Differential Equations

Abstract

Finding and interpreting the solutions of differential equations is a central and essential part of applied mathematics. This book aims to enable the reader to develop the required skills needed for a thorough understanding of the subject. The authors focus on the business of constructing solutions analytically, and interpreting their meaning, using rigorous analysis where needed. MATLAB is used extensively to illustrate the material. There are many worked examples based on interesting and unusual real world problems. A large selection of exercises is provided, including several lengthier projects, some of which involve the use of MATLAB. The coverage is broad, ranging from basic second-order ODEs and PDEs, through to techniques for nonlinear differential equations, chaos, asymptotics and control theory. This broad coverage, the authors' clear presentation and the fact that the book has been thoroughly class-tested will increase its attraction to undergraduates at each stage of their studies.

On the stability of two new two-step explicit methods for the numerical integration of second order initial value problem on a variable mesh

“…(1) arises in many fields of application; in particular, one such example is the telegraph equation which is obtainable from Maxwell's equations after separation of variables. Existence and uniqueness of (1) has been discussed in [1]. One numerical approach is to solve (1) as a system of coupled first order ordinary differential equations using, for example, the classical Runge-Kutta-Nystrom method [2][3][4]; alternatively we may, as we do here, solve the equation directly using two-step methods.…”

Analytical fundamentals of migration in reflection seismics

“…v ≡ v(z), it will still be possible to derive an asymptotic solution of Eq. (9), with the help of the WKB approximation [7,8]. We offer this treatment as a simple and a more elucidating alternative to the way this mathematical problem is treated even in very advanced texts on migration [1].…”

“…In this paper, we attempt to address certain lacunae in the way seismic migration is understood. The mathematical tools that we implement are quite common in mathematical physics, namely, second-order linear differential equations, Fourier transforms, and the Wentzel-Kramers-Brillouin (henceforth WKB) approximation [6][7][8]. To establish how these methods become relevant in migration-related studies, we first discuss the simple geometry-based approach to migration, if only to appreciate its inadequacies (Sections 2 & 3).…”

Beams

“…For an introduction see, for example, Kreyszig (1993) and for a more in depth discussion see King et al (2003). In this book, the discussion will be restricted to structural components with relatively simple geometry, where a governing partial differential equation can be approximated by a set of ordinary differential equations.…”