This second edition of Nonstandard Finite Difference Models of Differential Equations provides an update on the progress made in both the theory and application of the NSFD methodology during the past two and a half decades. In addition to discussing details related to the determination of the denominator functions and the nonlocal discrete representations of functions of dependent variables, we include many examples illustrating just how this should be done.Of real value to the reader is the inclusion of a chapter listing many exact difference schemes, and a chapter giving NSFD schemes from the research literature. The book emphasizes the critical roles played by the 'principle of dynamic consistency' and the use of sub-equations for the construction of valid NSFD discretizations of differential equations.
Finite Element Analysis is a very popular, computer-based tool that uses a complex system of points called nodes to make a grid called a "mesh. " The mesh contains the material and structural properties that define how the structure will react to certain loading conditions, allowing virtual testing and analysis of stresses or changes applied to the material or component design. This groundbreaking text extends the usefulness of finite element analysis by helping both beginners and advanced users alike. It simplifies, improves, and extends both the finite element method while at the same time advancing adaptive refinement procedures. These improvements are made possible due to a change in notation that embeds knowledge of solid continuum mechanics into the equations used to formulate the stiffness matrices; this allows the modeling characteristics of individual elements to be identified by visual inspection. The ability to visually relate the equations involved in element formulation to the physical process they represent is like having an x-ray of the inner workings of the finite element method; it is similar is to the effect that Graphical User Interfaces or GUI's had on computing. As a result, students at any level of finite element study are provided with an understanding of the capabilities and limitations of this powerful analytic tool. The book presents * A more simplified approach to finite element analysis based on computational continuum mechanics * Physically interpretable notation that identifies a common basis for the finite element and the finite difference methods. * New point-wise error estimators that identify errors in terms of quantities of direct interest in solid mechanics
Advances in the Analysis and Design of Marine Structures is a collection of papers presented at MARSTRUCT 2023, the 9th International Conference on Marine Structures, held in Gothenburg, Sweden, 3-5 April 2023. The conference was organised by the Division of Marine Technology, Department of Mechanics and Maritime Sciences at Chalmers University of Technology, in Gothenburg, Sweden. The MARSTRUCT Conference series deals with Ship and Offshore Structures, addressing topics in the fields of: Methods and tools for loads and load effects Methods and tools for strength assessment Experimental analysis of structures Materials and fabrication of structures Methods and tools for structural design and optimization Structural reliability, safety, and environmental protection The MARSTRUCT conferences series of started in Glasgow, UK in 2007, the second event of the series took place in Lisbon, Portugal in March 2009, the third in Hamburg, Germany in March 2011, the fourth in Espoo, Finland in March 2013, the fifth in Southampton, UK in March 2015, the sixth in Lisbon, Portugal in May 2017, the seventh in Dubrovnik, Croatia in May 2019, and the eighth event in Trondheim, Norway in June 2021. Advances in the Analysis and Design of Marine Structures is essential reading for academics, engineers and all professionals involved in the design of marine and offshore structures. The Proceedings in Marine Technology and Ocean Engineering series is devoted to the publication of proceedings of peer-reviewed international conferences dealing with various aspects of ‘Marine Technology and Ocean Engineering’. The Series includes the proceedings of the following conferences: the International Maritime Association of the Mediterranean (IMAM) Conferences, the Marine Structures (MARSTRUCT) Conferences, the Renewable Energies Offshore (RENEW) Conferences and the Maritime Technology (MARTECH) Conferences. The ‘Marine Technology and Ocean Engineering’ series is also open to new conferences that cover topics on the sustainable exploration and exploitation of marine resources in various fields, such as maritime transport and ports, usage of the ocean including coastal areas, nautical activities, the exploration and exploitation of mineral resources, the protection of the marine environment and its resources, and risk analysis, safety and reliability. The aim of the series is to stimulate advanced education and training through the wide dissemination of the results of scientific research.
This monograph is a detailed introduction to the nascent and ever-evolving fields of metamaterials and nanophotonics, with key techniques and applications needed for a comprehensive understanding of these fields all detailed. These include the 'standard' and high-accuracy 'nonstandard' FDTD techniques, finite-difference frequency-domain mode solvers, the transfer matrix method, analytic calculations for dielectric and plasmonic waveguides, dispersion, Maxwell-Bloch and density functional theory, as well as design methods for constructing metamaterials and nanolasers, and quantum plasmonics. The book is intended for final-year undergraduates, as well as postgraduates or active researchers who wish to understand and enter these fields in a 'user-friendly' manner, and who have a basic understanding of and familiarity with electromagnetic theory.
A study of difference equations and inequalities. This second edition offers real-world examples and uses of difference equations in probability theory, queuing and statistical problems, stochastic time series, combinatorial analysis, number theory, geometry, electrical networks, quanta in radiation, genetics, economics, psychology, sociology, and
This book provides a clear summary of the work of the author on the construction of nonstandard finite difference schemes for the numerical integration of differential equations. The major thrust of the book is to show that discrete models of differential equations exist such that the elementary types of numerical instabilities do not occur. A consequence of this result is that in general bigger step-sizes can often be used in actual calculations and/or finite difference schemes can be constructed that are conditionally stable in many instances whereas in using standard techniques no such schemes exist. The theoretical basis of this work is centered on the concepts of ?exact? and ?best? finite difference schemes. In addition, a set of rules is given for the discrete modeling of derivatives and nonlinear expressions that occur in differential equations. These rules often lead to a unique nonstandard finite difference model for a given differential equation.
This multi-volume handbook is the most up-to-date and comprehensive reference work in the field of fractional calculus and its numerous applications. This third volume collects authoritative chapters covering several numerical aspects of fractional calculus, including time and space fractional derivatives, finite differences and finite elements, and spectral, meshless, and particle methods.
Completely revised text focuses on use of spectral methods to solve boundary value, eigenvalue, and time-dependent problems, but also covers Hermite, Laguerre, rational Chebyshev, sinc, and spherical harmonic functions, as well as cardinal functions, linear eigenvalue problems, matrix-solving methods, coordinate transformations, methods for unbounded intervals, spherical and cylindrical geometry, and much more. 7 Appendices. Glossary. Bibliography. Index. Over 160 text figures.
Numerical Methods for Partial Differential Equations: Finite Difference and Finite Volume Methods focuses on two popular deterministic methods for solving partial differential equations (PDEs), namely finite difference and finite volume methods. The solution of PDEs can be very challenging, depending on the type of equation, the number of independent variables, the boundary, and initial conditions, and other factors. These two methods have been traditionally used to solve problems involving fluid flow. For practical reasons, the finite element method, used more often for solving problems in solid mechanics, and covered extensively in various other texts, has been excluded. The book is intended for beginning graduate students and early career professionals, although advanced undergraduate students may find it equally useful. The material is meant to serve as a prerequisite for students who might go on to take additional courses in computational mechanics, computational fluid dynamics, or computational electromagnetics. The notations, language, and technical jargon used in the book can be easily understood by scientists and engineers who may not have had graduate-level applied mathematics or computer science courses. - Presents one of the few available resources that comprehensively describes and demonstrates the finite volume method for unstructured mesh used frequently by practicing code developers in industry - Includes step-by-step algorithms and code snippets in each chapter that enables the reader to make the transition from equations on the page to working codes - Includes 51 worked out examples that comprehensively demonstrate important mathematical steps, algorithms, and coding practices required to numerically solve PDEs, as well as how to interpret the results from both physical and mathematic perspectives
