Mathematical Physics And Numerical Analysis

Stochastic Numerics for Mathematical Physics

This book is a substantially revised and expanded edition reflecting major developments in stochastic numerics since the first edition was published in 2004. The new topics, in particular, include mean-square and weak approximations in the case of nonglobally Lipschitz coefficients of Stochastic Differential Equations (SDEs) including the concept of rejecting trajectories; conditional probabilistic representations and their application to practical variance reduction using regression methods; multi-level Monte Carlo method; computing ergodic limits and additional classes of geometric integrators used in molecular dynamics; numerical methods for FBSDEs; approximation of parabolic SPDEs and nonlinear filtering problem based on the method of characteristics. SDEs have many applications in the natural sciences and in finance. Besides, the employment of probabilistic representations together with the Monte Carlo technique allows us to reduce the solution of multi-dimensional problems for partial differential equations to the integration of stochastic equations. This approach leads to powerful computational mathematics that is presented in the treatise. Many special schemes for SDEs are presented. In the second part of the book numerical methods for solving complicated problems for partial differential equations occurring in practical applications, both linear and nonlinear, are constructed. All the methods are presented with proofs and hence founded on rigorous reasoning, thus giving the book textbook potential. An overwhelming majority of the methods are accompanied by the corresponding numerical algorithms which are ready for implementation in practice. The book addresses researchers and graduate students in numerical analysis, applied probability, physics, chemistry, and engineering as well as mathematical biology and financial mathematics.

Fundamentals of Numerical Mathematics for Physicists and Engineers

Introduces the fundamentals of numerical mathematics and illustrates its applications to a wide variety of disciplines in physics and engineering Applying numerical mathematics to solve scientific problems, this book helps readers understand the mathematical and algorithmic elements that lie beneath numerical and computational methodologies in order to determine the suitability of certain techniques for solving a given problem. It also contains examples related to problems arising in classical mechanics, thermodynamics, electricity, and quantum physics. Fundamentals of Numerical Mathematics for Physicists and Engineers is presented in two parts. Part I addresses the root finding of univariate transcendental equations, polynomial interpolation, numerical differentiation, and numerical integration. Part II examines slightly more advanced topics such as introductory numerical linear algebra, parameter dependent systems of nonlinear equations, numerical Fourier analysis, and ordinary differential equations (initial value problems and univariate boundary value problems). Chapters cover: Newton’s method, Lebesgue constants, conditioning, barycentric interpolatory formula, Clenshaw-Curtis quadrature, GMRES matrix-free Krylov linear solvers, homotopy (numerical continuation), differentiation matrices for boundary value problems, Runge-Kutta and linear multistep formulas for initial value problems. Each section concludes with Matlab hands-on computer practicals and problem and exercise sets. This book: Provides a modern perspective of numerical mathematics by introducing top-notch techniques currently used by numerical analysts Contains two parts, each of which has been designed as a one-semester course Includes computational practicals in Matlab (with solutions) at the end of each section for the instructor to monitor the student's progress through potential exams or short projects Contains problem and exercise sets (also with solutions) at the end of each section Fundamentals of Numerical Mathematics for Physicists and Engineers is an excellent book for advanced undergraduate or graduate students in physics, mathematics, or engineering. It will also benefit students in other scientific fields in which numerical methods may be required such as chemistry or biology.

Numerical Methods in Mathematical Physics