This volume provides an introduction to the analytical and numerical aspects of partial differential equations (PDEs). It unifies an analytical and computational approach for these; the qualitative behaviour of solutions being established using classical concepts: maximum principles and energy methods. Notable inclusions are the treatment of irregularly shaped boundaries, polar coordinates and the use of flux-limiters when approximating hyperbolic conservation laws. The numerical analysis of difference schemes is rigorously developed using discrete maximum principles and discrete Fourier analysis. A novel feature is the inclusion of a chapter containing projects, intended for either individual or group study, that cover a range of topics such as parabolic smoothing, travelling waves, isospectral matrices, and the approximation of multidimensional advection-diffusion problems.
The underlying theory is illustrated by numerous examples and there are around 300 exercises, designed to promote and test understanding. They are starred according to level of difficulty. Solutions to odd-numbered exercises are available to all readers while even-numbered solutions are available to authorised instructors.
Written in an informal yet rigorous style, Essential Partial Differential Equations is designed for mathematics undergraduates in their final or penultimate year of university study, but will be equally useful for students following other scientific and engineering disciplines in which PDEs are of practical importance. The only prerequisite is a familiarity with the basic concepts of calculus and linear algebra.
John Dold is a professor of applied mathematics, having trained in physics, mathematics and (briefly) history. Apart from some purely experimental studies, particularly of fire behaviour, his research primarily makes use of partial differential equations to model practical problems, including water waves and combustion phenomena. He founded the journal Combustion Theory and Modelling and he has organised or helped to organise major conferences on combustion research. In his teaching of mathematical methods and their application to various physical phenomena he has written a number of coursework texts. One of these has been expanded and improved to create the current volume. David Griffiths trained originally as an applied mathematician and has spent his academic career as a numerical analyst at the University of Dundee specialising in the numerical solution of partial differential equations. He taught the subject at both undergraduate and postgraduate levels for many years and published over 50 scientific articles on finite difference and finite element methods. He has previously published two textbooks on finite difference methods for ordinary and partial differential equations. David was joint organising secretary of the Dundee Biennial Conferences on Numerical Analysis from 1983 to 2005. He was awarded a "Certificate of Recognition" from NASA in both 1991 and 1993 for his work on spurious solutions that can arise from approximations of nonlinear differential equations. David Silvester is a Professor in the School of Mathematics at The University of Manchester. His research concerns numerical solution of partial differential equations, computational fluid dynamics, uncertainty quantification, and high performance computing. He received his doctorate in mathematics from the University of Manchester Institute of Science and Technology in 1984 and has had visiting positions at Stanford University, the University of Maryland at College Park, and the Universite du Littoral, France. He has served on the editorial boards of SIAM Journal on Scientific Computing and the International Journal for Numerical Methods in Fluids.
Setting the scene.- Boundary and initial data.- The origin of PDEs.- Classification of PDEs.- Boundary value problems in R1.- Finite difference methods in R1.- Maximum principles and energy methods.- Separation of variables.- The method of characteristics.- Finite difference methods for elliptic PDEs.- Finite difference methods for parabolic PDEs.- Finite difference methods for hyperbolic PDEs.- Projects.