Spline Collocation Methods for Partial Differential Equations: With Applications in R

A comprehensive approach to numerical partial differential equations

Spline Collocation Methods for Partial Differential Equations combines the collocation analysis of partial differential equations (PDEs) with the method of lines (MOL) in order to simplify the solution process. Using a series of example applications, the author delineates the main features of the approach in detail, including an established mathematical framework. The book also clearly demonstrates that spline collocation can offer a comprehensive method for numerical integration of PDEs when it is used with MOL in which spatial (boundary value) derivatives are approximated with splines, including the boundary conditions.

R, an open–source scientific programming system, is used throughout for programming the PDEs and numerical algorithms, and each section of code is clearly explained. As a result, readers gain a complete picture of the model and its computer implementation without having to fill in the details of the numerical analysis, algorithms, or programming. The presentation is not heavily mathematical, and in place of theorems and proofs, detailed example applications are provided. 

Appropriate for scientists, engineers, and applied mathematicians, Spline Collocation Methods for Partial Differential Equations:

Introduces numerical methods by first presenting basic examples followed by more complicated applications Employs R to illustrate accurate and efficient solutions of the PDE models Presents spline collocation as a comprehensive approach to the numerical integration of PDEs and a valuable alternative to other, well established methods Discusses how to reproduce and extend the presented numerical solutions Identifies the use of selected algorithms, such as the solution of nonlinear equations and banded or sparse matrix processing Features a companion website that provides the related R routines

Spline Collocation Methods for Partial Differential Equations is a valuable reference and/or self–study guide for academics, researchers, and practitioners in applied mathematics and engineering, as well as for advanced undergraduates and graduate–level students. 

Dedication

Preface

Chapter 1: Introduction

Chapter 2: One–dimensional PDEs

Chapter 3: Multidimensional PDEs

Chapter 4: Navier–Stokes, Burgers Equations

Chapter 5: Korteweg–deVries Equation

Chapter 6: Maxwell Equations

Chapter 7: Poisson–Nernst–Planck Equations

Chapter 8: Fokker–Planck Equation

Chapter 9: Fisher–Kolmogorov Equation

Chapter 10: Klein–Gordon Equation

Chapter 11: Boussinesq Equation

Chapter 12: Cahn–Hilliard Equation

Chapter 13: Camassa–Holm Equation

Chapter 14: Burgers–Huxley Equation

Chapter 15: Gierer–Meinhardt Equations

Chapter 16: Keller–Segel Equations

Chapter 17: Fitzhugh–Nagumo Equations

Chapter 18: Euler–Poisson–Darboux Equation

Chapter 19: Kuramoto–Sivashinsky Equation

Chapter 20: Einstein–Maxwell Equations

Index

William E. Schiesser, PhD, ScD (hon.), is Emeritus McCann Professor of Biomolecular and Chemical Engineering and Professor of Mathematics at Lehigh University. He is the author of 17 books, including Method of Lines PDE Analysis in Biomedical Science and Engineering; Differential Equation Analysis in Biomedical Science and Engineering: Ordinary Differential Equation Applications with R and Differential Equation Analysis in Biomedical Science and Engineering: Partial Differential Equation Applications with R, published by Wiley.