Nonstandard Finite Difference Models Of Differential Equations

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Nonstandard Finite Difference Models of Differential Equations

Ronald E. Mickens
Nonstandard Finite Difference Models of Differential Equations Book PDF

Author : Ronald E. Mickens
Publisher : World Scientific
Page : 264 pages
File Size : 45,5 Mb
Release : 1994
Category : Mathematics
ISBN : 9789810214586

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Nonstandard Finite Difference Models of Differential Equations by Ronald E. Mickens Pdf

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.

Nonstandard Finite Difference Models of Differential Equations

Ronald E Mickens
Nonstandard Finite Difference Models of Differential Equations Book PDF

Author : Ronald E Mickens
Publisher : World Scientific
Page : 264 pages
File Size : 54,7 Mb
Release : 1993-12-23
Category : Mathematics
ISBN : 9789814518079

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Nonstandard Finite Difference Models of Differential Equations by Ronald E Mickens Pdf

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. Contents:IntroductionNumerical InstabilitiesNonstandard Finite-Difference SchemesFirst-Order ODE'sSecond-Order, Nonlinear Oscillator EquationsTwo First-Order, Coupled Ordinary Differential EquationsPartial Differential EquationsSchrädinger Differential EquationsSummary and DiscussionAppendices: Difference EquationsLinear Stability AnalysisDiscrete WKB MethodBibliographyIndex Readership: Applied mathematicians (numerical analysis and modeling). keywords:Finite Difference Techniques;Numerical Schemes for Differential Equations;Numerical Instabilities;Nonstandard Schemes;Exact Finite Difference Schemes;Best Finite Difference Schemes;Denominator Functions;Linear Stability Analysis;Discrete WKB Method “This book contains a clear presentation of nonstandard finite difference schemes for the numerical integration of differential equations. A set of rules for constructing nonstandard finite difference schemes is also presented. An important feature of the book is the illustration of the various discrete modeling principles, by their application to a large number of both ordinary and partial differential equations.” Mathematical Reviews

Applications of Nonstandard Finite Difference Schemes

Ronald E. Mickens
Applications of Nonstandard Finite Difference Schemes Book PDF

Author : Ronald E. Mickens
Publisher : World Scientific
Page : 268 pages
File Size : 51,7 Mb
Release : 2000
Category : Mathematics
ISBN : 981024133X

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Applications of Nonstandard Finite Difference Schemes by Ronald E. Mickens Pdf

The main purpose of this book is to provide a concise introduction to the methods and philosophy of constructing nonstandard finite difference schemes and illustrate how such techniques can be applied to several important problems. Chapter I gives an overview of the subject and summarizes previous work. Chapters 2 and 3 consider in detail the construction and numerical implementation of schemes for physical problems involving convection-diffusion-reaction equations, that arise in groundwater pollution and scattering of electromagnetic waves using Maxwell's equations. Chapter 4 examines certain mathematical issues related to the nonstandard discretization of competitive and cooperative models for ecology. The application chapters illustrate well the power of nonstandard methods. In particular, for the same accuracy as obtained by standard techniques, larger step sizes can be used. This volume will satisfy the needs of scientists, engineers, and mathematicians who wish to know how to construct nonstandard schemes and see how these are applied to obtain numerical solutions of the differential equations which arise in the study of nonlinear dynamical systems modeling important physical phenomena.

Nonstandard Finite Difference Schemes: Methodology And Applications

Ronald E Mickens
Nonstandard Finite Difference Schemes Methodology And Applications Book PDF

Author : Ronald E Mickens
Publisher : World Scientific
Page : 332 pages
File Size : 55,5 Mb
Release : 2020-11-11
Category : Mathematics
ISBN : 9789811222559

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Nonstandard Finite Difference Schemes: Methodology And Applications by Ronald E Mickens Pdf

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.

Advances in the Applications of Nonstandard Finite Diffference Schemes

Ronald E. Mickens
Advances in the Applications of Nonstandard Finite Diffference Schemes Book PDF

Author : Ronald E. Mickens
Publisher : World Scientific
Page : 668 pages
File Size : 50,5 Mb
Release : 2005
Category : Mathematics
ISBN : 9789812564047

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Advances in the Applications of Nonstandard Finite Diffference Schemes by Ronald E. Mickens Pdf

This volume provides a concise introduction to the methodology of nonstandard finite difference (NSFD) schemes construction and shows how they can be applied to the numerical integration of differential equations occurring in the natural, biomedical, and engineering sciences. These methods had their genesis in the work of Mickens in the 1990's and are now beginning to be widely studied and applied by other researchers. The importance of the book derives from its clear and direct explanation of NSFD in the introductory chapter along with a broad discussion of the future directions needed to advance the topic.

Advances in the Applications of Nonstandard Finite Difference Schemes

Ronald E Mickens
Advances in the Applications of Nonstandard Finite Difference Schemes Book PDF

Author : Ronald E Mickens
Publisher : World Scientific
Page : 664 pages
File Size : 47,9 Mb
Release : 2005-10-25
Category : Mathematics
ISBN : 9789814479868

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Advances in the Applications of Nonstandard Finite Difference Schemes by Ronald E Mickens Pdf

This volume provides a concise introduction to the methodology of nonstandard finite difference (NSFD) schemes construction and shows how they can be applied to the numerical integration of differential equations occurring in the natural, biomedical, and engineering sciences. These methods had their genesis in the work of Mickens in the 1990's and are now beginning to be widely studied and applied by other researchers. The importance of the book derives from its clear and direct explanation of NSFD in the introductory chapter along with a broad discussion of the future directions needed to advance the topic. Contents:Nonstandard Finite Difference Methods (R E Mickens)Application of Nonstandard Finite Difference Schemes to the Simulation Studies of Robotic Systems (R F Abo-Shanab et al.)Applications of Mickens Finite Differences to Several Related Boundary Value Problems (R Buckmire)High Accuracy Nonstandard Finite-Difference Time-Domain Algorithms for Computational Electromagnetics: Applications to Optics and Photonics (J B Cole)Nonstandard Finite Difference Schemes for Solving Nonlinear Micro Heat Transport Equations in Double-Layered Metal Thin Films Exposed to Ultrashort Pulsed Lasers (W Dai)Reliable Finite Difference Schemes with Applications in Mathematical Ecology (D T Dimitrov et al.)Applications of the Nonstandard Finite Difference Method in Non-Smooth Mechanics (Y Dumont)Finite Difference Schemes on Unbounded Domains (M Ehrhardt)Asymptotically Consistent Nonstandard Finite-Difference Methods for Solving Mathematical Models Arising in Population Biology (A B Gumel et al.)Nonstandard Finite Difference Methods and Biological Models (S R-J Jang)Robust Discretizations versus Increase of the Time Step for Chaotic Systems (C Letellier & E M A M Mendes)Contributions to the Theory of Nonstandard Finite-Difference Methods and Applications to Singular Perturbation Problems (J M-S Lubuma & K C Patidar)Frequency Accurate Finite Difference Methods (A L Perkins et al.)Nonstandard Discretization Methods on Lotka-Volterra Differential Equations (L-I W Roeger) Readership: Applied mathematicians, and researchers in numerical & computational mathematics and analysis & differential equations. Usable as a secondary text to a standard undergraduate or graduate course on numerical methods for differential equations. Keywords:Numerical Integration Methods;Finite Differences;Nonstandard Finite Difference Schemes;Differential Equations;Discrete Models;Numerical and Computational MathematicsKey Features:A collection of papers from renowned experts in their respective fieldsProvides the most recent work on the application of NSFD schemes and some of the mathematical analysis related to these schemes

Mathematical Modelling and Nonstandard Schemes for the Corona Virus Pandemic

Sarah Marie Treibert
Mathematical Modelling and Nonstandard Schemes for the Corona Virus Pandemic Book PDF

Author : Sarah Marie Treibert
Publisher : Springer Nature
Page : 260 pages
File Size : 43,5 Mb
Release : 2021-12-11
Category : Mathematics
ISBN : 9783658359324

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Mathematical Modelling and Nonstandard Schemes for the Corona Virus Pandemic by Sarah Marie Treibert Pdf

This book deals with the prediction of possible future scenarios concerning the COVID-19 pandemic. Based on the well-known SIR model by Kermack and McKendrick a compartment model is established. This model comprises its own assumptions, transition rates and transmission dynamics, as well as a corresponding system of ordinary differential equations. Making use of numerical methods and a nonstandard-finite-difference scheme, two submodels are implemented in Matlab in order to make parameter estimations and compare different scenarios with each other.

Fractional Partial Differential Equations and Their Numerical Solutions

Boling Guo,Xueke Pu,Fenghui Huang
Fractional Partial Differential Equations and Their Numerical Solutions Book PDF

Author : Boling Guo,Xueke Pu,Fenghui Huang
Publisher : World Scientific
Page : 348 pages
File Size : 53,9 Mb
Release : 2015-03-09
Category : Mathematics
ISBN : 9789814667067

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Fractional Partial Differential Equations and Their Numerical Solutions by Boling Guo,Xueke Pu,Fenghui Huang Pdf

This book aims to introduce some new trends and results on the study of the fractional differential equations, and to provide a good understanding of this field to beginners who are interested in this field, which is the authors' beautiful hope. This book describes theoretical and numerical aspects of the fractional partial differential equations, including the authors' researches in this field, such as the fractional Nonlinear Schrödinger equations, fractional Landau–Lifshitz equations and fractional Ginzburg–Landau equations. It also covers enough fundamental knowledge on the fractional derivatives and fractional integrals, and enough background of the fractional PDEs. Contents:Physics BackgroundFractional Calculus and Fractional Differential EquationsFractional Partial Differential EquationsNumerical Approximations in Fractional CalculusNumerical Methods for the Fractional Ordinary Differential EquationsNumerical Methods for Fractional Partial Differential Equations Readership: Graduate students and researchers in mathematical physics, numerical analysis and computational mathematics. Key Features:This book covers the fundamentals of this field, especially for the beginnersThe book covers new trends and results in this fieldThe book covers numerical results, which will be of broad interests to researchersKeywords:Fractional Partial Differential Equations;Numerical Solutions

Numerical Methods for Partial Differential Equations

Sandip Mazumder
Numerical Methods for Partial Differential Equations Book PDF

Author : Sandip Mazumder
Publisher : Academic Press
Page : 484 pages
File Size : 46,8 Mb
Release : 2015-12-01
Category : Technology & Engineering
ISBN : 9780128035047

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Numerical Methods for Partial Differential Equations by Sandip Mazumder Pdf

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

Finite Difference Methods for Ordinary and Partial Differential Equations

Randall J. LeVeque
Finite Difference Methods for Ordinary and Partial Differential Equations Book PDF

Author : Randall J. LeVeque
Publisher : SIAM
Page : 356 pages
File Size : 47,6 Mb
Release : 2007-01-01
Category : Mathematics
ISBN : 0898717833

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Finite Difference Methods for Ordinary and Partial Differential Equations by Randall J. LeVeque Pdf

This book introduces finite difference methods for both ordinary differential equations (ODEs) and partial differential equations (PDEs) and discusses the similarities and differences between algorithm design and stability analysis for different types of equations. A unified view of stability theory for ODEs and PDEs is presented, and the interplay between ODE and PDE analysis is stressed. The text emphasizes standard classical methods, but several newer approaches also are introduced and are described in the context of simple motivating examples.

Models of Delay Differential Equations

Francisco Rodríguez
Models of Delay Differential Equations Book PDF

Author : Francisco Rodríguez
Publisher : Unknown
Page : 248 pages
File Size : 42,8 Mb
Release : 2021-06-30
Category : Electronic
ISBN : 3036509321

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Models of Delay Differential Equations by Francisco Rodríguez Pdf

This book gathers a number of selected contributions aimed at providing a balanced picture of the main research lines in the realm of delay differential equations and their applications to mathematical modelling. The contributions have been carefully selected so that they cover interesting theoretical and practical analysis performed in the deterministic and the stochastic settings. The reader will find a complete overview of recent advances in ordinary and partial delay differential equations with applications in other multidisciplinary areas such as Finance, Epidemiology or Engineering

Advanced Numerical Methods for Differential Equations

Harendra Singh,Jagdev Singh,Sunil Dutt Purohit,Devendra Kumar
Advanced Numerical Methods for Differential Equations Book PDF

Author : Harendra Singh,Jagdev Singh,Sunil Dutt Purohit,Devendra Kumar
Publisher : CRC Press
Page : 336 pages
File Size : 55,8 Mb
Release : 2021-07-29
Category : Mathematics
ISBN : 9781000381085

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Advanced Numerical Methods for Differential Equations by Harendra Singh,Jagdev Singh,Sunil Dutt Purohit,Devendra Kumar Pdf

Mathematical models are used to convert real-life problems using mathematical concepts and language. These models are governed by differential equations whose solutions make it easy to understand real-life problems and can be applied to engineering and science disciplines. This book presents numerical methods for solving various mathematical models. This book offers real-life applications, includes research problems on numerical treatment, and shows how to develop the numerical methods for solving problems. The book also covers theory and applications in engineering and science. Engineers, mathematicians, scientists, and researchers working on real-life mathematical problems will find this book useful.

Exact Finite-Difference Schemes

Sergey Lemeshevsky,Piotr Matus,Dmitriy Poliakov
Exact Finite Difference Schemes Book PDF

Author : Sergey Lemeshevsky,Piotr Matus,Dmitriy Poliakov
Publisher : Walter de Gruyter GmbH & Co KG
Page : 246 pages
File Size : 40,7 Mb
Release : 2016-09-26
Category : Mathematics
ISBN : 9783110491326

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Exact Finite-Difference Schemes by Sergey Lemeshevsky,Piotr Matus,Dmitriy Poliakov Pdf

Exact Finite-Difference Schemes is a first overview of the topic also describing the state-of-the-art in this field of numerical analysis. Construction of exact difference schemes for various parabolic and elliptic partial differential equations are discussed, including vibrations and transport problems. After this, applications are discussed, such as the discretisation of ODEs and PDEs and numerical methods for stochastic differential equations. Contents: Basic notation Preliminary results Hyperbolic equations Parabolic equations Use of exact difference schemes to construct NSFD discretizations of differential equations Exact and truncated difference schemes for boundary-value problem Exact difference schemes for stochastic differential equations Numerical blow-up time Bibliography

Numerical Methods for Delay Differential Equations

Alfredo Bellen,Marino Zennaro
Numerical Methods for Delay Differential Equations Book PDF

Author : Alfredo Bellen,Marino Zennaro
Publisher : Numerical Mathematics and Scie
Page : 411 pages
File Size : 55,7 Mb
Release : 2013-01-10
Category : Business & Economics
ISBN : 9780199671373

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Numerical Methods for Delay Differential Equations by Alfredo Bellen,Marino Zennaro Pdf

This unique book describes, analyses, and improves various approaches and techniques for the numerical solution of delay differential equations. It includes a list of available codes and also aids the reader in writing his or her own.

Numerical Approximation of Partial Differential Equations

Sören Bartels
Numerical Approximation of Partial Differential Equations Book PDF

Author : Sören Bartels
Publisher : Springer
Page : 535 pages
File Size : 49,7 Mb
Release : 2016-06-02
Category : Mathematics
ISBN : 9783319323541

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Numerical Approximation of Partial Differential Equations by Sören Bartels Pdf

Finite element methods for approximating partial differential equations have reached a high degree of maturity, and are an indispensible tool in science and technology. This textbook aims at providing a thorough introduction to the construction, analysis, and implementation of finite element methods for model problems arising in continuum mechanics. The first part of the book discusses elementary properties of linear partial differential equations along with their basic numerical approximation, the functional-analytical framework for rigorously establishing existence of solutions, and the construction and analysis of basic finite element methods. The second part is devoted to the optimal adaptive approximation of singularities and the fast iterative solution of linear systems of equations arising from finite element discretizations. In the third part, the mathematical framework for analyzing and discretizing saddle-point problems is formulated, corresponding finte element methods are analyzed, and particular applications including incompressible elasticity, thin elastic objects, electromagnetism, and fluid mechanics are addressed. The book includes theoretical problems and practical projects for all chapters, and an introduction to the implementation of finite element methods.