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There are many excellent computational biology resources now available for learning about methods that have been developed to address specific biological systems, but comparatively little attention has been paid to training aspiring computational biologists to handle new and unanticipated problems. This text is intended to fill that gap by teaching students how to reason about developing formal mathematical models of biological systems that are amenable to computational analysis. It collects in one place a selection of broadly useful models, algorithms, and theoretical analysis tools normally found scattered among many other disciplines. It thereby gives the aspiring student a bag of tricks that will serve him or her well in modeling problems drawn from numerous subfields of biology. These techniques are taught from the perspective of what the practitioner needs to know to use them effectively, supplemented with references for further reading on more advanced use of each method covered. The text, which grew out of a class taught at Carnegie Mellon University, covers models for optimization, simulation and sampling, and parameter tuning. These topics provide a general framework for learning how to formulate mathematical models of biological systems, what techniques are available to work with these models, and how to fit the models to particular systems. Their application is illustrated by many examples drawn from a variety of biological disciplines and several extended case studies that show how the methods described have been applied to real problems in biology.

Additional features of this new edition include broader coverage of PDE methods and applications, with new chapters on the method of characteristics, Sturm-Liouville problems, and Green s functions, and a new section on the finite difference method for the wave equation. The author continues to emphasize Fourier series and finite element methods, which were the primary scope of the first edition.

The book also features emphasis on linear algebra, particularly the idea of best approximation; realistic physical parameters and meaningful experiments for many of the examples and exercises; and tutorials for the most popular software (MATLAB, Mathematica, and Maple) that can be used to reproduce the examples and solve the exercises.

Audience: This book is written for undergraduate courses usually titled Introduction to Partial Differential Equations or Fourier Series and Boundary Value Problems.

Contents: Preface; Chapter 1: Classification of Differential Equations; Chapter 2: Models in One Dimension; Chapter 3: Essential Linear Algebra; Chapter 4: Essential Ordinary Differential Equations; Chapter 5: Boundary Value Problems in Statics; Chapter 6: Heat Flow and Diffusion; Chapter 7: Waves; Chapter 8: First-Order PDEs and the Method of Characteristics; Chapter 9: Green's Functions; Chapter 10: Sturm-Liouville Eigenvalue Problems; Chapter 11: Problems in Multiple Spatial Dimensions; Chapter 12: More about Fourier Series; Chapter 13: More about Finite Element Methods; Appendix A: Proof of Theorem 3.47; Appendix B: Shifting the Data in Two Dimensions; Bibliography; Index.

The influence of scientific computing has become very wide over the last few decades: almost every area of science and engineering is greatly influenced by simulations - image processing, thin films, mathematical finance, electrical engineering, moving interfaces and combustion, to name but a few.

One half of this book focuses on the techniques of scientific computing: domain decomposition, the absorption of boundary conditions and one-way operators, convergence analysis of multi-grid methods and other multi-grid techniques, dynamical systems, and matrix analysis.

The remainder of the book is concerned with combining techniques with concrete applications: stochastic differential equations, image processing, thin films, and asymptotic analysis for combustion problems.