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• Fundamentals: Random variables Limit theorems Markov chains Monte Carlo methods Stochastic processes Wiener process Stochastic differential equations Fokker-Planck equation Advanced topics: Path integral Random fields Introduction to statistical mechanics Rare events Introduction to chemical reaction kinetics Appendix Bibliography Index.
• (source: Nielsen Book Data)

This is a textbook for advanced undergraduate students and beginning graduate students in applied mathematics. It presents the basic mathematical foundations of stochastic analysis (probability theory and stochastic processes) as well as some important practical tools and applications (e.g., the connection with differential equations, numerical methods, path integrals, random fields, statistical physics, chemical kinetics, and rare events). The book strikes a nice balance between mathematical formalism and intuitive arguments, a style that is most suited for applied mathematicians. Readers can learn both the rigorous treatment of stochastic analysis as well as practical applications in modeling and simulation. Numerous exercises nicely supplement the main exposition.
(source: Nielsen Book Data)

• Introduction
• A two-point boundary value problem
• Elliptic equations
• Finite difference methods for elliptic equations
• Finite element methods for elliptic equations
• The elliptic eigenvalue problem
• Initial-value problems for ODEs
• Parabolic equations
• Finite difference methods for parabolic problems
• The finite element method for a parabolic problem
• Hyperbolic equations
• Finite difference methods for hyperbolic equations
• The finite element method for hyerbolic equations
• Some other classes of numerical methods
• Some tools from mathematical analysis
• Orientation on numerical linear algebra.

This book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering. The main theme is the integration of the theory of linear PDE and the theory of finite difference and finite element methods. For each type of PDE, elliptic, parabolic, and hyperbolic, the text contains one chapter on the mathematical theory of the differential equation, followed by one chapter on finite difference methods and one on finite element methods. The chapters on elliptic equations are preceded by a chapter on the two-point boundary value problem for ordinary differential equations. Similarly, the chapters on time-dependent problems are preceded by a chapter on the initial-value problem for ordinary differential equations. There is also one chapter on the elliptic eigenvalue problem and eigenfunction expansion. The presentation does not presume a deep knowledge of mathematical and functional analysis. The required background on linear functional analysis and Sobolev spaces is reviewed in an appendix.
(source: Nielsen Book Data)