.

This title is part of the Pearson Modern Classics series. Pearson Modern Classics are acclaimed titles at a value price. Please visit www.pearsonhighered.com/math-classics-series for a complete list of titles. Applied Partial Differential Equations with Fourier Series and Boundary Value Problems emphasizes the physical interpretation of mathematical solutions and introduces applied mathematics while presenting differential equations. Coverage includes Fourier series, orthogonal functions, boundary value problems, Green's functions, and transform methods. This text is ideal for readers interested in science, engineering, and applied mathematics.

Superb introduction devotes almost half its pages to numerical methods for solving partial differential equations, while the heart of the book focuses on boundary-value and initial-boundary-value problems on spatially bounded and on unbounded domains; integral transforms; uniqueness and continuous dependence on data, first-order equations, and more. Numerous exercises included, with solutions for many at end of book. For students with little background in linear algebra, a useful appendix covers that subject briefly.

Written in a clear and accurate language that students can understand, Trench's new book minimizes the number of explicitly stated theorems and definitions. Instead, he deals with concepts in a conversational style that engages students. He includes more than 250 illustrated, worked examples for easy reading and comprehension. One of the book's many strengths is its problems, which are of consistently high quality. Trench includes a thorough treatment of boundary-value problems and partial differential equations and has organized the book to allow instructors to select the level of technology desired. This has been simplified by using symbols, C and L, to designate the level of technology. C problems call for computations and/or graphics, while L problems are laboratory exercises that require extensive use of technology. Informal advice on the use of technology is included in several sections and instructors who prefer not to emphasize technology can ignore these exercises without interrupting the flow of material.